Book of Abstracts

EASIAM 2011 The 7th East Asia SIAM Conference & RIMS Workshop on Methods in Industrial and Applied Mathematics

Waseda University, Kitakyushu Campus, June 27-29, 2011 http://www.oishi.info.waseda.ac.jp/~easiam2011/ The 7th East Asia SIAM Conference & RIMS Workshop on Methods in Industrial and Applied Mathematics EASIAM 2011

Kitakyushu Campus, Waseda University, Japan June 27-29, 2011

Book of Abstracts

Organaizer East Asia SIAM

Co-Sponsors Research Institute for Science and Engineering, Waseda University Research Institute for Mathematical Sciences (RIMS), Kyoto University Numerical Analysis Group, Department of Computational Science and Engineering, Nagoya University Kitakyushu City West Japan Industry and Trade Convention Association Scientiﬁc Committee President Hisashi Okamoto (Kyoto University) Vice-President Ming-Chih Lai (National Chiao Tung University) Secretary Shao Liang Zhang (Nagoya University) Ex-President Dongwoo Sheen (Seoul National University)

Member Victor Didenko (Universiti Brunei Darussalam) Zhong-Ci Shi (Chinese Academy of Sciences) Tao Tang (Hong Kong Baptist University) Wai-Ki Ching (University of Hong Kong) L. H. Wiryanto (Bandung Institute of Technology) Hendra Gunawan (Bandung Institute of Technology) Hyung-Chun Lee (Ajou University) Byeong-Chun Shin (Chonnam National University) Xiao-Qing Jin (University of Macau) Haiwei Sun (University of Macau) Mohd Omar (University of Malaya) Abdul Rahni (Universiti Sains Malaysia) Rafael P. Saldana (Ateneo de Manila University) Weizhu Bao (National University of Singapore) Zuowei Shen (National University of Singapore) I-Liang Chern (National Taiwan University) Yongwimon Lenbury (Mahidol Univeristy) Pham Ky Anh (Vietnam National University) Nguyen Van Huu (Hanoi University of Science)

1 Greeting

The 7th East Asia SIAM Conference & RIMS Workshop on Methods in Industrial and Applied Mathematics will be held at Kitakyushu Campus, Waseda University, Japan. The EASIAM conference gathers researchers in applied mathematics with respect to science, engineering and technology. The scope of the EASIAM conference covers all aspects of applied mathematics to science, industry, engineering and technology in the East and South East Asia. Previous EASIAM meetings were held in Hong Kong (1st), Sap- poro (2nd), Xiamen (3rd), Daejeon (4th), Brunei (5th) and Kuala Lumpur (6th). EASIAM is a section of Society for Industrial and Applied Mathemat- ics (SIAM). It promotes basic research and education in mathematics, which supports science and technology. Topics of interest include applications of partial diﬀerential equations; applied and numerical linear algebra; ﬂuid mechanics; mathematical analysis of engineering problems; mathematical aspects of geophysical problems; mathematical biology and/or medicine; mathematics of material sciences; numerical analysis; operations research; optimization. We are grateful to the distinguished SIAM keynote speakers & invited speakers, who have accepted to give plenary talks:

SIAM Keynote Speakers Mitchell Luskin (University of Minnesota, USA) • Andrea Bertozzi (University of California, Los Angeles, USA) • Invited Speakers Ji Hui (National University of Singapore, Shingapore) • Yasumasa Nishiura (Hokkaido University, Japan) • Tony Wen-Hann Sheu (National Taiwan University, Republic of China) • Roslinda Nazar (National University of Malaysia, Malaysia) • Eun-Jae Park (Yonsei University, Korea) • We wish you a productive, rich, instructive edition of the EASIAM conference and we thank you for your participation.

Hisashi Okamoto President of the East Asia section of SIAM Shin’ichi Oishi Chair of local organizing committee

2 Acknowledfements

It is my great pleasure to thank all the participants of the 7th East Asia SIAM conference, whose latest achievements are reported here. I am particularly grateful to them since they do that when our country faces an unprecedented diﬃculty caused by the earthquake in March 11th, 2011, and the nuclear plant disaster right after that. On behalf of the committee of East Asia SIAM, I thank very much the keynote lecturers and the invited speakers. Their determination to give a talk in Japan was vitally important in holding the conference in this hard time. Last but not least, I would like to express my deep gratitude to organizers of Waseda University, particularly Professor Shin ’ichi Oishi, without whose eﬀorts this conference would not be held.

Hisashi Okamoto President of the East Asia section of SIAM Kyoto University

On behalf of the local organizing committee, we would like to express out sincere thanks to Kitakyushu city for supporting the conference, and we are also grateful to Research Institute of Science and Engineering, Waseda University for supporting the conference.

Shin’ichi Oishi Chair of local organizing committee Waseda University

3 Local Organizing Committee

Chair Shin’ichi Oishi (Waseda University) Member Takeshi Ogita (Tokyo Woman’s Christian University) Katsuhisa Ozaki (Shibaura Institute of Technology) Xuefeng Liu (Waseda University) Naoya Yamanaka (Waseda University) Akitoshi Takayasu (Waseda University)

4 Free Shuttle Bus Information

Free shuttle buses are arranged for all conference participants.

Direction Hotel (Comfort Hotel Kurosaki) Conference (Collaboration Center) ↔ Departure Time Date Time Direction 27th, June 7:45–8:00 am Hotel Conference 19:45–20:00 pm Conference → Hotel 28th, June 8:30–8:45 am Hotel → Conference 17:40–18:00 pm Conference → Banquet Hall After Banquet Banquet Hall → Hotel 29th, June 8:30–8:45 am Hotel → Conference 19:15–19:30 pm Conference → Hotel →

5 Public Transportation

No. Direction Way Time Fare 1 Kitakyushu Airport Conference Place Kitakyushu Airport Access Bus 80 min 700 JPY 2 Kitakyushu Airport ↔ Kokura Sta. City Bus 40 min 600 JPY 3 Kokura Sta. ↔ Kurosaki Sta. Kagoshima Main Line 20 min 270 JPY *4 Kurosaki Sta. ↔ Conference Place Public Bus 30 min 270 JPY 5 Fukuoka Airport ↔ Hakata Sta. Fukuoka City Subway 5 min 250 JPY 6 Hakata Sta. ↔ Orio Sta. Kagoshima Main Line 47 min 910 JPY ↔ Limited Express Kirameki 30 min 1910 JPY 7 Orio Sta. Conference Place Public Bus 20 min 200 JPY 8 Orio Sta. ↔ Kurosaki Sta. Kagoshima Main Line 4 min 200 JPY * For more information,↔ see next page.

Taxi Direction Fare (about) Fukuoka Airport Conference Place 13,000 JPY Kitasyushu Airport ↔ Conference Place 14,000 JPY Hakata Sta. ↔ Conference Place 19,000 JPY Kokura Sta. ↔ Conference Place 8,000 JPY Kurosaki Sta. ↔ Conference Place 4,000 JPY Orio Sta. ↔ Conference Place 2,000 JPY ↔

6 *4. Kurosaki Station Conference Place ↔

Direction Kurosaki Station Gakkentoshi Hibikino (Conference Place) → Hour Departure Time 8 (Free Shuttle Bus Available) 9 47 10 11 12 12 52 13 14 22 15 52 16 17 07

Direction Gakkentoshi Hibikino (Conference Place) Kurosaki Station → Hour Departure Time 9 48 10 11 12 12 47 13 14 17 15 16 02 17 18 18 18 19 (Free Shuttle Bus Available)

7 Banquet

Date 28th June, 19:00–21:30 Place Hotel Harmonie Cinq ( TEL : 093-592-5401 ) Address 12-3 Otemachi Kokurakita Kitakyushu Fukuoka, Japan, 803-0814. From Conference Place • – Free Shuttle Bus (Direct only) Departure Time : 18:00 on 28th June @ Collabolation center From JR Kokura Station (2km away) • 1. Bus No.27, No.110. Bus Stop: ”Soleil-hall-move mae”, 10min. 2. Taxi 10min, about 1,000 JPY.

8 1 2 3 4 27th, June 9:00-- 9:30-- 10:30 10:40-- 12:00-- 13:20-- 15:20 15:40-- 18:00-- Keynote Lecture Invited Lecture Student Prize Media Center Room A Opening 27-1-A 27-2-A (1)(2) 27-3-A General Session Room B 27-4-B Collaboration General Session Room C Center 27-4-C General Session Room D 27-4-D

Conference Center Lunch Reception

28th, June 9:00-- 9:30-- 10:30 10:40-- 12:00-- 13:20-- 15:20 15:40-- 18:00-- Invited Lecture Invited Lecture Media Center Room A 28-1-A 28-2-A (1)(2) General Session General Session Room B 28-3-B 28-4-B Collaboration General Session General Session Banquet Room C Center 28-3-C 28-4-C Time Registration Desk General Session General Session Room D 28-3-D 28-4-D 27th, June 8:30--15:40 @RoomA 15:40--18:00 @Conference Center Conference Center Lunch

29th, June 9:00-- 9:30-- 10:30 10:40-- 12:00-- 13:20-- 15:20 15:40-- 18:00-- 28th, June 9:00--12:00 @RoomA Keynote Lecture 12:00--18:00 @Conference Center Media Center Room A 29-1-A General Session General Session General Session Room B 29-2-B 29-3-B 29-4-B 29th, June 9:00--10:40 @RoomA Collaboration General Session General Session General Session 10:40--18:00 @Conference Center Room C Center 29-2-C 29-3-C 29-4-C General Session General Session General Session Room D 29-2-D 29-3-D 29-4-D

Conference Center Lunch Dinner

1 2 3 4 Program

10 Monday, June 27, 2011

Opening Ceremony 09:00∼09:20 Room A

Chair: Shao-liang ZHANG

◦ Opening talk by the president

◦ Some remarks

27-1-A SIAM Key Note Speaker 09:30∼10:30 Room A

Chair: Hisashi OKAMOTO ◦ Mitchell Luskin : Atomistic-to-Continuum Coupling Methods ...... p.24

27-2-A(1) Invited Speaker 10:40∼11:20 Room A

Chair: I-Liang Chern ◦ Wen-Hann Sheu : On a symplectic scheme that optimizes the dispersion-relation equation of the Maxwell’s equations ...... p.26

27-2-A(2) Invited Speaker 11:20∼12:00 Room A

Chair: Xiao-qing Jin

◦ HUI JI : Sparsity-based regularizations for image restoration ...... p.27

11 27-3-A Student prize winner presentations 13:20∼15:20 Room A

Chair: Dongwoo Sheen ◦ Yongyong Cai, Weizhu Bao and Hanquan Wang : Eﬃcient numerical methods for computing ground states and dynamics of dipolar Bose-Einstein condensates ...... p.28

◦ Lei Du, Tomohiro Sogabe and Shao-Liang Zhang : A variant of the IDR(s) method with quasi-minimal residual strategy ...... p.30

◦ Chiun-Chang Lee : Limit Problems of Solutions for the Coupled Nonlinear Schr¨odinger Equations and Steady-state Solutions of the Poisson-Nernst-Planck Systems ...... p.32

◦ Akitoshi Takayasu and Shin’ichi Oishi : A method of computer assisted proof for nonlinear two-point boundary value problems using higher order ﬁnite elements ...... p.33

27-4-B Numerical Linear Algebra 15:40∼17:40 Room B

Chair: T. Miyata and K. Ozaki ◦ Takafumi Miyata, Lei Du, Tomohiro SOGABE, Yusaku YAMAMOTO and Shao-Liang ZHANG : A Projection Approach to Complex Eigenvalues of a Speciﬁed Absolute Value p.34

◦ Takeshi Fukaya, Yusaku Yamamoto and Shao-Liang Zhang : Automatic Performance Tuning for the Blocked Householder QR Algorithm ...... p.36

◦ Xiao-Qing Jin, Wei-ping Shen and Chong Li : A Ulm Method for Inverse Eigenvalue Problems ...... p.38

◦ Yung-Ta Li : Folding a linear system to a second-order form and its application to structure preserving model order reduction ...... p.39

◦ Roberd Saragih : Model reduction order of bilinear system using balanced singular perturbation ...... p.41

◦ Katsuhisa Ozaki, Takeshi Ogita and Shin’ichi Oishi : Simplified semi-static floating-point filter for 2D orientation Problem ...... p.42

12 27-4-C Mathematical Analysis 15:40∼17:40 Room C

Chair: C-A Wang and C-H. Cho ◦ Ching-An Wang and Un-Un Kuo : On the existence of steady ﬂows in a rectangular channel with slip eﬀect on one porous wall ...... p.44

◦ Yukihito Suzuki, Yukio Takemoto and Jiro Mizushima : Exchange of two modes of instability leading to sustained oscillations in the ﬂow past a cylinderical obstacle . . . . p.47

◦ Chien-Hong Cho and Marcus Wunsch : Global and singular solutions to the generalized Proudman-Johnson equation ...... p.49

◦ Hendra Gunawan and Yoshihiro Sawano : Application of RKHS Theory to a Minimization Problem with Prescribed Nodes ...... p.51

◦ Kwok Kin Wong, Robert Conte and Tuen-Wai Ng : Exact meromorphic solutions of the real cubic Swift-Hohenberg equation ...... p.53

27-4-D Mathematics for Flow 15:40∼17:40 Room D

Chair: L.H.Wiryanto and D. Victor ◦ Leo Hari Wiryanto : Free-surface ﬂow under a sluice gate of an inclined wall from deep water ...... p.55

◦ Syakila Binti Ahmad, Ioan Pop and Kuppalapalle Vajravelu : Free Convection Boundary Layer Flow from a Vertical Cone in a Porous Medium Filled with a Nanoﬂuid ...... p.56

◦ Victor Didenko and Johan Helsing : Features of the Nystroem Method for the Sherman- Lauricella Equation on Piecewise Smooth Contours ...... p.58

◦ Changtong Luo, Shao-Liang Zhang and Zonglin Jiang : A New Evolutionary Algorithm for Ground to Flight Data Correlation ...... p.60

◦ Swaroop Nandan Bora and Santu Das : Ocean wave scattering within or over a porous medium ...... p.62

13 Tuesday, June 28, 2011

28-1-A Invited Speaker 09:30∼10:20 Room A

Chair: Hiroshi Kokubu ◦ Yasumasa Nishiura, Takeshi Watanabe and Makoto Iima : Spatially localized traveling structures and the asymptotic behavior in binary ﬂuid convection ...... p.64

28-2-A(1) Invited Speaker 10:40∼11:20 Room A

Chair: Hyung-Chun Lee

◦ Eun-Jae Park : Adaptive mixed FEM and new Discontinuous Galerkin methods . . . p.66

28-2-A(2) Invited Speaker 11:20∼12:00 Room A

Chair: Mohd Omar ◦ Roslinda Mohd Nazar : Numerical investigation of mixed convection ﬂow over a horizontal circular cylinder and a solid sphere in nanoﬂuids ...... p.68

14 28-3-B Numerical Algorithm I 13:20∼15:20 Room B

Chair: H.Yasuda and C.Luo ◦ Hidenori Yasuda : Lamellar pattern and two-phase shallow water equations ...... p.70

◦ Tanay Deshpande : Implementing the Finite Volume Method on a Rectangular Grid p.72

◦ Naoya Yamanaka, Masahide Kashiwagi and Shin’ichi Oishi : Accurate and Rigorous Exponential Algorithm in Round to Nearest ...... p.73

28-3-C Phenomena, Model, Simulation I 13:20∼15:20 Room C

Chair: Tomoeda and C. Mendoza ◦ Mohd Omar : An integrated manufacturing system for time-varying demand process p.75

◦ Takanori Katsura : Numerical analysis of a stationary transport equation by upwind scheme ...... p.76

◦ Akiyasu Tomoeda, Ryosuke Nishi, Kazumichi Ohtsuka and Katsuhiro Nishinari : Stable Traﬃc Jam Occurs from Nonlinear Saturation ...... p.78

◦ Carolina Mendoza and A.M. Mancho : Lagrangian Skeleton on the Gulf of Mexico Oil Spill ...... p.80

◦ Keita Iida, Hiroyuki Kitahata, Satoshi Nakata, Masaharu Nagayama, : Mathematical studies on the self-motion of surfactant scrapings at the air-water interface ...... p.82

◦ Hisashi Okamoto : A curious bifurcation diagram arising in a scalor reaction-diﬀusion equation with an integral constraint ...... p.84

15 28-3-D Applied Mathematics I 13:20∼15:20 Room D

Chair: K. Matsue and T. Mizuguchi ◦ Hiroshi Kokubu : Application of topological computation method for global dynamics of an associative memory model of the Hopﬁeld type ...... p.85

◦ Kaname Matsue : Rigorous veriﬁcation of equilibria for evolutionary equations - Existence, Uniqueness and Hyperbolicity - ...... p.86

◦ Tsuyoshi Mizuguchi, and Makoto Yomosa : Role of unstable symmetric solutions in symmetry restoring process ...... p.88

◦ Edi Cahyono and Buyung Sarita : A Diﬀusion Equation Representing the Dynamics of the Jakarta Composite Index (JSX) ...... p.90

◦ Buyung Sarita, Femy Puspita Arisanty and Edi Cahyono : The Trend and Probability Density Function of the Jakarta Composite Index (JSX) ...... p.91

◦ Masataka Kuwamura : Dormancy of predators in prey-predator systems ...... p.92

28-4-B Numerical Algorithm II 15:40∼17:40 Room B

Chair: M. Kimura and K. H. Kwon ◦ Masato Kimura and Kiyohiro Ishijima : Truncation error analysis for particle methods p.94

◦ Elliott Ginder, Seiro Omata and Karel Svadlenka : A numerical method for multiphase volume-preserving mean curvature ﬂow ...... p.96

◦ Min-Hung Chen and Rong-Jhao Wu : A High-Order Discontinuous Galerkin Method for Elliptic Interface Problems ...... p.98

◦ Teng-Yao Kuo, Hsin-Chu Chen and Tzyy-Leng Horng : A Coarse-Grain Parallel Scheme for Solving Poisson Equation by Chebyshev Pseudospectral Method ...... p.99

◦ Haiwei Sun and Qin Sheng : On the stability of an oscillation-free ADI method in laser beam propagation computations ...... p.101

16 28-4-C Phenomena, Model, Simulation II 15:40∼17:40 Room C

Chair: H. Suito and S.R. Pudjaprasetya

◦ Hiroshi Suito, Takuya Ueda and Daniel Sze : Flows in thoracic aorta with torsion p.103

◦ Alex Chang : A Pore-scale Network Flow Model for Two Phase Flow: The Genesis and Migration of Gas Phase ...... p.105

◦ Santiago Madruga and Santiago Madruga : Convective transport and stability in ﬁlms of binary mixtures ...... p.106

◦ Sri Redjeki Pudjaprasetya, and Elis Khatizah : Longshore Wave Breaker with Reﬂected Beach ...... p.108

◦ Ikha Magdalena and S.R. Pudjaprasetya : Wave Energy Dissipation in Porous Media p.110

◦ G Sharma and Madhu Jain : Multi-Compartment Modeling of Tumor Cells Interacting With Dynamic Chemotherapeutic Drug ...... p.112

28-4-D Applied Mathematics II & Networks 15:40∼17:40 Room D

Chair: K. Miura and W-K Ching ◦ Matthew Min-Hsiung Lin : On LaSalle’s Invariance Principle and Its Application to Synchronize Hyperchaotic Systems ...... p.113

◦ Keiji Miura : An unbiased estimator of noise correlations under signal drift ...... p.115

◦ Wai-Ki CHING : Construction of Probabilistic Boolean Networks: A Maximum Entropy Rate Approach ...... p.117

◦ Inseok Yang, Donggil Kim and Dongik Lee : Compensation Technique for Transmission Delay and Packet Loss in Networked Control System based on Lagrange Interpolation p.119

17 Wednesday, June 29, 2011

29-1-A SIAM Key Note Speaker 09:30∼10:30 Room A

Chair: Ming-Chih Lai

◦ Andrea Bertozzi : Mathematics of Crime ...... p.121

29-2-B Numerical Algorithms III 10:40∼12:00 Room B

Chair: T. Ichinomiya and C-T. Wu

◦ David Ni : Numerical Studies of Lorentz Transformation ...... p.122

◦ Chin-Tien Wu, Ming-Chiang Jiang and Yu-Lin Tsai : Numerical Studies on Monge Ampere Equation Arising from Free-Form Design of Geometric Optics ...... p.124

◦ Takashi Ichinomiya : Renormalization approach to solve Langevin’s equation . . . . . p.126

◦ Kil H Kwon and Lee Jaekyu : Consistent approximation-sampling with multi pre and post ﬁltering ...... p.128

29-2-C Error Analysis 10:40∼12:00 Room C

Chair: K. Kobayashi and T. Kinoshita ◦ Yintzer Shih and R. Bruce Kellogg : A Tailored Finite Point Method for Convection Diﬀusion Reaction Problems with Variable Coeﬃcients ...... p.130

◦ Kenta Kobayashi : On the interpolation constants over triangular elements ...... p.131

◦ Takehiko Kinoshita, T. Kimura and M.T. Nakao : A posteriori estimates of inverse linear ordinary diﬀerential operators ...... p.132

◦ Xuefeng LIU , Akitoshi Takayasu and Shin’ichi Oishi : Numerical veriﬁcation for solution existence of elliptic PDE on arbitrary polygonal domain ...... p.133

18 29-2-D Applied Mathematics III 10:40∼12:00 Room D

Chair: C. Y. Han and K.K. Viswanathan ◦ Naoki Wada : Reconstruction of the density of tree-shaped networks from boundary measurements of waves ...... p.135

◦ Chang Yong Han and Song-Hwa Kwon : Cubic helical splines with Frenet-frame continuity p.137

◦ Viswanathan Kodakkal Kannan, Zainal Abdul Aziz and Saira Javed : Free vibration of symmetric angle-ply laminated cylindrical shells of variable thickness including shear deformation theory: spline method ...... p.139

29-3-B Bifurcation 13:20∼15:20 Room B

Chair: T. Sakajo and M. Shoji

◦ Takashi Sakajo : Point vortex equilibria enhancing forces over two parallel plates . p.141

◦ Tomoyuki Miyaji, Isamu Ohnishi and Yoshio Tsutsumi : Bifurcation analysis for the Lugiato-Lefever equation in a square ...... p.143

◦ Kuo-Chih Hung and Shin-Hwa Wang : A theorem on S-shaped bifurcation curve for a positone problem with convex-concave nonlinearity and its applications to the perturbed Gelfand problem ...... p.145

◦ Shin-Hwa Wang, Kuo-Chih Hung and Chien-Shang Yu : On the existence of a double S-shaped bifurcation curve with six solutions for a combustion problem ...... p.146

◦ Mayumi Shoji and Chika Shimizu : Appearance and disappearance of non-symmetric progressive capillary-gravity waves of deep water ...... p.148

◦ Takeshi WATANABE, Makoto IIMA and Yasumasa NISHIURA : Bifurcation structure and spontaneous pattern formation in binary ﬂuid convection ...... p.150

19 29-3-C Optimization I 13:20∼15:20 Room C

Chair: W.J.Leong and W-H Kuo ◦ Wen-Hung Kuo : Single-machine group scheduling problems with time-dependent learning eﬀect and position-based setup time learning eﬀect ...... p.152

◦ Seyoung Park, Bong-Gyu Jang and Yuna Rhee : Optimal Retirement with Unemployment Risks and Market Completion ...... p.154

◦ Wah June Leong, Mahboubeh Farid and Malik Abu Hassan : Rescaled Gradient-based Methods with Fixed Stepsize for Large-scale Optimization ...... p.156

29-3-D Numerical Analysis I 13:20∼15:20 Room D

Chair: T. Ishiwata and H. Notsu ◦ Tetsuya Ishiwata : Behavior of polygonal curves by crystalline curvature ﬂow with a driving force ...... p.157

◦ Guanyu Zhou and Norikazu Saito : Some remarks on the ﬁctitious domain method with penalty for elliptic problems ...... p.159

◦ Norikazu Saito : Analysis of the ﬁnite volume approximation for a degenerate parabolic equation ...... p.161

◦ Takahito Kashiwabara : FEM analysis of the Stokes equations under boundary conditions of friction type ...... p.163

◦ Hirofumi Notsu, Masahiro Yamaguchi and Daishin Ueyama : A mesh generator using a self-replicating system ...... p.165

20 29-4-B Accurate and High Performance Computing 15:40∼17:40 Room B

Chair: H. Fujiwara and D. Sheen ◦ Hiroshi Fujiwara : A Remark on Numerical Instability of Complex Inverse Laplace Trans- forms using Multiple-Precision Arithmetic ...... p.167

◦ Dongwoo Sheen and Jiwoon Kim : Numerical Laplace inversion using mult-precision p.169

◦ Chenhan Yu, Weichung Wang and Dan’l Pierce : CPU-GPU Hybrid Approaches in Multifrontal Methods for Large and Sparse Linear System ...... p.170

◦ Shugo Manabe : Design and implementation of a multiple-precision system on GPU p.172

29-4-C Optimization II 15:40∼17:40 Room C

Chair: G. M. Lee and M. Jain ◦ Gue Myung Lee : On Optimality Theorems for Robust Multiobjective Optimization problems ...... p.174

◦ Sangho Kum and Yongdo Lim : The Resolvent average on symmetric cones ...... p.176

◦ Dai-Ni Hsieh, Ray-Bing Chen, Ying Hung and Weichung Wang : Optimizing Latin Hy- percube Designs by Particle Swarm with GPU Acceleration ...... p.178

◦ Madhu Jain : Maximum Entropy Approach For Optimal Repairable Mx/G/1 Queue With Bernoulli Feedback And Setup ...... p.180

◦ Afshin Ghanbarzadeh and Abbas : Application of the Bees Algorithm to Multi-Objective Optimization Engineering Problems ...... p.181

21 29-4-D Numerical Analysis II 15:40∼17:40 Room D

Chair: C. Park and S-M. Chang ◦ Chunjae Park and Dongwoo Sheen : A quadrilateral Morley element for biharmonic equations ...... p.183

◦ Shu-Ming Chang : Applying Snapback Repellers in Ecology ...... p.185

◦ Yusuke Morikura, Katsuhisa Ozaki and Shin’ichi Oishi : Veriﬁed solutions of linear systems on GPU ...... p.186

Abstracts

Atomistic-to-Continuum Coupling Methods Mitchell Luskin

School of Mathematics, University of Minnesota Minneapolis, MN 55455, USA

e-mail: [email protected] http://www.umn.edu/~luskin/

Abstract Materials often have highly singular strain fields at crack tips, dislocations, and grain boundaries that require the accuracy of atomistic modeling in small regions surrounding these defects. However, these localized defects interact through long-ranged elastic fields with a much larger region where the strain gradients are sufficiently small to allow accurate approximation by coarse-grained continuum finite element models. The size of the region requiring atomistic modeling is typically several orders of magnitude smaller than the size of the region that can be adequately approximated by coarse- grained continuum finite element models [6], which has motivated the development of numerical methods that couple atomistic regions with continuum regions to compute up to length scales that are sufficiently large for accurate and reliable scientific and engineering application.

Figure 1: The atomistic (red) region surrounding an edge dislocation is coupled to a continuum (grey) region. The O(1/r2) magnitude of the strain gradient (where r is the distance to the dislocation) implies that a small region (the core) around the dislocation requires atomistic modeling, but a coarse-grained continuum model can be accurately used outside the dislocation core. In the continuum region, the deformation of the atoms are approximated by piecewise linear interpolation from the deformation of the blue atoms, thus reducing the computational degrees of freedom and allowing signiﬁcantly increased length scales to be computed.

During the past several years, we have developed an analysis of the stability, modeling error, and iterative solution of prototypical atomistic-to- continuum coupling methods [1–3,5,6]. The formation and motion of defects such as cracks, dislocations, or grain boundaries, occurs when the conﬁgura- tion loses stability, that is, when an eigenvalue of the Hessian of the energy functional becomes negative. It is thus essential for the evaluation of the predictive capability of atomistic-to-continuum coupling methods that our theoretical analyses have investigated accuracy and stability for strains up to the onset of instability of the atomistic energy [2,6]. We have initiated a computational study of the accuracy of atomistic- to-continuum coupling methods for several benchmark problems [4]. Our theoretical analysis can successfully explain most of the computational results for these multi-dimensional benchmark problems. However, we also observe the need for additional theoretical analysis and benchmark problems to more thoroughly understand the accuracy of atomistic-to-continuum methods. Joint work with M. Dobson, B. Van Koten, Xingjie Li, and C. Ortner.

References [1] M. Dobson, M. Luskin, and C. Ortner. Stability, instability, and error of the force-based quasicontinuum approximation. Archive for Rational Mechanics and Analysis, 197:179–202, 2010. [2] M. Dobson, M. Luskin, and C. Ortner. Accuracy of quasicontinuum approximations near instabilities. Journal of the Mechanics and Physics of Solids, 58:1741–1757, 2010. arXiv:0905.2914v2. [3] M. Dobson, M. Luskin, and C. Ortner. Iterative methods for the force- based quasicontinuum approximation. Computer Methods in Applied Mechanics and Engineering, to appear. arXiv:0910.2013v3.

[4] B. Van Koten, Xingjie Li, M. Luskin, and C. Ortner. A computational and theoretical investigation of the accuracy of QC methods. In Num. Anal. of Multi. Prob.. Springer, to appear. arXiv:1012.6031. [5] M. Luskin and C. Ortner. Linear stationary iterative methods for the force-based quasicontinuum approximation. In Numerical Analysis and Multiscale Computations, volume 82 of Lect. Notes Comput. Sci. Eng. Springer Verlag, to appear. arXiv:1104.1774. [6] B. Van Koten and M. Luskin. Development and Analysis of Blended Quasicontinuum Approximations. arXiv:1008.2138.

On a symplectic scheme that optimizes the dispersion- relation equation of the Maxwell’s equations Tony W. H. Sheu123*, L. Y. Liang1, C. M. Mei1, J. H. Li1 1. Department of Engineering Science and Ocean Engineering, National Taiwan University, No. 1, Sec. 4, Roosevelt Road, Taipei, Taiwan, Republic of China 2. Taida Institute of Mathematical Science (TIMS), National Taiwan University 3. Center for Quantum Science and Engineering (CQSE), National Taiwan University

Abstract

In this talk I will present an explicit finite-difference time-domain scheme in Cartesian and curvilinear coordinates for solving the Maxwell’s equations in non-staggered grids. The proposed scheme for solving the Faraday’s and Ampere’s equations theoretically preserves the discrete zero-divergence for the electrical and magnetic fields. The inherent local conservation laws in Maxwell’s equations are also preserved discretely all the time using the explicit second-order accurate symplectic partitioned Runge-Kutta temporal scheme. The remaining spatial derivative terms in the semi-discretized Faraday’s and Ampere’s equations are then discretized to provide an accurate functional relation that governs the numerical angular frequency and the wave numbers in two space dimensions. To achieve the goal of getting a better dispersive characteristics, a fourth-order accurate space centered scheme which minimizes the difference between the exact and numerical dispersion relation equations is proposed. Through the computational exercises, the proposed dual-preserving solver is computationally demonstrated to be efficient for use to predict the long-term accurate Maxwell’s solutions.

Sparsity-based regularizations for image restoration Hui Ji Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, Singapore 119076 E-mail: [email protected] Abstract Image restoration is about how to recover images distorted and degraded in the formation or transmission stage. Most image restoration problems are challenging ill-posed inverse problems that require certain prior information of images to obtain satisfactory solutions. In recent years, the sparsity assumption of image/video data under suitable dictionaries, i.e. most of important information of images can be represented by only few coefficients in some system, has emerged as one powerful prior for most nature images. Com- bined with rapid progresses of `1 norm related minimization approaches for finding the sparse approximation of a given signal in redundant systems (e.g. split Bregman iteration), sparsity-based regularization has been one highly efficient technique for solving many challenging image restoration problems. In this talk, we will begin with a brief review on mathematical theory and numerical algorithms of sparse approximation under redundant tight frame. Then, several applications of sparsity-based regularization in image restoration will be presented, including: blind motion deblurring ([4]), blind image inpainting ([1]), surface fitting in range imaging ([3]) and action analysis in video ([2]).

References

[1] B. Dong, H. Ji, J. Li and Z. Shen and Y.-H. Xu: Wavelet Frame Based Blind Image Inpainting, Technical report, NUS, 2011.

[2] Y. Li, C. Fermuller, Y. Aloimonos and H. Ji: Learning shift-invariant sparse representation of actions, IEEE Conf. on Computer Vision and Pattern Recog- nition (CVPR), San Francisco, 2010. [3] H. Ji, Z. Shen and Y.-H. Xu: Wavelet frame based scene reconstruction from range data, Journal of Computational Physics, 229 (6), March 2010, 2093– 2018. [4] J. Cai, H. Ji, C. Liu and Z. Shen: Blind motion deblurring from a single image using sparse approximation, IEEE Conf. on Computer Vision and Pattern Recognition (CVPR), Miami, 2009.

Eﬃcient numerical methods for computing ground states and dynamics of dipolar Bose-Einstein condensates

1),2) 2), 4) Weizhu Bao , Yongyong Cai ∗, Hanquan Wang

1) Department of Mathematics, National University of Singapore, 117543, Singapore 2) Center for Computational Science and Engineering, National University of Singapore, 117543, Singapore 3) School of Statistics and Mathematics, Yunnan University of Finance and Economics, PR China

e-mail: [email protected] ∗ Abstract New efficient and accurate numerical methods are proposed to compute ground states and dynamics of dipolar Bose-Einstein condensates (BECs) described by a three-dimensional (3D) Gross-Pitaevskii equation (GPE) with a dipolar interaction potential. Due to the high singularity in the dipolar interaction potential, it brings significant difficulties in mathematical analysis and numerical simulations of dipolar BECs. In this paper, by decoupling the two-body dipolar interaction potential into short-range (or local) and long-range interactions (or repulsive and attractive interactions), the GPE for dipolar BECs is reformulated as a Gross-Pitaevskii-Poisson type system. Based on this new mathematical formulation, we prove rigorously existence and uniqueness as well as nonexistence of the ground states, and discuss the existence of global weak solution and finite time blowup of the dynamics in different parameter regimes of dipolar BECs. In addition, a backward Euler sine pseudospectral method is presented for computing the ground states and a time-splitting sine pseudospectral method is proposed for computing the dynamics of dipolar BECs. Due to the adaption of new mathematical formulation, our new numerical methods avoid evaluating integrals with high singularity and thus they are more efficient and accurate than those numerical methods currently used in the literatures for solving the problem. Extensive numerical examples in 3D are reported to demonstrate the efficiency and accuracy of our new numerical methods for computing the ground states and dynamics of dipolar BECs.

References

[1] W. Bao, Y. Cai and H. Wang, Eﬃcient numerical methods for computing ground states and dynamics of dipolar Bose-Einstein condensates, J. Comput. Phys., 229 (2010), 7874-7892.

A variant of the IDR(s) method with quasi-minimal residual strategy Lei Du1),∗, Tomohiro Sogabe2), Shao-Liang Zhang1)

1) Department of Computational Science and Engineering Nagoya University 2) Graduate School of Information Science and Technology Aichi Prefectural University

∗ e-mail: [email protected]

Abstract We consider the solution of large, sparse and nonsymmetric linear systems of the form Ax = b (1) where A is a nonsingular n × n real matrix, and b is a real vector of order n. This problem arises in numerous applications such as discretizations of partial diﬀerential equations, Newton-type methods for nonlinear linear systems and optimization. The IDR(s) method proposed by Sonneveld and van Gijzen [4] can solve problem (1) eﬀectively. Recently, it has received extensive attention and several variants have been developed. In this talk, we will describe a new variant of the IDR(s) method. The IDR(s) algorithm can converge faster than most product-type methods, but also usually with irregular convergence behavior. We reformulate the relations of residuals and their auxiliary vectors generated by the IDR(s) method in matrix form. Then, by this new formulation and motivated by methods in [1, 2, 3], we present the variant of IDR(s), named QMRIDR(s), to improve the demerit of its irregular convergence behavior. Both the fast and smooth convergence behaviors of the QMRIDR(s) method can be exhibited. Two numerical results are reported to show the comparison between IDR(s) and QMRIDR(s) in Figs. 1–2.

References

[1] T.F. Chan, E. Gallopoulos, V. Simoncini, T. Szeto and C. Tong, A quasi- minimal residual variant of the Bi-CGSTAB algorithm for nonsymmetric systems, SIAM J. Sci. Comput., 15(1994), 338–347.

[2] R.W. Freund, QMR: a quasi-minimal residual method for non-Hermitian linear systems, Numer. Math., 60(1991), 315–339. [3] R.W. Freund, A transpose-free quasi-minimal residual algorithm for non-Hermitian linear systems, SIAM J. Sci. Comput., 14(1993), 470– 482.

[4] P. Sonneveld and M.B. van Gijzen, IDR(s): a family of simple and fast algorithms for solving large nonsymmetric systems of linear equations, SIAM J. Sci. Comput., 31(2008), 1035–1062.

Figure 1: PDE2961(n=2961, nnz=14585)

Figure 2: SHERMAN4(n=1104, nnz=3786)

LIMIT PROBLEMS OF SOLUTIONS FOR THE COUPLED NONLINEAR SCHRODINGER¨ EQUATIONS AND STEADY-STATE SOLUTIONS OF THE POISSON-NERNST-PLANCK SYSTEMS CHIUN-CHANG LEE DEPARTMENT OF MATHEMATICS, NATIONAL TAIWAN UNIVERSITY

Abstract. In this talk we present results from my Ph.D. thesis, which is joint works with my advisor Professor Tai-Chia Lin and my Host Professor Chun Liu, when I visited the Institute of Mathematics and Its Applications (IMA) in Minnesota. The Poisson-Boltzmann (PB) equation is conventionally used to model the equilibrium of bulk ionic species in diﬀerent media and solvents. We study a New Poisson-Boltzmann (PB n) equation in 1-dimensional interval (−1, 1):

N N 1 akφ 2 −blφ 2 00 X akαke X blβle φ = 1 − 1 R akφ(y) R −blφ(y) k=1 −1 e dy l=1 −1 e dy with a small dielectric parameter 2 and nonlocal nonlinearity which takes into consideration of the preservation of the total amounts of each individual ion. This equation can be derived from the original Poisson-Nernst-Planck (PNP) system. Under Robin type boundary conditions with various coefficient scales, we demonstrate the asymptotic behaviors of one dimensional solutions of PB n equations as the parameter approaches zero. In particular, we show that in case of electro- PN1 PN2 neutrality, i.e., k=1 akαk = l=1 blβl, we prove that φ’s solutions of 1-D PB n equations may tend to a nonzero constant c at every interior point as goes to zero. The value c can be uniquely determined by ak, bl’s valences of ions, αk, βl’s total concentrations of ions and the limit of φ’s at the boundary x = ±1. Such a result can not be found in conventional 1-D Poisson-Boltzmann (PB) PN1 PN2 equations. On the other hand, as k=1 akαk 6= l=1 blβl (non-electroneutrality), solutions of 1-D PB n equations may have blow-up behavior which also may not be obtained in 1-D PB equations. We also study the compressible and incompressible limits of the solution of the non-linear Schrödingersystem. Recently, a rich variety of dynamical phenomena and a turbulent relaxation have been observed in rotating Bose-Einstein condensates depicted by Gross-Pitaevskii equations coupled with rotating fields and trap potentials. The dynamical phenomena range from shock-wave formation to anisotropic sound propagation. The turbulent relaxation leads to the crystallization of vortex lattices. To see the dynamical phenomena and the turbulent relaxation of two-component rotating Bose-Einstein condensates, we study the incompressible and the compressible limits of two- component systems of Gross-Pitaevskii equations. We define ”H-function” a modulated energy functional which may control the propagation of densities and linear momentums under the effect of rotating fields and trap potentials.

A method of computer assisted proof for nonlinear two-point boundary value problems using higher order ﬁnite elements

1), 2,3) Akitoshi Takayasu ∗, Shin’ichi Oishi 1) Graduate School of Fundamental Science and Engineering,

Waseda University 2) Department of Applied Mathematics, Faculty of Science and Engineering, Waseda University 3) CREST, JST e-mail: [email protected] ∗ Abstract Present authors have presented with Takayuki Kubo at University of Tsukuba a method of a computer assisted proof for the existence and uniqueness of solutions to two-point boundary value problems of nonlinear ordinary differential equations in the paper submitted for NOLTA, IEICE. This method uses piecewise linear finite element base functions and sometimes requires fine mesh. To overcome this difficulty, in this paper, an improved method is presented for the norm estimation of the residual to the operator equation. In this refined formulation, piecewise quadratic finite element base functions are used. A kind of the residual technique works sophisticatedly well. It is stated that the estimation of the residual can be expected smaller than that of the previous method. Finally, four examples are presented. Each result demonstrates that a remarkable improvement is achieved in accuracy of the guaranteed error estimation.

References

[1] S. Oishi: Numerical verification of existence and inclusion of solutions for nonlinear operator equations, Journal of Computational and Applied Mathematics, 60 (1995), 171–185. [2] M. Plum: Computer-assisted existence proofs for two-point boundary value problems, Computing, 46 (1991), 19–34. [3] N. Yamamoto and M.T. Nakao: Numerical verifications for solutions to elliptic equations using residual equations with a higher order finite element, Journal of Computational and Applied Mathematics, 60 (1995), 271–279.

A Projection Approach to Complex Eigenvalues of a Speciﬁed Absolute Value

1), 1) 2) Takafumi MIYATA ∗, Lei DU , Tomohiro SOGABE , Yusaku YAMAMOTO3), Shao-Liang ZHANG1)

1) Graduate School of Engineering, Nagoya University 2) Graduate School of Information Science & Technology, Aichi Prefectural University 3) Graduate School of System Informatics, Kobe University

e-mail: [email protected] ∗ Abstract Given n n matrices A and B, we consider computing a few eigenpairs (λ , x ×n) satisfying ∈ C ∈ C A x = λ B x ( x = 0 ). ̸

This problem arises in many applications of scientific computing, such as structural analysis and stability analysis of fluid dynamics [1]. It is often needed to find a few eigenvalues near a specified point in the complex plane and for the problem several iterative methods are available [2, 3]. In recent years, so called the photonic crystals [4] have attracted much attention due to their potential applications in future integrated circuits based on light, instead of electrons. In these practical problems, it is needed to find all the complex eigenvalues such that λ = 1, see Fig. 1. The number of desired eigenvalues are small. However,| they| are usually scattered, not located near one point in the complex plane. In this talk, we present an approach to complex eigenvalues of a specified absolute value. This approach extends the projection method of Sakurai and Sugiura [5] to a new method for the practical problems of photonic crystals. In contrast to existing methods, our method can avoid computing almost all the undesired eigenvalues and thus can be efficient.

0 Im.

-1

-1 0 1 Re. Figure 1: Distribution of eigenvalues in the complex plane. The desired eigenvalues on the unit circle are shown by , the other ones are shown by . • ◦

References [1] Z. Bai, D. Day, J. W. Demmel, J. J. Dongarra, A test matrix collection for non-Hermitian eigenvalue problems, 1996.

[2] Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, H. van der Vorst, eds., Tem- plates for the solution of Algebraic Eigenvalue Problems: A Practical Guide, SIAM, Philadelphia, 2000.

[3] D. R. Fokkema, G. L. G. Sleijpen, H. A. van der Vorst, Jacobi-Davidson style QR and QZ algorithms for the reduction of matrix pencils, SIAM J. Sci. Comput., 20 (1998), 94–125.

[4] Y. Huang, Y. Y. Lu, S. Li, Analyzing photonic crystal waveguides by Dirichlet-to-Neumann maps, J. Opt. Soc. Am. B, 24 (2007), 2860–2867.

[5] T. Sakurai and H. Sugiura, A projection method for generalized eigenvalue problems using numerical integration, J. Comput. Appl. Math., 159 (2003), 119–128.

Automatic Performance Tuning for the Blocked Householder QR Algorithm

1), 2) 1) Takeshi FUKAYA ∗, Yusaku YAMAMOTO , Shao-Liang ZHANG 1) Graduate School of Engineering, Nagoya University 2) Graduate School of System Informatics, Kobe University

e-mail: [email protected] ∗ Abstract In developing software for scientific computations, it is important to optimize the software in order to fully exploit the potential performance of hardware. Such optimization has conventionally been done by humans based on their empirical knowledge. However, as the computer architecture becomes complex, cost for optimizing becomes too high to be done only by humans. In addition, such optimization is very difficult for researchers in application fields who lack the knowledge about hardware. These issues indicate the necessity of systematic and nonempiri- cal frameworks for the performance optimization. Under this situation, a framework named Software Automatic Tuning has recently been extensively researched [1]. In this framework, the target software is required to be tunable, that is, it has controllable variability or adaptability in it. Additionally a tuning mechanism tunes the target software automatically so that it runs in good performance under given conditions. Considering these two points, we have studied on the automatic tuning especially for the dense matrix computations such as LU or QR decomposition [2]. In this talk, we introduce our approach to automatic tuning for the blocked Householder QR algorithm. The QR decomposition of A using the Householder transformations is written in T T (I tny y ) (I t1y y )A = R − n n ··· − 1 1 where ti is a scalar, yi is a vector, and R is an upper triangular matrix. In the blocked algorithm, we first partition the target matrix into blocks and compute partial decomposition. Next, we aggregate the Householder transformations used in the partial decomposition into a matrix form as T T T (I tpy y ) (I t1y y ) = I Y1T1Y1 . − p p ··· − 1 1 − After that, we update the remaining blocks using matrix-matrix multiplication. An overview of these computations is shown in Figure 1. Though we pay additional computation cost for aggregating the transformations, by using matrix- matrix multiplications, which have high data reusability, we can obtain advantage on recent architecture where floating point operation is much faster than data transfer.

©¨ ¢¡

¢ ¢£

¢¤£ ¢¦¥ §¨£©¢¦¥ ¢

¡ ¡

¤ ¤ ¤ ¥¦¥¦¥ ¥¦¥¦¥

Figure 1: Overview of the blocked algorithm Figure 2: Binary tree

The performance of the blocked algorithm depends on how to partition the target matrix into blocks. This is because total computation cost and the performance of each matrix-matrix multiplication which affected by the matrix size are ﬁxed by the partitioning way. Therefore, it is important to optimize how to partition the matrix. For this subject, optimization by man power has been done [3], however, there are few studies on systematic optimization. Our goal is automatic tuning of the way of partitioning. In our approach, a binary tree shown in Figure 2 represents the way of partitioning. By using a binary tree as a parameter, various kinds of partitioning way can be controlled systematically. Moreover taking advantage of the recursive structure in binary trees, we construct the mechanism based on the dynamic programming to ﬁnd the optimal binary tree. Through the performance evaluation on some computational environments, it is shown that the performance obtained by our approach is as good as that obtained by human tuning.

References

[1] K. Naono, K. Teranishi, J. Cavazos, and R. Suda: Software Automatic Tun- ing: From Concepts to State-of-the-Art Results, Springer, 2010.

[2] T. Fukaya, Y. Yamamoto, and S. L. Zhang: A Dynamic Programming Ap- proach to Optimizing the Blocking Strategy for the Householder QR De- composition, Proceedings of IEEE Cluster 2008, 2008, 402–410.

[3] E. Elmroth, and F. Gustavson: Applying Recursion to Serial and Parallel QR Factorization Leads to Better Performance, IBM Journal of Research and Development, 44 (2000), 605–624.

A Ulm-like Method for Inverse Eigenvalue Problems

1), 2) 3) Xiao-qing Jin ∗, Wei-ping Shen , Chong Li ,

1) Department of Mathematics, University of Macau 2) Department of Mathematics, Zhejiang Normal University 3) Department of Mathematics, Zhejiang University

e-mail: [email protected] ∗ Abstract We propose a Ulm-like method for solving inverse eigenvalue problems, which avoids solving approximate Jacobian equations comparing with other known methods. A convergence analysis of this method is provided and the root quadratic convergence property is proved under the assumption of the distinction of given eigenvalues. Numerical experiments are given in comparison with the inexact Newton-like method. This paper is concerned with [1].

References

[1] W. P. Shen, C. Li, X. Q. Jin: A Ulm-like method for inverse eigenvalue problems, Applied Numerical Mathematics, 61(3) (2011), 356-367.

Folding a linear system to a second-order form and its application to structure preserving model order reduction Yung-Ta Li

Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan 30010

e-mail: [email protected]

Abstract A strategy that folds a linear system to a second-order representation is presented in [1]. The strategy begins with the observation that the inverse matrix of the transformation matrix permits a specific form. However, not all matrices that promise the specific form are of full rank. Consequently one might need to search for all possibilities. In other words, the construction of the transformation matrix is combinatorially hard. We introduce a new method to extract second-order system matrices by applying a similarity transformation to a linear state space system. Instead of taking advantage of the specific form of the inverse of the transformation matrix, we obtain the transformation matrix in a factored form. The new method solves the hard combinatorial problem for the construction of the transformation matrix proposed in [1]. An application of the folding technique is for structure preserving reduction of large-scale second-order systems. The main idea is to reduce a linear state space system and then fold it into a second-order form. A second-order system with the output being a linear combination of displacement and velocity is considered in [1]. The output associated with displacement or velocity only is considered in [2] or [3], respectively. Focusing on the folding process, we find that the construction of the transformation matrices in [2] and [3] suffers the same challenge as in [1]. We will show that a combination usage of the new method and the procedure in [2] or [3] solves the hard combinatorial problem.

References

[1] D. G. Meyer and S. Srinivasan: Balancing and Model Reduction for Second-Order Form Linear Systems, IEEE Trans. Aut. Control, 41 (1996), 1632–1644.

[2] B. Salimbahrami and B. Lohmann: Order reduction of large scale second-order systems using Krylov subspace methods, Linear Algebra Appl., 415 (2006), 385–405.

[3] B. Salimbahrami, B. Lohmann and A. Bunse-Gerstner: Passive reduced order modelling of second-order systems, Mathematical and Computer Modelling of Dynamical Systems, 14 (2008), 407–420

Model Order Reduction for Bilinear System using Balanced Singular Perturbation

Roberd Saragih

Industrial and Financial Mathematics Research Group, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, INDONESIA e-mail: [email protected]

Abstract Many problems in science and engineering have model in the bilinear system such as electrical networks, mechanical links, nuclear fission, cardiovascular regulator, urban process, hormone regulation, predator-prey models, etc. In some applications, it has the high–order degree of freedom (state). In such case, it is difficult to implement the designed controller since it can cause the numerical difficulties and high computational cost. Hence, it is desired the low-order controller. Meanwhile, the low-order controller can be obtained from the low-order model. In this paper, we concerned with reduction order of the bilinear system model by using the balanced singular perturbation approach. Some approximation methods for the bilinear system, such as the balanced truncation and the projection method, have been developed by authors [2,3]. The singular perturbation approach has been applied to the linear system by Liu [1]. As given in [2], we first transform the bilinear system into the balanced system. The concept of the singular perturbation is based on the rejection of very fast modes from a model. Based on this concept, we then divide the state of the balanced bilinear system into the fast modes and the slow modes. The low-order model can be obtained by replacing the small parameter by zero to the fast mode. The advantage of the singular perturbation is better than the balanced truncation in the low frequencies.

References

1] Liu. Y. and Anderson, B.D.O. (1989), Singular perturbation approximation of balanced system, International Journal of Control, Vol.50, pp. 1379-1405. 2] Liqian Zhang and James Lam, (2002), On H2 model reduction of bilinear systems, Automatica, 38, pp. 205-216. [3] Zhaojun Bai and Daniel Skoogh (2006), A projection method for model reduction of bilinear dynamical systems, Linear Algebra and Its Applications, 415, pp. 406-425

Simplified semi-static floating-point filter for 2D orientation problem

1),4) 2).4) 3),4) Katsuhisa Ozaki ∗, Takeshi Ogita , Shin’ichi Oishi ,

1) Department of Mathematical Sciences, Shibaura Institute of Technology 2) Department of Mathematical Sciences, Tokyo Woman’s Christian University 3) Faculty of Science and Engineering, Waseda University 4) Japan Science and Technology Agency, CREST

e-mail: [email protected] ∗ Abstract This talk is concerned with robustness problems in computational geometry, especially 2D orientation problem is focused on. The 2D orientation problem is one of basic problems in the field of computational geometry. The following is brief explanation of this problem: An oriented line and a point are given on the two-dimensional surface. The aim is to clarify that the point is left or right, or on the line. It can be boiled down to a sign of the 3-by-3 matrix determinant. If the sign of the determinant is positive / negative, the point lies on the left / right to the line. If the determinant is zero, the point is on the line. Based on this problem, segment intersection and a point-in polygon problem can be solved. Although the computation is small-scale, the 2D orientation problem plays important roles in computational geometry. Floating-point arithmetic is fast performed by modern architectures. How- ever, the number of significand bits of a floating-point number is finite so that it may cause rounding errors in each operation. In the worst case, unreliable results may be obtained due to accumulation of rounding errors. As for the 2D orientation problem, the computed determinant may be opposite in sign to the exact determinant. The accuracy is especially worried when the point is very close to the line. Classical algorithms in computational geometry correctly work if the all computations are exact. Namely, it is assumed that all expressions are evaluated by rational arithmetic (symbolic computations). Once an inexact result is obtained, then an algorithm may output a meaningless result. For examples, there are several algorithms outputting a convex hull which is the minimum convex enclosing all points. However, meaningless results are sometimes output, for example, a resultant convex does not contain all points or the result is not convex. Such problems are called robustness problems and many examples can be shown in [1].

If the accuracy of the computed result is a concern, using multi-precision arithmetic or rational arithmetic is an option. However, these approaches take much computing time. In addition, many problems are correctly solved by floating-point arithmetic. Therefore, straightforward use of robust computations is not effective. There are so-called floating-point filters. They quickly guarantee that the computed sign of the determinant is correct if the problem is well-conditioned. It is a kind of self-validating methods. If the filter cannot guarantee the correctness of the result, then robust computations should be applied. As for earlier researches for the floating-point filters, for example, semi-static filters [2, 3] and dynamic filters [4, 5] have been developed. We suggest a simplified semi-static floating-point filter for the 2D orientation problem. When the filter handles floating-point exceptions like overflow and underflow, then the checks of the order of the magnitude for input coordinates seem to be necessary. Several branches are required so that it makes the performance slow down. Our filter handles any floating-point exceptions with only a branch so that it is possible to work fast. Finally, numerical examples are shown to illustrate the efficiency of the proposed filter.

References

[1] L. Kettner, K. Mehlhorn, S. Pion, S. Schirra, C. Yap: Classroom Ex- amples of Robustness Problems in Geometric Computations, Computa- tional Geometry, 40, 61–78, 2008. [2] J. R. Shewchuk: Adaptive Precision Floating-point Arithmetic and Fast Robust Geometric Predicates, Discrete & Computational Geometry, 18, 305–363, 1997. [3] G. Melquiond, S. Pion: Formally certified floating-point filters for ho- mogenous geometric predicates, Theoretical Informatics and Applica- tions, Special issue on Real Numbers, 41:57–69, 2007. [4] V. Y. Pan, Y. Yu: Certified Computation of the Sign of a Matrix Deter- minant, Proc. 10th Annual ACM-SIAM Symposium on Discrete Algo- rithms, ACM Press, New York, and SIAM Publications, Philadelphia, 715-724, 1999. [5] H. Brönnimann,C. Burnikel, S. Pion: Interval Arithmetic Yields Ef- ficient Dynamic Filters for Computational Geometry, Discrete Applied Mathematics, 109:25–47, 2001.

On the existence of steady ﬂows in a rectangular channel with slip eﬀect on one porous wall Un-Un Kuo and Ching-An Wang∗

Department of Mathematics, National Chung Cheng University, TAIWAN

∗ e-mail: [email protected]

Abstract We study a boundary layer problem arising from the study of steady laminar flows in channels with one porous wall with the slip effect. By applying a similarity transformation [1], the governing equation of the Navier- Stokes system for steady, incompressible laminar flows in this channel could be reduced to f 000(η) + R((f 0(η))2 − f(η)f 00(η)) = K. (1) Here K is an integration constant and R corresponds to the cross flow

Reynolds number based on wall velocity (filtration Reynolds number), while positive (negative) R represents the suction (injection) through the walls. The function f is related to the stream function, η is the normalized transverse coordinate, namely, η = 1 at the the porous wall, while the position at the insulated wall is denoted by η = −1. By imposing the slip effect on the porous wall, the steady flows could be studied from the boundary value problem (BVP) of (1), subjects to the boundary conditions:

f(−1) = f 00(−1) = 0, f(1) = 1, f 0(1) + ϕf 00(1) = 0, (2) where positive ϕ is the slip coeﬃcient. Following the frame work in [9], it is our purpose to explore the solutions of (BVP). It is clear that the given problem possesses the unique solution when R = 0. For nonzero R, as in [2, 4], suppose we set f(η) = bG(ξ)/2R, ξ = b(η + 1)/2 for some positive b which is to be determined. Then, G(ξ) satisﬁes the following associated boundary value problem(BVP1):

G000(ξ) + (G0(ξ))2 − G(ξ)G00(ξ) = 16RK/b4 ≡ β, (3)

G(0) = G0(0) = 0, (4) G(b) = 2R/b, G0(b) + ϕbG00(b)/2 = 0. (5)

By assigning values to

G00(0) = α, G000(0) = β, (6) we let G(ξ; α, β) be the solution of the initial value problem (IVP) of (??), (??) and (??). Suppose, for a prescribed positive ϕ, G0(b; α, β)+ϕbG00(b; α, β)/2 = 0 holds at ξ = b∗, then (BVP) will possess a solution with R = b∗G(b∗; α, β)/2 and K = (b∗)4β/16R. From the homogeneity of G(ξ; α, β),

R(α, β) = R(α/λ3, β/λ4), (7)

K(α, β) = K(α/λ3, β/λ4), (8) for all λ > 0, we are able to verify that given any position ϕ > 0, (BVP) can only possess either nonnegative and monotone, or non-monotone solutions. In particular, there exists a positive number R(ϕ) such that (BVP) possesses at least one nonnegative and monotone solutions for R < R(ϕ), and at least non-monotone solution for R ≥ R(ϕ). Moreover, the given problem possesses no solution for R ≤ 0, K ≥ 0.

References

[1] A.S. Berman, Laminar ﬂow in channels with porous walls, J. Appl. Phy., 1953, 24, 1232-1235.

[2] F. M. Skalak and C. Y. Wang, On the nonunique solutions of laminar ﬂow through a porous tube or channel SIAM J. Appl. Math., 1978, 34, 535-544.

[3] S. M. Cox, Analysis of steady ﬂow in channel with porous walls, or with accelerating walls, SIAM J. Appl. Math., 1991, 51, 429-438.

[4] T.-W. Hwang and C.-A. Wang, On multiple solutions for Berman’s problem, Proc. Roy. Soc. Edinburgh, 1992, 121A: 219-230.

[5] S. Chellaml, M. Wiesnerl and C. Dawson2,Slip at a unifromly porous boundary: effect on fluid flow and mass transfer, Journal of Engineering Mathematics, 1992, 26, 481-492.

[6] C.-A. Wang and Y.-Y. Chen, On existence of similarity solutions for Laminar ﬂow in a channel with one porous wall, Bulletin Inst. Math. Academia Sinica, 1992, 20, 299-320.

[7] C.-A. Wang and T.-C. Wu, Similarity solution of steady ﬂows in a channel with accelerating walls, Computers Math. Applic., 1995, 30, 1-16.

[8] S. Chellam and M. Liu, Eﬀect of slip on existence, uniqueness, and behavior of similarity solutions for steady incompressible laminar ﬂow in porous tubes and channels, Phys. Fluid, 18(8), 2006, DOI:10.1063/1.2236302.

[9] U.-U. Kuo and C.-A. Wang, Multiple solutions of steady ﬂows in a rectangular channel with slip eﬀect on two equally porous walls, to appear in Taiwanese J. Math.

Exchange of two modes of instability leading to sustained oscillations in the ﬂow past a cylinderical obstacle

1), 2) 2) Yukihito Suzuki ∗, Yukio Takemoto , Jiro Mizushima

1) School of Fundamental Science and Engineering, Waseda University 2) Faculty of Science and Engineering, Doshisha University

e-mail: [email protected] ∗ Abstract Flows past cylindrical obstacles often exhibit staggered arrangement of localized vortices, called Bénard-Kármán’svortex street. The most typical and simplest example is the flow around a circular cylinder placed in a uniform flow. The vortex street was conjectured to arise due to instability of the otherwise steady flow and the classical stability theory, based on a parallel flow approximation, was applied to the flow[1], which predicted a so small value such as Rec = 3.2 for the critical Reynolds number of instability. Here the Reynolds number is defined with the cylinder diameter and the uniform velocity. On the other hand, experiments and numerical simulations show that the flow becomes oscillatory for Re & 40 50. In fact, the global stability analysis of the flow without resorting the− parallel flow approximation revealed that the critical Reynolds number Reg is 46.184[2]. The difference between Rec = 3.2 and Reg = 46.184 is beyond allowance of error arising from the parallel flow approximation in the classical stability theory. In order to resolve the disagreement between the classical theory and the global stability analysis, a packet of disturbance is considered instead of a monochromatic wave[3, 4, 5]. For disturbances having a packet shape, two distinct instability modes are identified, one of which is called convective instability, and the other absolute instability. The flow is said convectively unstable when a packet of growing disturbances has a positive group velocity, and the absolute instability is defined if there exists a packet of growing disturbance with zero group velocity. Many numerical works have been reported on the convective and absolute instabilities of the cylinder wake based on the parallel flow approximation and it was found that the value Rec = 3.2 from the classical stability theory indicates the critical condition for the convective instability, while the criticality for the absolute instability is determined as Rea 25[4]. It is a common understanding at present that the global instability∼ occurs after the region of the absolute instability has a greater extent in the streamwise direction than 3.5 times the cylinder diameter[4] so that there exists a ‘pocket’ between Rea 25 and Reg = 46.184[5]. ∼ 47

We aim to resolve the gap between the absolute and global instabilities and clarify the mechanism of sustained oscillations in the flow, in which im- pulsive force is added at a certain point in the flow and subsequent spatiotem- poral development of induced disturbance is observed. In order to clarify the underlying physics of the transition from globally stable to unstable flow, we introduce the notions of passive and active modes of instability, extending the convective and absolute instabilities to non-parallel flows, respectively. Our numerical analysis brought us a conclusion that the oscillation is sustained by superiority of the growth of disturbance due to instability over the decrease due to advection of the packet. According to our definitions of the passive and active instabilities, the flow becomes unstable to active mode of disturbance simultaneously in the whole flow field at the critical Reynolds number Reg 47 for the global instability, whereas some extent of the region ∼ is passively unstable above a critical Reynolds number Rep. When the flow is passively unstable, the flow can be oscillatory if some external disturbance is continuously added anywhere in the flow field. Otherwise, any localized disturbance is swept out to far downstream at Re < 47. At supercritical Reynolds numbers, any packet of disturbance added in any location will expand over the entire region behind the cylinder, proceeding upstream to the region just behind the cylinder as well as being advected far downstream. Once the packet of disturbance arrives at the vicinity of the cylinder, it acti- vates oscillation leading to the global instability and becomes an oscillation source. We will propose an one-dimensional model equation, like a Ginzburg- Landau equation, which illustrates the mechanism of sustained oscillation and the transition from the passively unstable to actively (globally) unstable flow. The equation and its solutions will be presented at the conference.

References

[1] S. Taneda, J. Phys. Soc. Japan., 18, (1963), 288. [2] C. P. Jackson, J. Fluid Mech., 182, (1987), 23. [3] H. Oertel, Jr, Annu. Rev. Fluid Mech., 22, (1990), 539. [4] P. Huerre and P. A. Monkewitz, Annu. Rev. Fluid Mech., 22, (1990), 473. [5] J. M. Chomaz, Annu. Rev. Fluid Mech., 37, (2005), 357. [6] Y. Takemoto and J. Mizushima, Phys. Rev. E, 82, (2010), 056316.

Global and singular solutions to the generalized Proudman-Johnson equation Chien-Hong Cho1),∗, Marcus Wunsch2)

1) Department of Mathematics, National Chung Cheng University 2) Research Institute for Mathematical Sciences, Kyoto University

∗ e-mail: [email protected]

Abstract We consider the Generalized Proudman-Johnson equation

f + ff = af f , x ∈ , t > 0 txx xxx x xx R , (1) f(0, x) = f0(x) on the whole real line as well as the periodic Generalized Proudman-Johnson equation

f + ff = af f , x ∈ = / , t > 0 txx xxx x xx S R Z . (2) f(0, x) = f0(x), x ∈ S The generalized Proudman-Johnson equation (1), (2) can be motivated in several ways. The first motivation comes from the study of the two- dimensional incompressible Euler equations. For parameters a ∈ R ∪ {∞}, Eq. (1) was first derived by Okamoto and Zhu [7]. Remarkably, the parameter a interconnects several well-studied equations within the framework of the generalized Proudman-Johnson equation (1), (2). For a = 1, it reduces to the original Proudman-Johnson equation [8], which is obtained via the separation of spatial variables ψ(t, x, y) = yf(t, x) for the stream function ψ of the two-dimensional incompressible Euler equations, where x is contained in a finite interval, and y ∈ . Moreover, the generalized Proudman-Johnson R equation encompasses the well-known Burgers equation for a = −3, which is a successful model in gas dynamics; if a = −2, the generalized Proudman- Johnson equation becomes the Hunter-Saxton equation [5] arising in the study of nematic liquid crystals.

The existence of global and singular solutions were widely investigated recently, see for instance [1], [2], [6], [9]. In this talk, we construct a weak solution for (1) for −2 ≤ a < −1 by the modiﬁed characteristic method. Then by using an abstract lemma in [3], we explore the blow-up solutions for (2) for a < −1 and their exact blow-up rate.

References

[1] A. Bressan, A. Constantin: Global solutions of the HunterVSaxton equation, SIAM J. Math. Anal., 37 (2005) 996-1026.

[2] X.-F. Chen and H. Okamoto: Global existence of solutions to the generalized Proudman-Johnson equation, Proc. Japan Acad. Ser. A, 78 (2002) 136-139.

[3] A. Constantin and J. Escher: Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998) 229-243.

[4] C.-H. Cho and M. Wunsch: Global and singular solutions to the generalized Proudman-Johnson equations, J. Diﬀerential Equations, 249 (2010) 392-413.

[5] J.K. Hunter, R. Saxton: Dynamics of director ﬁelds, SIAM J. Appl. Math., 51 (1991) 1498-1521.

[6] H. Okamoto: Well-posedness of the generalized ProudmanVJohnson equation without viscosity, J. Math. Fluid Mech., 11 (2009) 46-59.

[7] H. Okamoto, J. Zhu: Some similarity solutions of the NavierVStokes equations and related topics, in: Proceedings of 1999 International Con- ference on Nonlinear Analysis (Taipei), Taiwanese J. Math., 4 (2000) 65-103.

[8] I. Proudman, K. Johnson: Boundary-layer growth near a rear stagnation point, J. Fluid Mech., 12 (1962) 161-168.

[9] R. Saxton, F. Tiglay: Global existence of some inﬁnite energy solutions for a perfect incompressible ﬂuid, SIAM J. Math. Anal., 4 (2008) 1499- 1515.

Application of RKHS Theory to a Minimization Problem with Prescribed Nodes Hendra GUNAWAN1),∗, Yoshihiro SAWANO2) 1) Department of Mathematics, Institut Teknologi Bandung, Indonesia 2) Department of Mathematics, Kyoto University, Japan ∗ e-mail: [email protected]

Abstract d Let 2 < α < ∞. We deﬁne a Hilbert space Hα to be the set of functions f on [0, 1]d of the form X

f(x1, . . . , xn) := am1···md sin(m1πx1) ··· sin(mdπxd) m1,...,md∈N for which π2α X kfk := (m2 + ··· + m2)α|a |2 < ∞. Hα 2d 1 d m1···md m1,...,md∈N The above norm is induced from the inner product π2α X hf, gi = (m2 + ··· + m2)αa b , Hα 2d 1 d m1···md m1···md m1,...,md∈N where am1···md and bm1···md are the coeﬃcients of f and g, respectively. We are then interested in studying the following problem:

Minimize kfkHα subject to the prescribed nodes:

f(pk) = ck, k = 1,...,N,

d where pk := (pk1, . . . , pkd) ∈ (0, 1) and ck ∈ R are given. [Note here that d the points pk’s are ‘inside’ the unit cube [0, 1] .] The 1- and 2-dimensional case have been studied by Gunawan et al. [1, 2]. In this talk, we shall present the use the theory of reproducing kernel Hilbert spaces to study the problem (in a more general setting). One of our results is the following: Proposition. The solution to the minimization problem

Minimize kfkHα

51 subject to f(p1, . . . , pd) = 1, is given by F (x1, . . . , xd) :=

X sin(m πp ) ··· sin(m πp ) A 1 1 d d sin(m πx ) ··· sin(m πx ). (m2 + ··· + m2)α 1 1 d d m1,...,md∈N 1 d

P 2 −1 (sin(m1πp1)··· sin(mdπpd)) where A := 2 2 α . (m1+···+md) m1,...,md∈N A more general result will be presented. In addition, we shall also consider the H¨oldercontinuity of the solution, by using properties of Besov spaces.

References

[1] H. Gunawan, F. Pranolo and E. Rusyaman, “An interpolation method that minimizes an energy integral of fractional order”, in D. Kapur (ed.), ASCM 2007, LNAI 5081 (2008), 151–162.

[2] H. Gunawan, E. Rusyaman and L. Ambarwati, “Surfaces with prescribed nodes and minimum energy integral of fractional order”, submitted.

[3] G.G. Lorentz, Approximation of Functions, AMS Chelsea Publishing, Providence, 1966.

[4] M. Sugimoto and N. Tomita, “The dilation property of modulation spaces and their inclusion relation with Besov spaces”, J. Funct. Anal. 248 (2007), 79–106.

[5] M. H. Taibleson, “On the theory of Lipschitz spaces of distributions on Euclidean n-space. I. Principal properties”, J. Math. Mech. 13 (1964), 407–479.

Exact meromorphic solutions of the

real cubic Swift-Hohenberg equation

Robert Conte1,2∗, Tuen-Wai Ng1†and Kwok-Kin Wong1‡

1. Department of Mathematics, The University of Hong Kong. 2. LRC MESO, Centre de math´ematiqueset de leurs applications (UMR 8536) et CEA-DAM, Ecole´ normale sup´erieurede Cachan.

E-mail: [email protected], [email protected], [email protected]

Abstract

We considered the real cubic Swift-Hohenberg (RCSH) equation

∂u ∂2 ∂2 2 = εu − 1 + + u − u3, ε ∈ . (1) ∂t ∂x2 ∂y2 R The equation and its generalizations have been used in various areas, such as laser [4] and nonlinear optics [5] and the theory of pattern formation [2], [6]. The equation (1) admits the following traveling wave reduction [7],

0000 00 3 0 d U + aU + U − U = 0, := . (2) dZ In our paper, we considered the equation (2) to be deﬁned on complex domain. By using a method of Eremenko [3], we show that all meromorphic solutions of the traveling wave reduction (2) of (1) are elliptic or degenerate

∗Partially supported by PROCORE - France/Hong Kong joint research grant F- HK29/05T and RGC grant HKU 703807P. †Partially supported by PROCORE - France/Hong Kong joint research grant F- HK29/05T and RGC grant HKU 703807P. ‡Partially supported by RGC grant HKU 703807P and a post-graduate studentship at HKU.

531 elliptic. The method is based on the analysis using Nevanlinna theory from complex analysis. We then obtain all the meromorphic solutions explicitly by the subequation method of Conte and Musette [1]. This is an algorithmic method, so is systematic. One of the solutions appears to be a new elliptic solution.

References

[1] R. Conte and M. Musette, Analytic solitary waves of nonintegrable equations, Phys. D 181 (2003) pp. 70–79.

[2] M.C. Cross and P.C. Hohenberg, Pattern formation outside of equilibrium, Rev. Modern Phys. 65 (1993) pp. 851–1112.

[3] A.E. Eremenko, Meromorphic traveling wave solutions of the Kuramoto- Sivashinsky equation, J. Math. Phys., Anal. Geom. 2 (2006) pp. 278– 286.

[4] J. Lega, J.V. Moloney and A.C. Newell, Swift-Hohenberg equation for lasers, Phys. Rev. Lett. 73 (1994) pp. 2978–2981.

[5] S. Longhi and A. Geraci, Swift-Hohenberg equation for optical parametric oscillators, Phys. Rev. A 54 (1996) pp. 4581–4584.

[6] R. Hoyle, Pattern formation: an introduction to methods (Cambridge University Press, Cambridge, 2006).

[7] L.A. Peletier and W.C. Troy, Spatial patterns: higher order models in physics and mechanics (Springer, 2001).

Free-surface flow under a sluice gate of an inclined wall from deep water

L.H. Wiryanto

Faculty of Mathematics and Natural Sciences, Bandung Institute of Technology e-mail: [email protected]

Abstract Nonlinear solutions of free surface flow under a sluice gate are studied in this paper. Upstream, the fluid is assumed to be infinitely in depth, and the gate makes an angle β to the horizontal axis. Therefore, the flow emerges from the gate and produces uniform stream far downstream. The problem is solved numerically by a boundary element method derived from an integral equation along the free surface. An analytical function is constructed, relating to the upstream flow, so that the integral equation is solvable. As the result, a free surface flow with smooth detachment from the edge of the gate is obtained for relatively large upstream Froude numbers, otherwise a free surface with back flow near the edge of the gate is indicated, and it tends to a stagnation point for a certain Froude number.

Free Convection Boundary Layer Flow from a Vertical Cone in a Porous Medium Filled with a Nanoﬂuid S. Ahmad1),∗, I. Pop2), K. Vajravelu3)

1) School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM, Penang, Malaysia 2) Faculty of Mathematics, University of Cluj, R-400082 Cluj, CP 253, Romania 3) Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA

∗ e-mail: [email protected]

Abstract

A study of convective flow in porous media filled with conventional heat transfer fluids has been widely conducted in the recent years due to its wide applications in engineering as post-accidental heat removal in nuclear reac- tors, solar collectors, drying processes, heat exchangers, geothermal and oil recovery, building construction, etc. (see [1], [2], [3], [4]). It is well known that conventional heat transfer fluids including water, oil and ethylene glycol mixture are poor heat transfer fluids, since the thermal conductivity of these fluids plays an important role on the heat transfer coefficient between the heat transfer medium and the heat transfer surface. In order to improve the efficiency of the fluids in the term of heat transfer, a new kind of heat transfer fluids known as nanofluid which contains a small quantity of nanosized particles (usually less than 100nm) that are uniformly and stably suspended in a liquid has been introduced by Choi [5]. In this paper, a problem of steady free convection boundary layer flow near a vertical cone (with half angle σ) embedded in a saturated porous medium filled with a nanofluid is theoretically studied using three types of metallic and non-metallic nanoparticles such as copper (Cu), alumina (Al2O3) and titania (T iO2) in the base fluid of water. It appears that this problem have been investigated by Cheng et al. [6], Yih [7] and Sohouli et al. [8] but only for the conventional heat transfer fluids. Therefore, our aim is to extend the problem by considering a nanofluid with constant surface heat flux. Under the assumption of using the model proposed by Tiwari and Das [9] along with the assumptions of the Darcy-Boussinesq and boundary layer approximations, the basic partial equations along with the boundary conditions are obtained. Then, these equations are reduced to ordinary 56 differential equations (ODEs) by introducing the similarity variables. The ODEs are solved numerically using shooting method for some values of the nanoparticle volume fraction parameter ϕ. The local Nusselt number and the temperature profiles are presented and discussed in detail. It is found that the local Nusselt number and the temperature profiles increase as the volume fraction parameter increases.

References

[1] D.A. Nield, A. Bejan: Convection in Porous Media, 3rd ed., Springer, New York, 2006. [2] D.B. Ingham, I. Pop (Eds.): Transport Phenomena in Porous Media, Vol. III, Elsevier, Oxford, 2005. [3] K. Vafai (Ed.): Handbook of Porous Media, 2nd ed., Taylor & Frabcis, New York, 2005.

[4] P. Vadasz: Emerging Topics in Heat and Mass Transfer in Porous Media, Springer, New York, 2008. [5] S.U.S. Choi: Enhancing Thermal Conductivity of Fluids with Nanopar- ticles, in: D.A. Siginer, H.P. Wang (Eds.), Development and Applica- tions of non-Newtonian Flows, ASME MD-vol. 231 and FED-vol. 66 (1995), 99–105. [6] P. Cheng, T.T. Le, I. Pop: Natural Convection of a Darcian Fluid about a Cone, International Communications in Heat and Mass Transfer 12 (1985), 705–717. [7] K.A. Yih: Uniform Transpiration Effect on Combined Heat and Mass Transfer by Natural Convection over a Cone in Saturated Porous Media: Uniform Wall Temperature/Concentration or Heat/Mass Flux, Interna- tional Journal of Heat and Mass Transfer 42 (1999), 3533–3537. [8] A.R. Sohouli, M. Famouri, A. Kimiaeifar, G. Domairry: Application of Homotopy Analysis Method for Natural Convection of Darcian fluid about a vertical full cone embedded in Porous Media Prescribed Surface Heat Flux, 15 (2010), 1691–1699. [9] R.K. Tiwari, M.K. Das: Heat Transfer Augmentation in a Two-sided Lid-driven differentially Heated Square Cavity Utilizing Nanofluids, In- ternational Journal of Heat and Mass Transfer 50 (2007), 2002–2018. 57

Features of the Nystr¨omMethod for the Sherman-Lauricella Equation on Piecewise Smooth Contours

1), 2),⋆ Didenko Victor ∗, Helsing Johan

1) Faculty of Science, University of Brunei Darussalam 2) Centre for Mathematical Sciences, Lund University

e-mail: [email protected] ⋆ ∗e-mail: [email protected]

Abstract Let Γ be a simple closed contour in the complex plane C. The Sherman- Lauricella equation 1 τ t 1 τ t ω(t) + ω(τ) d ln − ω(τ) d − = f(t), t Γ. 2πi Γ τ t − 2πi Γ τ t ∈ ∫ ( − ) ∫ ( − ) (1) plays an important role in various applications. Thus it is used in radar imaging, in theory of viscous incompressible ﬂows, as well as in plane elasticity and in other problems of mathematical physics [1, 2]. However, the solution of equation (1), if it exists, is not unique and there is no analytic formula to obtain it. In this work we study the stability of an approximation method in the space L2(Γ) for equation (1) in the case where contour Γ has corner points. More precisely, we consider the Nystr¨ommethod based on composite Gauss- Legendre quadrature formula

1 n 1 d 1 − − u(s) ds wpu(slp)/n, (2) ≈ ∫0 l=0 p=0 ∑ ∑ where l + εp slp = , l = 0, 1, . . . , n 1; p = 0, 1, . . . , d 1, (3) n − − and wp and 0 < ε0 < ε1, . . . < εd < 1 are the weights and the Gauss-Legendre points on the interval [0, 1]. It is worth noting that for smooth contours Γ the integral operators in the left-hand side of (1) are compact, and the method proposed is always stable. On the other hand, if Γ possesses angular points τk, k = 1, 2, . . . , m, the stability of the method under consideration is connected to the invertibility of certain operators Aτk from an algebra of Toeplitz operators. The operators

Aτk depend on the above parameters εp and on the magnitude of the corresponding angles at the points τk. These operators Aτk arise if one applies quadrature formulas similar to formulas (2)-(3) to Mellin operators with the kernels deﬁned by the kernels of the integral operators in equation (1). They have a complicated structure and, at the moment, there is no analytic tools to study their invertibility. However, each such operator can be associated with the corresponding Nystr¨ommethod for the Sherman-Laurichella equation on a special model curve Γ0 with only one corner point. On the other hand, the stability of the method in those model situations is connected to the behaviour of condition numbers of systems of algebraic equations and it can be tested numerically. Thus, considering the sequences of the condition numbers we obtain certain information about the invertibility of the operators Aτk .

References

[1] S. G. Mikhlin: Integral Equations and Their Applications to Certain Problems in Mechanics, Mathematical Physics and Technology, 2nd ed.,

Pergamon Press, 1964.

[2] N. I. Muskhelishvili: Fundamental Problems in the Theory of Elasticity, Nauka, 1966.

A New Evolutionary Algorithm for Ground to Flight Data Correlation Changtong Luo1),∗, Shao-Liang Zhang2), Zonglin Jiang1) 1) Institute of Mechanics, Chinese Academy of Sciences 2) Department of Computational Science and Engineering, Nagoya University ∗ e-mail: [email protected]

Abstract In hypersonic vehicle design, it is very difficult/expensive to simulate the environment in flight. Therefore, it is of key important to predict the aerodynamic coefficients of the flight vehicle using wind-tunnel and flight- test data. The task is called ground to flight data correlation, also shorten as ground/flight correlation. A number of methods have been applied to ground/flight correlation, including extrapolation [1], least squares regression [2], artificial neural network [3, 4] and maximum likelihood method [5]. However, all these methods are based on a certain model structure and optimize the parameters in the model. This is, in many cases, unreliable. To find an optimal model structure will definitely improve the predicted results. In this study, we suggest searching the best model structure from a func- N tional space F with collected data {{xi1, xi2, ··· , xin; yi}}i=1, which can be N ∗ X described as f = arg min kf(xi1, xi2, ··· , xin) − yik . f∈F i Genetic programming [6] is a candidate for this optimization problem. However, its tree-based representation makes it difficult (although not impossible) to be implemented in general-purpose programming languages such as C/C++ and Fortran. To overcome this difficulty, M. O’Neill and C. Ryan proposed a grammatical evolution (GE) [7] in 2001. However, GE is still not so easy to use. First, the implementation of GE is complicated because it needs an additional function parser for the encoding and decoding process. Next, the incomplete mapping and extra codons problems [7, 8] are common but difficult to handle. In this paper, a new evolutionary algorithm, parse-matrix evolution (PME), for flight data correlation is proposed. A chromosome in PME is a parse- matrix with integer entries. The mapping process from a chromosome to its model function is based on a mapping table. PME can easily be implemented in any programming language and free to control. Furthermore, it does not need any additional function parsing process. Numerical results show that PME can solve the symbolic regression problems effectively (see Table 1).

Table 1: Test models and performance of PME no. Dim Target model Domain no.samples ave no.eval ave no.start 1 1 x2 − sin x [1, 3] 8 3230 4.8 2 1 sin x + 2x [1, 3] 8 2071 1.1 3 1 cos x2 − x [1, 3] 8 2603 2.3 4 1 sin x + cos x − x [1, 3] 8 3646 7.4 2 5 2 x1 + 2x2 [1, 3] 16 1047 1.0 2 6 2 (x1 + x2)/x2 [1, 3] 16 1701 1.3 2 7 2 sin(x1 + x2) [1, 3] 16 1537 1.0 2 8 2 cos(x1 + x2) − x2 [1, 3] 16 2798 1.5 2 9 2 ln(x1 + x2) [1, 3] 16 1584 1.2 x1+x2 2 10 2 x1 − e [1, 3] 16 3593 1.8 2 11 2 ln(x1 + x2) + sin(x1 + x2) [1, 3] 16 4911 5.7 2 2 12 2 sin(x1 − x2) + ln(x1 + x2) [1, 3] 16 7637 13.2 3 13 3 x1 + x2 − x3 [1, 3] 64 4505 1.9 3 14 3 x1x2 − x3 [1, 3] 64 3164 2.7 3 15 3 sin(x1x2) + x3 [1, 3] 64 3302 11.6

References

[1] J.B. Peterson, Jr., M.J. Mann, R.B. Sorrells III, W.C. Sawyer, and D.E. Fuller: Extrapolation of Wind-Tunnel Data to Full-Scale Conditions, NASA TP-1515, 1980. [2] E.A. Morelli, R. DeLoach: Wind Tunnel Database Development Using Mod- ern Experiment Design and Multivariate Orthogonal Functions, AIAA paper, AIAA 2003-0653, 2003. [3] M. Norgaard, C. Jorgensen, J. Ross: Neural Network Prediction of New Aircraft Design Coefficients, NASA Technical Memorandum 112197, 1997. [4] T. Rajkumar, Jorge Bardina: Prediction of Aerodynamic Coefficients using Neural Networks for Sparse Data, in: Proc. 15th Int’l Florida Artificial Intel- ligence Research Society Conf., 242-246, 2002.

[5] J.-H. Lee, E.T. Kim, B.-H. Chang, I.-H. Hwang, D.-S. Lee, The Accuracy of the Flight Derivative Estimates Derived from Flight Data, World Academy of Science, Eng. Technol., 58 (2009), 843-849. [6] J. R. Koza, Genetic programming: on the programming of computers by means of natural selection. Cambridge, MA: MIT Press, 1992. [7] M. O’Neill and C. Ryan, Grammatical evolution, IEEE Trans. Evol. Comput. 5 (2001), 349-358. [8] M. O’Neill, A. Brabazon, Grammatical swarm: the generation of programs by social programming. Nat. Comput. 5 (2006), 443-462.

Ocean wave scattering within or over a porous medium Swaroop Nandan Bora1),∗, Santu Das2)

1) Department of Mathematics, Indian Institute of Tech. Guwahati, India 2) Department of Mathematics, Indian Institute of Tech. Guwahati, India

∗ e-mail: [email protected]

Abstract Water wave scattering within a porous medium on a sea-bed is considered based on linear wave theory. Using the model of wave-induced flow within a porous medium and Galerkin eigenfunction expansions, refraction-diffraction equations for surface waves are derived whenever needed. The reflection and transmission of waves are considered mainly for two cases: (i) the porous medium is obstructed on the right by a solid wall which does not allow any transmission for the case of horizontal bottom, (ii) and when the bottom is approximated by a line of constant slope or by a raised horizontal bottom. Further, by introducing a non-porous medium again after the porous medium, subsequent scattering is investigated. The fluid region is divided into two or three sub-regions according to the points of discontinuity. For the first case, the wave motion inside the porous structure tends to decay as it propagates through the pores. The wave, on encountering the solid wall, gets reflected into the porous structure. For the second case, refraction-diffraction equations for surface waves are derived for the porous region. The scattering is now due to both the porous structure and the structure at the bottom. A boundary value is set up for a flat sea-bed and the analytical solution obtained and Galerkin eigenfunction method is introduced in order to expand the spatially dependent potential in terms of depth-dependent functions. The oblique incident wave propagates at an angle θ to the x-direction. By appropriate matching of the properties such as velocity potential, velocity components of two nearby regions, the reflection phenomenon is studied. For a water depth of h, which may be uniform (even bed) or non-uniform (uneven bed), the potential φ in each region satisfies Laplace’s equation

∇2φ = 0, −h ≤ z ≤ 0. (1)

The free surface boundary condition will be ∂φ − µφ = 0, at z = 0, in the non-porous medium, and, (2) ∂z

∂φ − iµR = 0, at z = 0, in the porous medium. (3) ∂z where µ = σ2/g with σ as the wave frequency, g the gravity constant, R = f − iS the impedance of the porous medium with f as friction factor and S as inertial coefficient. Considering the depth-dependent part of the potential cosh kn(hj + z) as Zj,n(z) = , n = 0, 1, 2,... with j = 1, 2,... denoting the cosh knhj region number and the wave number kn satisfying the complex dispersion relation iRµ = kn tanh knhj. In the first region prior to the location of the porous structure, the velocity potential is X∞ iK1,0 x −iK1,0 x −iK1,n x φ(x, z) = (e + R0e )Z1,0(h1, z) + Rne Z1,n(h1, z), (4) n=1 q 2 2 where K1,n = k1,n − λ , with λ = k0 sin θ and R0 the reflection coefficient. The potentials in the other regions can be found by looking at the relevant physics. Ultimately the reflection coefficient can be obtained with the help of coefficients through the orthogonal functions Zn(z). Some values for the relevant parameters are assumed and the reflection is studied for various depths, porosity factor etc. When the wave is allowed to propagate over an uneven porous sea-bed, perturbation technique is used to reduce the original problem to a simpler one for the first order correction of the potentials. The solution of this problem is obtained by an approximate use of Green’s integral theorem. Sinusoidal ripples is considered as an example of the sea-bed. The reflection and transmission coefficients are evaluated against various values of parameters and it is observed that very few ripples are needed to generate significant reflection.

References

[1] Martha SC, Bora SN, Chakrabarti A. Oblique water wave scattering by small undulations on a porous sea-bed. Applied Ocean Research, 2007; 29: 86-90. [2] Dalrymple RA, Losada MA, Martin PA. Reﬂection and transmission from porous structures under oblique wave attack. J. Fluid Mech., 1991; 224: 625-644. [3] Zhu S. Water waves within a porous medium on an undulating bed. Coastal Engineering, 2001; 42: 87-101.

Spatially localized traveling structures and the asymptotic behavior in binary fluid convection 1),* 2) 2) Yasumasa Nishiura , Takeshi Watanabe , Makoto Iima

1) Research Institute for Electronic Science, Hokkaido University 2) Department of Mathematical and Life Sciences, Hiroshima University * e-mail: [email protected]

Abstract We study spontaneous pattern formation and its asymptotic behavior in binary fluid flow driven by a temperature gradient [1]. When the conductive state is unstable and the size of the domain is large enough, finitely many number of spatially localized time-periodic traveling structures (PTP) [3] are generated spontaneously in the conductive state. They collide and merge via strong collisions, and finally, they are arranged at non-uniform intervals and move in the same direction as in Fig.2. Although there are many other stable patterns, it is remarkable that PTPs end up the winner for survival via interactions as in Fig.1. We found that the role of PTP solutions and their strong interactions (collision) [2] are important to characterize the asymptotic state and hence, pulse-pulse interactions are investigated in detail.

Fig.1: Collision between stable stationary convective pattern (middle) and PTP (from right). They merge into one body, but PTP persists after strong interaction.

Fig.2: Time evolution for two generic initial data. A point disturbance is added at t = 0, x = 500 (right), and small random disturbances are added in the whole domain (left). Time interval of the visualization is t = 250.

References

[1] P. Kolodner, Collisions between pulses of traveling-wave convection, Phys. Rev. A 44, (1991) 6466–6479. [2] Y. Nishiura, T. Teramoto, and K. Ueda, Scattering of traveling spots in dissipative system, Chaos 15, (2005) 047509. [3] T. Watanabe, K. Toyabe, M. Iima and Y. NishiuraTime-periodic traveling solutions of localized convection cells in binary fluid mixture, Theoretical and Applied Mechanics Japan, 59, (2010), 211–219.

Adaptive Mixed Finite Element Methods and New Discontinuous Galerkin methods

1), Eun-Jae Park ∗,

1) Department of Computational Science and Engineering, Yonsei University, Seoul 120-749, Korea

e-mail: [email protected] ∗ Abstract This extended abstract deals with locally conservative methods and consists of two parts: first part on adaptive mixed finite element methods and second on new discontinuous Galerkin methods. First part is based on joint work with Dongho Kim [2]. We study adaptive mixed finite element discretizations for nonlinear elliptic problems. The mixed finite element method has two important features; it conserves the mass locally and produces accurate flux even for highly nonhomogeneous media with large jumps in the physical properties. In this paper, we take an approach based on the Brezzi-Rappaz-Raviart framework rather than using the Brouwer fixed point theory. Compared with existing literature several improvements are made. First, the Raviart-Thomas finite element space of all orders including the lowest order case is treated. Existence and uniqueness of the approximate solution to the model problem is proved. Next, we point out that compared with the work [5], the required H5/2(Ω) regularity of the solution weakens to H2+ϵ(Ω). Next, we drive an optimal order a priori error estimate measured in the Lm(Ω)-norm in the framework of Brezzi-Rappaz-Raviart. The Brezzi- Rappaz-Raviart theory requires of the C1,1-regularity of the nonlinear function; we extend the theory to allow more general C1,α class of the function for 0 < α 1. Finally,≤ we derive reliable and efficient a posteriori error estimators in Lm(Ω) for the error control of our approximation to the nonlinear problem under consideration. To the best of the authors’ knowledge, this is the first result obtaining a posteriori error estimator measured in the Lm(Ω)-norm which is reliable and efficient for a mixed approximation of nonlinear elliptic problems.

Second part is based on joint work with Youngmok Jeon [4]. A new class of hybrid discontinuous Galerkin methods is introduced and analyzed for second-order elliptic equations. Recently, the hybridizable discontinuous

Galerkin (HDG) method is introduced and developed by Cockburn and his collaborators [1]. The main feature of the HDG method is that their approximate solutions can be expressed in an element-by-element fashion in terms of an approximate trace uh satisfying a global weak formulation, which reduces globally coupled degrees of freedom dramatically. In this sense, our method can be understood asb a HDG method.

On the other hand, our method can be viewed as a generalization of so called cell boundary element (CBE) by Jeon and his coauthors [3], therefore, the method is named as GCBE. The CBE method is a flux conserving method without a dual partition, but it has the same disadvantages as the finite volume method, namely that high-order accurate methods are not easy to construct [3]. The motivation of the GCBE method is to construct a flux conserving method with high order approximation property. More details can be found in the recent paper [4].

References

[1] , Uniﬁed hy- B. Cockburn, J. Gopalakrishnan and R. Lazarov bridization of discontinuous Galerkin, mixed and continuous Galerkin methods for second order elliptic problems, SIAM J. Numer. Anal. 47 (2009), pp. 1319–1365.

[2] D. Kim and E.-J. Park, A Priori and A Posteriori Analysis of Mixed Fi- nite Element Methods for Nonlinear Elliptic Equations. SIAM J. Numer. Anal. Volume 48 (2010), no. 3, pp. 1186-1207.

[3] Y. Jeon, E.-J. Park, Nonconforming cell boundary element methods for elliptic problems on triangular mesh, Appl. Numer. Math. 58 (6) (2008) 800–814.

[4] Y. Jeon and E.-J. Park, A hybrid discontinuous Galerkin method for elliptic problems, SIAM J. Numer. Anal. Volume 48 (2010), no. 5, pp. 1968-1983

[5] F. A. Milner, E.-J. Park, Mixed ﬁnite element methods for Hamilton- Jacobi-Bellman-type equations, IMA J. Numer. Anal. Vol.16 (1996), pp. 399–412.

Numerical Investigation of Mixed Convection Flow over a Horizontal Circular Cylinder and a Solid Sphere in Nanoﬂuids Roslinda Nazar

School of Mathematical Sciences, Faculty of Science & Technology Universiti Kebangsaan Malaysia

e-mail: [email protected]

Abstract The recent discovery of nanofluids by Choi [1], which is a new kind of fluid suspension consisting of uniformly dispersed and suspended nanometer-sized (10-50 nm) particles and fibers in the base fluid, marks the next approach as a cooling technology. This is because nanofluids exerting nano effects have special features. One such characteristic of nanofluids is the anomalous high thermal conductivity at very low concentration of nanoparticles and the considerable enhancement of forced convective heat transfer, as it was pointed out by many researchers. Nanofluids usually contain the nanoparticles such as metals, oxides, or carbon nanotubes, whereby these nanoparticles have unique chemical and physical properties. In this study, the steady mixed convection boundary layer flow of nanofluids, formed by the dilution of nanoparticles, past a horizontal circular cylinder and about a solid sphere with constant surface temperature is investigated numerically for both cases of assisting and opposing flows. The nanofluid equations model proposed by Tiwari and Das [2] is used for the analysis. The resulting system of nonlinear partial differential equations is solved using an implicit finite-difference scheme known as the Keller-box method. The solutions for the flow and heat transfer characteristics are evaluated numerically for various values of the parameters, namely the nanoparticle volume fraction ϕ and the mixed convection parameter λ at Prandtl number Pr =

0.7, 1, 6.2 and 7. Three different types of nanoparticles, namely Al2O3, Cu and T iO2 (by using water-based fluid with Pr = 6.2) have been considered in this study and representative results for the local skin friction coefficient and the local heat transfer coefficient or local Nusselt number have been obtained for the following range of nanoparticle volume fraction: 0 ≤ ϕ ≤ 0.2 (see Abu-Nada and Oztop [3]). We have used data related to thermophysical properties of fluid and nanoparticles as presented in [3] to compute each case of nanofluid. In order to verify the accuracy of the present method, the values of the local skin friction coefficient and the local Nusselt number are compared

68 with those reported by Merkin [4] for ϕ = 0 (regular Newtonian fluid) at Pr = 1 and various values of x and λ, for the horizontal circular cylinder case; while the values of the local skin friction coefficient and the local heat transfer coefficient are compared with those reported by Nazar et al. [5] for ϕ = 0 at Pr = 0.7 and Pr = 7 and various values of x and λ, for the solid sphere case. The comparison of the previous results obtained in [4, 5] and the present ones are found to be in excellent agreement. It is found that for each particular nanoparticle, as the nanoparticle volume fraction ϕ increases, the skin friction coefficient and the heat transfer rate at the surface also increase, and it also leads to the increase of the value of the mixed convection parameter λ which first gives no separation and the value of λ below which a boundary layer solution does not exist. It is also found that an increase in the Prandtl number Pr leads to an increase of the local skin friction coefficient and the local Nusselt number. Finally, it is found that the velocity and temperature profiles satisfy the far field boundary conditions asymptotically, and as such, this support the validity of the numerical results obtained. The results obtained can be used to explain the characteristics and applications of nanofluids, which are widely used as coolants, lubricants, heat exchangers and micro-channel heat sinks.

References

[1] S. Choi: Enhancing thermal conductivity of ﬂuids with nanoparticles, ASME International Mechanical Engineering Congress and Exposition, FED 231/MD, 66 (1995) 99–105.

[2] R.K. Tiwari, M.K Das: Heat transfer augmentation in a two-sided lid- driven diﬀerentially heated square cavity utilizing nanoﬂuids, Interna- tional Journal of Heat and Mass Transfer, 50 (2007), 2002–2018.

[3] E. Abu-Nada, H.F. Oztop: Eﬀects of inclination angle on natural convec-

tion in enclosures ﬁlled with Cu-water nanoﬂuids, International Journal of Heat and Mass Transfer, 30 (2009), 669–678.

[4] J.H. Merkin: Mixed convection from a horizontal circular cylinder, In- ternational Journal of Heat and Mass Transfer, 20 (1977), 73–77.

[5] R. Nazar, N. Amin, I. Pop: Mixed convection boundary layer ﬂow about an isothermal sphere in a micropolar ﬂuid, International Journal of Ther- mal Sciences, 42 (2002) 283–293.

Lamellar pattern and two-phase shallow water equations

1) Hidenori Yasuda ∗

1) Department of Mathematics, Josai University

e-mail: [email protected] ∗ Abstract We propose a numerical method to simulate the phase separation of block copolymer in one-dimensional thin liquid film. Phase separation of polymer blend: a mixture of homopolymers, finally becomes a single circle [1]. However, phase separation of block copolymer: chemically connected two homopolymers, is qualitatively different. In one- dimensional system, periodic microscopic scale pattern called lamellar pattern appears. Phenomena of phase separation of copolymer is modeled by two-phase shallow water equations and Ohta-Kawasaki potential. Ohta-Kawasaki potential governs the phase separation of block copolymer [2]. However, some difficulty appears in the computation of Ohta-Kawasaki potential. A modification of the potential is proposed to circumvent the difficulty. Two- phase shallow water equations are discretized by an invariant finite difference method. And the potential is integrated by the trapezoidal rule. Lamellar pattern and wavy surface developed using the proposed method [3].

Models and numerical method 1. Two-phase shallow water equations Two-phase ﬂow in the liquid ﬁlm is described by two-phase shallow water equations, ∂ (α h) ∂ d + (α hu ) = 0, (1) ∂t ∂x d d ∂ (α h) ∂ c + (α hu ) = 0, (2) ∂t ∂x c c 2 ∂ud ∂ 1 ∂ ∂ + ud ud = (ρmgh) + ν 2 ud + fd, (3) ∂t ∂x −ρd ∂x ∂x 2 ∂uc ∂ 1 ∂ ∂ + uc uc = (ρmgh) + ν 2 uc + fc, (4) ∂t ∂x −ρc ∂x ∂x

αd + αc = 1. (5)

Here, α is the volume fraction, u the velocity, h surface height, ρ density, d is the suﬃx of minor phase, c is the suﬃx of major phase, and ρm is the density of mixture:

Two-phase shallow water equations is discretized by an invariant ﬁnite diﬀer- ence scheme which highly resolve the interface region of phase separations [1].

2. Ohta-Kawasaki potential Phase separation of block copolymer is governed by Ohta-Kawasaki potential [2], F ψ = Fs ψ + Fl ψ (6) { } { } { } ψ = αd αc (7) − a 2 b 4 γ 2 Fs ψ = dx ψ + ψ + ( ψ) (8) { } ∫ [−2 4 2 ∇ ] ε ¯ ¯ Fl ψ = dx dx0G (x, x0) ψ (x) ψ ψ (x0) ψ (9) { } 2 − − ∫ ∫ ( )( ) ∆G (x, x0) = δ (x x0) . (10) − − Here, F means free energy of block copolymer; ψ is the order parameter: ψ¯ is the average of ψ. G is the fundamental solution of Laplace equation.

Ohta and Kawasaki potential has some difficulty to compute numerically. For lamellar patterns, we propose a modification to circumvent the difficulty. Numerical integration of the integration of the potential is performed by the trapezoidal rule.

In simulations using the proposed method, lamellar pattern and wavy surface consistent with the speciﬁed period developed [3].

References

[1] H. Yasuda, Two-phase shallow water equations and phase separation in thin immiscible liquid ﬁlms, J. Sci. Comput. 43 (2010) 471-487.

[2] T.Ohta, K. Kawasaki, Equilibrium morphology of block copolymer melts, Macromolecules, 19 (1986) 2621-2632.

[3] H. Yasuda, Simulation of micro-phase separation of block copolymer in one-dimensional thin liquid ﬁlms, submitted.

IMPLEMENTING THE FINITE VOLUME METHOD ON A RECTANGULAR GRID

Tanay Milind Deshpande* Undergraduate, B.E.(Hons.) Mechanical Engineering Birla Institute of Technology and Science (BITS) – Pilani K. K. Birla Goa Campus Goa, India *[email protected]

Abstract

The Finite Volume Method (FVM) is commercially the most widely used CFD technique to convert the non-linear partial differential equations that govern fluid flow into a linear algebraic system of equations. This paper formulates an algorithm to solve steady-state, laminar and incompressible fluid flow equations with a structured collocated grid arrangement in a rectangular domain and describes its implementation using C++. The chief aim of this algorithm is to considerably simplify and optimize existing FVM techniques such as the Semi Implicit Method for Pressure Linked Equations (SIMPLE) algorithm (Patankar, 1972) for low Peclet number problems in regular geometries. After setting boundary conditions for driven-cavity flow as an instance, the Central Differencing scheme and the divergence theorem are invoked. The Navier-Stokes equations are discretized at each cell in the grid into a system of coefficients of neighbouring cell velocities. The algebraic equations representing each line in the grid are then fed into a solver subroutine that implements the Tridiagonal Matrix Algorithm. After two Alternating Direction Implicit traversals for velocities, the continuity equation is discretized in a similar manner for evaluating pressure correction terms which update the pressure and velocities at every node for the next iteration. At this crucial juncture where the SIMPLE takes contributions of pressure errors at neighbouring cells to be zero while calculating that at a single cell, this algorithm departs from the procedure and instead accommodates for neighbouring pressure errors.

References-

1. Versteeg H., Malalasekera W., ‘Introduction to Computational Fluid Dynamics: The Finite Volume Method’, 1995, Longman Scientific & Technical 2. Anderson J., ‘Computational Fluid Dynamics: The Basics with Applications’, 1995, McGraw-Hill 3. Patankar S., ‘Numerical Fluid Flow and Heat Transfer’, Hemisphere Publishing Corporation, 1980 4. Ferziger J., Peric M., ‘Computational Methods for Fluid Dynamics, Springer-Verlag, 2001 5. The SIMPLE Algorithm, Patakar S., Spalding B., 1972 & The Tridiagonal Matrix Algorithm, Thomas, 1949

Accurate and Rigorous Exponential Algorithm in Round to Nearest

1), 2) 2) Naoya Yamanaka ∗, Masahide Kashiwagi , Shin’ichi Oishi

1) Research Institute for Science and Engineering, Waseda University 2) Faculty of Science and Engineering, Waseda University

e-mail: naoya [email protected] ∗ Abstract This talk is concerned with numerical algorithms retaining high reliability, high accuracy and high portability. In this talk, an algorithm with high reliability means a numerical algorithm which outputs a mathematically- rigorous result. An algorithm with high accuracy represents an algorithm which returns a result with high accuracy. Furthermore, an algorithm with high portability indicates an algorithm which calculates a result without relying on any numerical environment. In this talk, numerical exponential algorithm retaining high reliability, high accuracy and high portability is discussed. Todays libraries for the approximation of elementary functions are very fast and the results are mostly of very high accuracy [1, 2]. The achieved accuracy does not exceed one or two ulp for almost all input arguments; however, there is no proof for that. In this talk, we discuss a reliable, accurate, portable implementation of the exponential function. We intend to utilize the marvelous accuracy by a table approach. Proposed algorithm delivers rigorous bounds for the result for all floating point input arguments. The order of evaluation of the formula is carefully chosen to diminish accumulation of rounding errors. As a result we obtain the relative accuracy of the exponential function value is better than 1ulp. Throughout this talk, we assume floating-point arithmetic adhering to IEEE standard 754-1985. IEEE standard 754-1985 is one of a technical standard established by the IEEE and the most widely-used standard for floating point computation, followed by many hardware and software imple- mentations. Proposed algorithm is based on computer interval arithmetic adhering to the standard. Interval arithmetic is a numerical method developed by mathematicians since the 1950s and 1960s as an approach to putting bounds on rounding errors and measurement errors in mathematical computation and thus developing numerical methods that yield reliable results. It represents each value as a range of possibilities. The interval arithmetic on floating-point arithmetic with changing rounding mode is widely used.

Rounding mode is one of defined parameter in IEEE standard 754-1985. The standard defines four rounding algorithms: rounding to nearest, round toward zero, round toward plus-infinity, round toward minus-infinity. However, changing rounding mode takes some computational costs, and the commands to change the mode vary widely depending on the numerical environment. Besides, numerical environments which do not have the commands to change the mode exist. For these problems, proposed algorithm only in rounding to nearest is proposed.

References

[1] J. M. Muller: Elementary Functions, Algorithm and Implementation, Second Edition, Birkh¨auser,2006.

[2] S. M. Rump: Rigorous and portable standard functions. BIT Numerical Mathematics, 41(3):540–562, 2001.

An integrated manufacturing system for time-varying demand process M.Omar

Institute of Mathematical Sciences, University of Malaya, Kuala Lumpur

e-mail: [email protected]

Abstract This paper considers a manufacturing system in which a single-manufacturer procures raw material from a single-supplier at multiple installments and process them to make ﬁnished products and deliver to a single-buyer at multiple shipments to satisfy a time-varying demand rate. In this system, the manufacturer must deliver the products in small quantities to minimize the buyer’s holding cost, and accept the supply of small quantities of raw material to minimize its own holding cost. We develop a mathematical model for this problem and illustrate the eﬀectiveness of the model with numerical examples.

Numerical analysis of a stationary transport equation by upwind scheme Takanori Katsura Graduate School of Informatics, Kyoto University e-mail: [email protected] Abstract The aim of this study is to simulate the stationary transport equation numerically as a fundamental study of the diffuse optical tomography(DOT). DOT is a new biomedical imaging technique, which is hoped to come after X-ray CT and MRI, and it has been extensively studied for practical use in medical sciences. It is considered that light propagation in biomedical tissue is described as photon density by the transport equation, and DOT is mathematically formulated as an inverse problem of the equation [1, 2]. This mathematical model of DOT is commonly accepted but should be more verified in detail to put DOT to practical use. To this end, we propose a new numerical scheme based on the finite difference method which is helpful for a fundamental study of the equation. The comparison of the numerical simulations and the experimental measurements will give us some suggestions for the mathematical modeling. In this presentation, we consider the following boundary value problem of stationary transport equation:

ξ xu(x, ξ) + µt(x)u(x, ξ) = µs(x) p(ξ ξ0)u(x, ξ0) dσξ0 + f(x)(x, ξ) Q ·∇ 1 · ∈ ∫S u(x, ξ) = α u(x, ξ0) + q(x, ξ) (x, ξ) Γ−. ∈ Here Ω is a square domain (0, 1) (0, 1) and n(x) is the outward unit normal + × vector on ∂Ω, Γ is the outward boundary, Γ− is the inward boundary; 1 Γ± := (x, ξ) ∂Ω S ξ n(x) < 0 , α is reﬂectance, ξ0 = ξ 2 (ξ n(x)) n(x). To{ discretize∈ the× transport| · equation,} we apply the trapezoidal− · formula to the integral term, and the upwind diﬀerence to x. Let K be the divi- ∇ sion number of velocity and h be the the spatial mesh size. We set ξk := 2π 2π 1 (cos K , sin K ) S , xi,j := (ih, jh) Ω, s(k, i) := sgn(ξk,i). Then we propose the following∈ upwind scheme (1)∈ (4): − ui,j,k ui s(k,1),j,k ui,j,k ui,j s(k,2),k ξ − − + ξ − − + µ (x )u k,1 s(k, 1)h k,2 s(k, 2)h t i,j i,j,k K 1 (1) 2π − = µs(xi,j) p(ξk ξl)ui,j,l + f(xi,j)(xi,j, ξk) Q K · ∈ l=0 ∑ 76

ui,j,k ui 1,j,k ui 1,j,k ui 1,j s(k,2),k ξk,1 − ± + ξk,2 ± − ± − + µt(xi 1,j)ui 1,j,k h s(k, 2)h ± ± ∓ K 1 2π − (2) = µs(xi 1,j) p(ξk ξl)ui 1,j,l + f(xi 1,j) ± K · ± ± l=0 ∑ + (xi,j, ξk) Γ (x1 = 0, x1 = 1) ∈ ui,j 1,k ui s(k,1),j 1,k ui,j,k ui,j 1,k ξk,1 ± − − ± + ξk,2 − ± + µt(xi,j 1)ui,j 1,k s(k, 1)h h ± ± K ∓1 2π − (3) = µs(xi,j 1) p(ξk ξl)ui,j 1,l + f(xi,j 1) ± K · ± ± l=0 ∑ + (xi,j, ξk) Γ (x2 = 0, x2 = 1) ∈ u = α u + q(x , ξ ) i,j,k i,j,l i,j k (4) ξl = ξk 2 (ξk n(xi,j)) n(xi,j), (xi,j, ξk) Γ−. − · ∈ We prove the convergency of this upwind scheme. Moreover we propose a iterative scheme based on the upwind scheme and prove the convergency. The following ﬁgures are numerical examples of the light intensity U(x) by our proposed scheme. Light scattering is described by the scattering kernel p(ξ, ξ0) and the anisotropy factor g controls how strongly the scattering is forward-peaked; g = 0, isotropic scattering; g = 0.9, strongly forward-peaked scattering.

g = 0.0 g = 0.5 g = 0.9

Fig. 1 Numerical examples of the light intensity U(x)

References

[1] S. R. Arridge, M. Schwiger: Image reconstruction in optical tomography Philosophical Transactions of the Royal Society B, 352 (1997), 717-726

[2] A. D. Klose, E. W. Larsen: Light transport in biological tissue based on the simpliﬁed spherical harmonic equations, Journal of Computational Physics, 220 (2006), 441-470

Stable Traﬃc Jam Occurs from Nonlinear Saturation Akiyasu Tomoeda1,2),∗, Ryosuke Nishi3,4), Kazumichi Ohtsuka5,6), Katsuhiro Nishinari3,5)

1) Meiji Institute for Advanced Study of Mathematical Sciences, Meiji University 2) JST, CREST 3) School of Engineering, The University of Tokyo 4) Japan Society for the Promotion of Science 5) Research Center for Advanced Science and Technology, The University of Tokyo 6) Economics and Social Research Institute, Cabinet oﬃce, Government of Japan

∗ e-mail: [email protected]

Abstract

Among various kinds of jamming phenomena, a traffic jam of vehicles is a very familiar phenomenon and causes several losses in our daily life such as decreasing efficiency of transportation, waste of energy, serious environmental degradation, etc. In particular, the dynamics of traffic flow on highway has attracted many researchers and has been investigated as a non-equilibrium system of interacting particles for the last few decades [1, 2] . The dynamics of traffic flow is often treated as an effectively one-dimensional compressible fluid by focusing on the collective behavior of vehicles, so-called macroscopic models [3, 4] , which consist of two equations: conservation law and motion equation of traffic vehicles. In previous fluid models, one does not have any choice to introduce the diffusion term into the models such as Kerner and Konhäuser model [4] , in order to represent the stabilized density wave, which indicates the formation of traffic jam. However, it emerges as a serious problem that some vehicles move backward even under heavy traffic, as mentioned in [5] . Thus, unfortunately we would have to conclude that traffic models which include the diffusion term are not reasonable for the realistic expression of traffic flow. Given these factors, we suppose that traffic jam forms as a result of the plateaued growth of small perturbation by the nonlinear saturation effect. In this contribution, we have proposed a new compressible fluid model for the one-dimensional traffic flow taking into account a variation of the reaction time of drivers, which is based on the actual measurements. Our new model

78 is a generalization of the Payne model [3] by introducing a density-dependent function of reaction time as a reasonable assumption, and given by ∂ρ ∂ + (ρv) = 0, (1) ∂t ∂x ∂v ∂v 1 1 dV (ρ) ∂ρ + v = V (ρ) − v + opt , (2) ∂t ∂x τ(ρ) opt 2ρτ(ρ) dρ ∂x where ρ(x, t) and v(x, t) correspond to the spatial vehicle density and the average velocity at position x and time t, respectively. τ(ρ) and Vopt(ρ) are the reaction time of drivers and the optimal velocity function under the current density ρ, respectively. The linear stability analysis of this new model shows the instability of a free flow state caused by the enhancement of fluctuations around a certain critical density of vehicles, which is observed in real traffic flow [6] . Moreover, the condition of the nonlinear saturation of density against small perturbation is theoretically derived from the reductive perturbation method.

References

[1] D. Chowdhury, L. Santen and A. Schadschneider: Statistical physics of vehicular traﬃc and some related systems, Phys. Rep. 329 (2000), 199–329.

[2] D. Helbing: Traﬃc and related self-driven many-particle systems, Rev. Mod. Phys. 73 (2001), 1067–1141.

[3] H. J. Payne: Mathematical Models of Public Systems 1, ed G. A. Bekey (La Jolla,CA: Simulation Council) (1971), 51–61.

[4] B. S. Kerner and P. Konhäuser:Cluster effect in initially homogeneous traffic flow, Phys. Rev. E 48 (1993), R2355–R2338. [5] C. F. Daganzo: Requiem for second-order fluid approximation of traffic flow, Trans. Res. B 29B (1995), 277-286.

[6] Yuki Sugiyama, et al.: Traﬃc jams without bottlenecks – experimental evidence for the physical mechanism of the formation of a jam, New J. Phys. 10 (2008), 033001.

Lagrangian Skeleton on the Gulf of Mexico Oil Spill C. Mendoza1),∗, A.M. Mancho2)

1) Universidad Polit´ecnica de Madrid. ETSI Navales. Av. Arco de la Victoria s/n. 28040 Madrid, Spain 2) Instituto de Ciencias Matem´aticas. CSIC-UAM-UC3M-UCM. Serrano 121. 28006 Madrid, Spain

∗ e-mail: [email protected]

Abstract The understanding of the circulation of ocean currents, the exchange of CO2 between atmosphere and oceans, and the influence of the oceans on the distribution of heat on a global scale is key to our ability to predict and assess the future evolution of climate [1, 2]. Global climate change is affecting sea breathing through mechanisms not yet understood [3]. The ocean is important in the regulation of heat and moisture fluxes, and oceanic physical and bio-geochemical processes are major regulators of natural greenhouse gases. Understanding how oceans mix their waters is key to provide sound fore- casts on the climate [1]. Global change also affects marine biodiversity and threatens the survival of ecosystems and exploitable resources. To predict not only the effects of global change on the oceans, but also the response time of climate feedback requires to improve detection systems and to open new lines of research. We use a novel Lagrangian descriptor (function M, introduced in [4, 5]). It is based on the measure of the arclength of particle trajectories on the ocean surface at a given time. In [6, 7, 8] this technique has been proven to be successful for characterizing the Kuroshio current. We employ this tool on velocity data sets on the Gulf of Mexico obtained from HYCOM project. The tool identifies the underlying Lagrangian structures in the oceanic currents. In particular invariant manifolds, hyperbolic and non-hyperbolic flow regions are detected. We study the influence of these structures on particle motions on the oil spill. We acknowledge to CESGA for support with the supercomputer FINIS TERRAE. Thanks for support by grants: CSIC-OCEANTECH, I-Math C3- 0104, MICINN-MTM2008-03754 and MTM2008-03840-E, and C. Madrid- SIMUMAT.

References

[1] S. Bowen, M.S. Lozier, S. F. Gary, and C.W. Bning. Nature 459, 243- 247, 2009.

[2] K. Katija, and J.O. Dabiri. Nature 460, 624-626, 2009.

[3] Q. Schiermeier. Nature 447, 522-524, 2007.

[4] J. A. Jim´enezMadrid, A.M. Mancho. Distinguished trajectories in time dependent vector ﬁelds. Chaos 19 (2009), 013111-1-013111-18.

[5] C. Mendoza, A.M. Mancho. The hidden geometry of ocean ﬂows. Phys- ical Review Letters 105 (2010), 3, 038501-1-038501-4.

[6] A.M. Mancho, S. Wiggins, A. Turiel, E. Hern´andez-Garca,C. L´opez, E. Garc´ıa-Ladona.Nonlinear Processes in Oceanic and Atmospheric Flows. Nonlinear Proc. Geoph 17 (2010), 3, 283-285.

[7] C. Mendoza, A.M. Mancho, Marie-H´eleneRio. The turnstile mechanism across the Kuroshio current: analysis of dynamics in altimeter velocity ﬁelds. Nonlinear Proc. Geoph 17 (2010), 2, 103-111.

[8] C. Mendoza, A.M. Mancho. The Lagrangian description of aperiodic ﬂows: a case study of the Kuroshio Current. preprint (2011).

Mathematical studies on the self-motion of surfactant scrapings at the air-water interface

1), 2,5) Keita Iida ∗, Hiroyuki Kitahata , Satoshi Nakata3) and Masaharu Nagayama4,6)

1) Graduate School of Natural Science and Technology, Kanazawa University, 2) Graduate School of Science, Chiba University, 3) Graduate School of Science, Hiroshima University, 4) Institute of Science and Engineering, Kanazawa University, 5) PRESTO, Japan Science and Technology Agency, 6) CREST, Japan Science and Technology Agency

e-mail: [email protected] ∗ Abstract Several types of autonomous motors are known to exhibit various manners of “self-motion” at air/water interfaces. Spontaneous motion of surfactant scrapings at the water surface (henceforth surfactant-water system) is a representative example of self-motion, which involves a variety of rich topics and which was introduced in 1997 [1,2]. As presented in [2], camphor scrapings (diameter, ca. 1 mm) exhibit differing manners of sustained motion on water under almost isothermal and non-equilibrium conditions, of which the working mechanisms are similar to living organisms. Hence, from the physicochemical point of view, they have been considered as a novel efficient chemo-mechanical transducer which mimics living things. From the analytical point of view, we would like to discover the mathematics and physics that underlie the surfactant-water system and to understand the essential mechanisms of the self-motion.*1) One proposition on a camphor-water system which has been verified to be true, by means of experiments and analyses, is that a pitch-fork bifurcation of travelling pulse solution occurs depending on the viscosity of the aqueous phase in a one- dimensional problem [3]. For the one-dimensional problem, we can now see such mathematical structure at the point of demarcation between self-motion and the rest state. How about the two-dimensional problem? The next step we discuss below is to consider both translational and rotational motion induced by the * 1)Our real goal is to understand the principles and structures that living and nonliving systems have in common and to realize the autonomic motor system in living things at their most fundamental level.

82 anisotropic shape of the camphor scraping. Although a mathematical model consisting of translational and rotational equations, coupled with a reaction- diffusion equation, has been proposed in 2005 [4], the characteristics of the two-dimensional camphor motion depending on its shape has not yet been clarified. For example, let us consider a camphor scraping with elliptical shape. Which direction does the camphor move, in the long axis or in the short axis? Or, does the elliptical camphor rotate? In fact, we found it difficult to investigate the above question by brief experiments and simulations due to difficulties originating from the elliptical shape (e.g., we have to compute the elliptic integral in the model). To tackle the above questions, we apply bifurcation theory and conduct proper experiments to determine the mode of motion of elliptical camphor scrapings. In the presentation, we will introduce our mathematical findings, numerical simulations and experiments of the camphor-water system.

References

[1] A. Mikhailov and D. Meink¨ohn,Self-Motion in Physico-Chemical Sys-

tems Far from Thermal Equilibrium, Springer, Berlin Heidelberg, 484 (1997).

[2] S. Nakata and Y. Iguchi, S. Ose, M. Kuboyama, T. Ishii and K. Yoshikawa, Self-Rotation of a Camphor Scraping on Water: New In- sight into the Old Problem, Langmuir, 13 (1997), 4454–4458.

[3] M. Nagayama, S. Nakata, Y. Doi and Y. Hayashima, A theoretical and experimental study on the unidirectional motion of a camphor disk, Physica D (Amsterdam), 194 (2004), 151–165.

[4] H. Kitahata and K. Yoshikawa, Chemo-mechanical energy transduc- tion through interfacial instability, Physica D (Amsterdam), 205 (2005), 283–291.

A curious bifurcation diagram arising in a scalar reaction-diffusion equation with an integral constraint

Hisashi OKAMOTO

Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502 * e-mail: [email protected]

Abstract

2 π We consider uxx + Ru = constant + f (x), together with ∫ u(x) dx = 0. A -π strange bifurcation diagram was found. The external force f is chosen so that u = cos kx becomes a solution (k = 2, 3, 4). Solving the equation by the spectral method, we obtained non-trivial solutions. In the figure, where k = 2, the solution branch comes back to the trivial solution at the points indicated by the arrows. As R increases, the profile of the solution becomes complicated, but they also lose their complexity when they come back to the trivial solution. In my talk, I will explain these phenomena, emphasizing the contrast with the bifurcating solutions appearing in the Kolmogorov flows of 2D Navier-Stokes equations [1].

References

[1] Sun-Chul Kim & Hisashi Okamoto, Vortices of large scale appearing in the 2D stationary Navier-Stokes equations at large Reynolds numbers, Japan J. Indust. Appl. Math., vol. 27 (2010), 47--71. 84

Application of topological computation method for global dynamics of an associative memory model of the Hopﬁeld type

1), Hiroshi Kokubu ∗,

1) Department of Mathematics, Kyoto University / JST CREST

e-mail: [email protected] ∗ Abstract A new rigorous topological computation method for global analysis of dynamical systems was introduced in [1]. I would like to revisit the classical associative memory model of the Hopﬁeld type, and study its global phase space structure from the point of view of this topological computation approach.

References [1] Z. Arai, W. Kalies, H. Kokubu, K. Mischaikow, H. Oka, and P. Pilar- czyk: A database schema for the analysis of global dynamics of mul- tiparameter systems, SIAM Journal on Applied Dynamical Systems, 8 (2009), 757–789.

Rigorous veriﬁcation of equilibria for evolutionary equations - Existence, Uniqueness and Hyperbolicity -

1), Kaname Matsue ∗,

1) Department of Mathematics, Kyoto University

e-mail: [email protected] ∗ Abstract The aim of this talk is to provide a method for investigating a precise structure of equilibria of (infinite dimensional) dynamical systems, such as the local uniqueness and the hyperbolicity. Recently several rigorous computation methods for verifying stationary solutions of partial differential equations are studied (e.g. [8] or [10]). Ex- istence and local uniqueness are mainly investigated therein. Moreover, in [2], one of applications of [10], the global structure of all bounded global solutions for a partial differential equations is studied by using computer assisted proof and an algebraic-topological invariant of invariant sets, called the Conely index. This invariant gives us the information of an invariant set, like the existence and its stability. Hyperbolicity, which is related to the structural stability is known as one of the most important properties of dynamical systems, and is stronger than the robustness provided by the Conley index. An equilibrium u of a PDE is hyperbolic if the linearized operator at u has no spectrum on the imaginary axis. This property is important from the viewpoint of dynamical systems, such as the verification of local or global bifurcations (e.g. [3]).

In order to verify the hyperbolicity, one may need analytic methods for investigating the spectrum of a linear operator in general. It is sometimes the case, however, that a linearized operator is self-adjoint, in which case the invertibility of the linearized operator may be sufficient to yield the hyperbolicity of an equilibrium. Even if the linearized operator is not self-adjoint, we can prove the hyperbolicity of equilibria for a broad class of infinite dimensional dynamical systems, called dynamical systems generated by parabolic evolutionary equations ([5]). The author shows that the specific Conely index and the suitable hypoth- esis of the linearized operator yield the local uniqueness and the hyperbolicity of an equilibrium. Using the preceding work by Nakao and his collaborators ([7]) and by Zgliczyńskiand Mischaikow ([10]) or the author ([6]), we can

86 verify the hyperbolicity of equilibria of parabolic evolutionary equations by computer assisted analysis. At the end of this talk, the author will show several sample rigorous numerical results about the existence, local uniqueness and hyperbolicity of equilibria.

References

[1] C.Conley, Isolated Invariant Sets and the Morse Index, CBMS Reg. Con. Ser. Math., Vol.38, Amer. Math. Soc. Providence, RI, 1978.

[2] S. Day, Y. Hiraoka, K. Mischaikow and T. Ogawa, Rigorous Numerics for Dynamics: A Study of the Swift-Hohenberg Equation, SIAM J. Applied Dynamical Systems, 4(2005), 1–31.

[3] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Sys- tems, and Bifurcations of Vector Fields, Springer, 1983.

[4] J.K. Hale, Asymptotic behavior of dissipative systems, Mathematical Surveys and Monographs, 25. AMS, Providence, 1988.

[5] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, 1981.

[6] K. Matsue, Rigorous veriﬁcation of equilibria for evolutionary equations - Existence, Uniqueness and Hyperbolicity - Part 1. Gradient case, preprint.

[7] M.T. Nakao, K. Hashimoto and Y. Watanabe, A Numerical Method to Verify the Invertibility of Linear Elliptic Operators with Applications to Nonlinear Problems, Computing, 75(2005), 1–14.

[8] M.T. Nakao and N. Yamamoto, Numerical veriﬁcations for solutions to elliptic equations using residual iterations with higher order ﬁnite element, J. Comput. Appl. Math., 60(1995), 271–279.

[9] K.P. Rybakowski, The Homotopy Index and Partial Diﬀerential Equa- tions, Springer-Verlag, 1987.

[10] P. Zgliczy´nskiand K. Mischaikow, Rigorous numerics for partial diﬀer- ential equations: The Kuramoto-Sivashinsky Equation, Found. Comput. Math. 1(2001), 255–288.

Role of unstable symmetric solutions in symmetry restoring process

1),2) 1) Tsuyoshi Mizuguchi ∗, Makoto Yomosa

1) Department of Mathematical Sciences, Osaka Prefecture University. 2) PRESTO, Japan Science and Technology Agency (JST).

e-mail: [email protected] ∗ Abstract In the study of dynamical systems, symmetries are one of the most important concepts for characterising and/or classifying systems and solutions. For example, some bifurcations involve spontaneous breakdown or restoration of symmetry. Several chaotic systems having a discrete symmetry often exhibit a bifurcation that breaks or restores the symmetry while maintaining its aperiodic behaviour, i.e., an attractor merging crisis (AMC). On the asymmetric side of the AMC bifurcation point, a pair of strange attractors exists, one of which is chosen by an initial condition. At the bifurcation point, they merge with each other. On the symmetric side, a single strange attractor that satisfies the symmetry is observed. Let us focus on singularities near the AMC bifurcation point. Intermittent behaviours are reported in several systems. Namely, the inter-burst interval time tends to diverge when the system approaches the bifurcation point. If a state is translated to a motion in space, the singularity corresponds to divergence of the diffusion coefficient of the motion. Therefore, one can forecast the bifurcation by observing these quantities. The singularity described above is observed when the system approaches from the symmetric side. On the other hand, from the asymmetric side, a specific unstable symmetric solution so-called mediating unstable periodic orbit, or simply a mediating solution M plays an important role to characterize the symmetry restoration process. The mediating solution is defined as that with which a pair of strange attractors simultaneously collides at the AMC point when the system approaches from the asymmetric side. Near this AMC point, the outermost part of the strange attractor in each parameter passes close to solution M. Therefore, the distance dM between the strange attractor and M is an appropriate quantity to characterise the closeness to the bifurcation point. This approach is also characterised by a duration in which a state is close to solution M. We define a state whose distance to M is less than a threshold value as a temporarily sticking state, and we focus on the maximum

88 duration τM of this temporarily sticking state during the measurement time. A logarithmic singular behaviour of τM with periodic modulation is predicted near the bifurcation point. In this paper, we exemplify several dynamical systems exhibiting AMC transition, such as Ginzburg Landau type equation with external force term or coupled Stuart Landau type equation. The distance d and the maxi- M mum duration τM are directly measured near the bifurcation point. A linear decrease in dM and the logarithmic divergence of τM are conﬁrmed numerically. We suggest a time series analysis method to measure these quantities without knowing the precise bifurcation point or concrete function form of the mediating solution M.

References

[1] M. U. Kobayashi, T. Mizuguchi, “Chaotically oscillating interfaces in a parametrically forced system”, Phys. Rev. E, 73 (2006) 016212.

[2] M. U. Kobayashi, T. Mizuguchi, “Chaotic interfaces in a parametrically forces system”, Prog. Theor. Phys. Supplement, 161 (2006) 228–231.

[3] Y. Morita, N. Fujiwara, M. U. Kobayashi and T. Mizuguchi, “Scytale decodes chaos: A method for estimating unstable symmetric solutions”, Chaos, 20 (2010) 013126.

A Diffusion Equation Representing the Dynamics of the Jakarta Composite Index (JSX)

1),2)* 1), 3) Edi Cahyono , Buyung Sarita 1) Graduate Program, Universitas Haluoleo 2) Department of Mathematics FMIPA, Universitas Haluoleo 3) Faculty of Economics, Universitas Haluoleo * e-mail: [email protected]

Abstract We cosider the dynamics of the Jakarta Composite Index (JSX). The dynamics is presented as time series. The dynamics may be charaterized by the trend and the probability density function (pdf) based on the assumption that at a fixed time it is normally distributed with respect to ‘spatial’ variable. Details of the trend and the pdf for the case of the dynamics of JSX will be presented in the succesive article [2]. Following the work of [1], we seek a type of diffusion equation which has fundamental solution representing the trend and the probability density function of such dynamics.

References

[1] E. Cahyono, J.R. Juliana, R. Raya: Asset value dynamics and the fundamental solution of a modified heat equation, 4th Intl Conf. Math. Stat., Bandar Lampung, Indonesia (2009), 312--318. [2] B. Sarita, F. Arisanti, E. Cahyono: The trend and probability density function of Indonesia Stock Index (IHSG), submitted.

The Trend and Probability Density Function of the Jakarta Composite Index (JSX)

1),2)* 3) 1), 3) Buyung Sarita , Femy Puspita Arisanty , Edi Cahyono 1) Graduate Program, Universitas Haluoleo 2) Faculty of Economics, Universitas Haluoleo 3) Department of Mathematics FMIPA, Universitas Haluoleo * e-mail: [email protected]

Abstract In this paper we discuss the dynamics of the Jakart Composite Index (JSX), which is related to the work of [3]. The dynamics is presented as time series. To predict the dynamics is still difficult. In general it is almost impossible to predict such dynamics for the case of high frequency data [2]. Hence, we do not predict the dynamics. Rather, we seek the trend and the probability density function based on the assumption that the dynamics is normally distributed with respect to time variable. According to [1], the trend and the probability density function of such dynamics is the fundamental solution of a type of diffusion equation.

References

[1] E. Cahyono, J.R. Juliana, R. Raya: Asset value dynamics and the fundamental solution of a modified heat equation, Proc. 4th Intl Conf. Math. & Stat., Bandar Lampung, Indonesia (2009), 312--318. [2] T. Ito, K. Sato: Exchange rate change and inflation in post-crisis Asian economies: Vector autoregression analysis of exchange rate pass-through, J. Money, Credit and Banking, (2008) 40(7), 1407--1437. [3] B. Sarita, Agusrawati, L.D. Suryadi, S. Wahab, E. Cahyono: Dynamics of gold price relative to Indonesian rupiah, Proc. 3rd Intl. Conf. Quantitative Methods, Bandar Lampung, Indonesia (2010) 42--46.

ÓÖÑÒÝ Ó ÔÖ ØÓÖ× Ò ÔÖ Ý¹ÔÖ ØÓÖ

×Ý×Ø Ñ×

½µ£

Å×ØÃÙÛÑÙÖ

½µ ÙÐØÝ Ó ÀÙÑÒ ÚÐÓÔÑÒØ Ò ÒÚÖÓÒÑÒØ¸ ÃÓ ÍÒÚÖ×ØÝ

¹ÑÐ ÙÛÑÙÖ ÑÒº"ºÓ ¹Ùº º#Ô

×ØÖØ

ÁØ × ÛÐÐ ÒÓÛÒ Ø"Ø ÞÓ ÓÔÐÒØÓÒ ÑÒÐÝ ÔÖÓ Ù ×ÙØÒÓÙ× &&× Ò

ÓÑÓÖØÐ ÒÚÖÓÒÑÒØ×' "ÓÛÚÖ¸ Ø"Ý ÔÖÓ Ù ÖØÐÞ &&× ´Ö×ØÒ&

&&×¸ ÓÖÑÒØ ×ØØµ ØÓ × Ô Ô ÖÓ × Ó "Ö×" ÐÚÒ& ÓÒØÓÒ×º Ì"× Ñ¹

ÔÐ× Ø"Ø Ø" Ö×ØÒ& && ÔÖÓ Ù ØÓÒ Ó ÞÓ ÓÔÐÒØÓÒ "× ÖÙ Ð « Ø ÓÒ Ø"

Ô ÓÔÙÐØÓÒ ÝÒÑ × Ó Ô"ÝØÓÔÐÒØÓÒ¹ÞÓ ÓÔÐÒØÓÒ ÔÖÝ¹ÔÖØÓÖ ×Ý×ØÑ×º

ÁÒ Ø"× ØÐ¸ Û ÔÖÓÔ Ó× ×ÑÔÐ ÔÖÝ¹ÔÖØÓÖ ÑÓ Ð Ø"Ø ÓÒ×Ö× Ø" ×

Ò Û" " ÔÖØÓÖ× ÒØÖ ÓÖÑÒ Ý Ò Ø" ÚÒØ Ó ×"ÓÖØ& Ó ÔÖÝº ÍÒÖ

Ø" ××ÙÑÔØÓÒ Ø"Ø Ø" ÒØÖÒ& Ó ÓÖÑÒ Ý Ý ÔÖØÓÖ× × Ô ÒÒØ ÓÒ

Ø" ÙÖÖÒØ ÔÖÝ Ò×ØÝ¸ Ø × ×"ÓÛÒ Ø"Ø Ø" ÓÖÑÒ Ý Ó ÔÖØÓÖ× Ò

×ØÐÞ Ø" Ô ÓÔÙÐØÓÒ ÝÒÑ × Ó ÔÖÝ¹ÔÖØÓÖ ×Ý×ØÑ× +½,º ÅÓÖÓÚÖ¸

Û×"ÓÛ Ø"Ø ÓÖÑÒ Ý Ó ÔÖØÓÖ× Ò ÒÙ ÑÜ¹ÑÓ Ó× ÐÐØÓÒ× Ò

"Ó× Ò Ô ÓÔÙÐØÓÒ ÝÒÑ × Ó Ø" ÔÖÝ¹ÔÖØÓÖ ÑÓ Ð +¾,º ÆÜØ¸ Û ÓÒ¹

×Ö Ø" ×ØÐÞÒ& « Ø Ó Ø" ÓÖÑÒ Ý Ó ÔÖØÓÖ× Ò ÔÖÝ¹ÔÖØÓÖ

ÑÓ Ð Ø"Ø Ö Ø× ××ÓÒÐ "Ò&× Ò ÒÚÖÓÒÑÒØ× +¿,º ÒÐÐÝ¸Û ÓÒ×Ö

Ø" ×ØÐÞÒ& « Ø Ó ÓÖÑÒ Ý Ó ÔÖØÓÖ× Ò ÔÖÝ¹ÔÖØÓÖ ÑÓ Ð ÙÒ¹

Ö Ø" ××ÙÑÔØÓÒ Ø"Ø Ø" ÒØÖÒ& Ó ÓÖÑÒ Ý Ý ÔÖØÓÖ× × Ô ÒÒØ

ÓÒ Ø" ÖØ Ó ÚÖØÓÒ Ò Ø" ÔÖÝ Ò×ØÝ +2,º Ì"× Ö×ÙÐØ× ÑÝ "ÐÔ ÙÒÖ¹

×ØÒ Ø" Ô ÓÔÙÐØÓÒ ÝÒÑ × Ó Ô"ÝØÓÔÐÒØÓÒ¹ÞÓ ÓÔÐÒØÓÒ´¹Ö×ØÒ& &&×µ

ÓÑÑÙÒØ× Ò ÒØÙÖº Ê Ö Ò ×

+½, Åº ÃÙÛÑÙÖ¸ Ìº ÆÞÛ Ò Ìº Ç&Û¸ 4 ÑÒÑÙÑ ÑÓ Ð Ó ÔÖÝ¹

ÔÖØÓÖ ×Ý×ØÑ ÛØ" ÓÖÑÒ Ý Ó ÔÖØÓÖ× Ò Ø" ÔÖÓÜ Ó ÒÖ "¹

ÑÒØ¸ ÂÓÙÖÒÐ Ó ÅØ"ÑØ Ð 6ÓÐÓ&Ý¸ÚÓÐº 78¸ ÔÔº279¹2:9 ´¾¼¼9µ

+¾, Åº ÃÙÛÑÙÖ Ò Àº <"¸ ÅÜ¹ÑÓ Ó× ÐÐØÓÒ× Ò "Ó× Ò

ÔÖÝ¹ÔÖØÓÖ ×Ý×ØÑ ÛØ" ÓÖÑÒ Ý Ó ÔÖØÓÖ×¸ <"Ó×¸ ÚÓÐº ½9¸

¼2¿½¾½ ´¾¼¼9µ

+¿, Ìº ÆÞÛ¸ Åº ÃÙÛÑÙÖ Ò Æº =ÑÑÙÖ¸ ÁÑÔÐ ØÓÒ× Ó Ö×ØÒ&

&&× Ó ÞÓ ÓÔÐÒØÓÒ ÓÖ Ø" ÔÖÓÜ Ó ÒÖ "ÑÒØ¸ ÈÓÔÙÐØÓÒ ÓÐÓ&Ý¸

ÚÓÐº 7¿¸ ÔÔº¿2½¹¿7¼ ´¾¼½½µ

+2, Åº ÃÙÛÑÙÖ Ò Ìº ÆÞÛ¸ ÓÖÑÒ Ý Ó ÔÖØÓÖ× Ô ÒÒØ ÓÒ

Ø" ÖØ Ó ÚÖØÓÒ Ò ÔÖÝ Ò×ØÝ¸ ËÁ4Å ÂÓÙÖÒÐ ÓÒ 4ÔÔÐ ÅØ"¹

ÑØ ×¸ ÚÓÐº :½¸ ÔÔº½@9¹½:9 ´¾¼½½µ

Truncation error analysis for particle methods Masato Kimura1),∗, Kiyohiro Ishijima2)

1) Institute of Mathematics for Industry, Kyushu University 2) Graduate School of Mathematics, Kyushu University

∗ e-mail: [email protected]

Abstract Meshfree particle methods such as moving particle semi-implicit (MPS) method and smoothed particle hydrodynamics (SPH) method have been much developed in recent years and established as powerful tools in the fields of science and engineering, since they require no troublesome mesh generation and have flexible adaptation to changing shape of domains. On the other hand, there are few mathematical error analysis on their accuracy as discretization methods for partial differential equations. In [1], we proposed a new quantity concerning the distribution of the particles and gave a truncation error estimate for a discrete gradient operator of meshfree particle methods with weight functions, particularly one of MPS method. Let Ω be a subset of Rm (m ≥ 2) with a piecewise smooth boundary. We suppose that N X = {xi}i=1 ⊂ Ω, xi 6= xj (i 6= j) is given, and we call xi a particle. In order to measure the uniformness of the distribution of X in Ω, we introduce the following quantities. If a division N ω = {ωi}i=1 of Ω satisfies the condition: 1 ω ⊂ Ω, ω : open, ω ∩ ω = ∅ (i 6= j), |ω | = |Ω| i i i j i N we call ω N-equivolume partition of Ω. For given N particles X and an N-equivolume partition ω in Ω, we define the equivolume partition radius δ∞(X) by

δ∞(X) := inf δ∞(X, ω), (1) ω where δ∞(X, ω) := max sup |x − xj|, 1≤j≤N x∈ωj and the inﬁmum in (1) is took over all N-equivolume partition of Ω. The equivolume partition radius δ∞(X) is a quantity which stands for a kind of

94 distance between particles. If X distributes in Ω almost uniformly, there exist C1 > 0 and C2 > 0 independent of N such that − 1 − 1 C1N m ≤ δ∞(X) ≤ C2N m holds. In MPS method, we use a weight function

1 w (r) = C max − 1, 0 (r > 0), MPS r where C > 0 is chosen as R w(|x|) dx = 1. Moreover, for a > 0, we Rm a −m r a define w (r) := a w( a ). The support of w is the ball Ba(xi) := {x ∈ m R ; |x − xi| ≤ a}. We call a the influence radius. Using the scaled weight a function w , we define the particle density di(X, a) and the particle density deviation qi(X, a) as follows: |Ω| X d (X, a) := wa(|x − x |), q (X, a) := d (X, a) − 1. (2) i N j i i i j6=i

For a function φ deﬁned on X with φi = φ(xi)(i = 1, 2, ··· ,N), the following discrete gradient operator is used in MPS method [2]:

m|Ω| X (φj − φi) h∇φi := (x − x ) wa(|x − x |) i N 2 j i j i j6=i |xj − xi|

Even for a function u(x) deﬁned on Ω, we deﬁne h∇ui similarly with ui = u(xi). Our main result for m = 3 is as follows. N Theorem 1. Let m = 3 and let X = {xj}j=1 be a set of particles in Ω. If 3 xi ∈ X and a > 0 satisfy the condition Ba(xi) ⊂ Ω, for u ∈ C (Ω), the following inequality holds: δ (X) |h∇ui − ∇u(x )| ≤ C ∞ + a2 + C q (X, a). i i 1 a 2 i

References

[1] K. Ishijima, M. Kimura: Truncation error analysis of finite difference formulae in meshfree particle methods, Trans. Japan Soc. Indust. Appl. Math, 20 (2010), 165-182. (in Japanese) [2] S. Koshizuka, Y. Oka: Moving particle semi-implicit method for frag- mentation of incompressible fluid, Nuclear Science and Engineering, 123 (1996), 421434.

A numerical method for multiphase volume-preserving mean curvature ﬂow

1), 2) 3) Elliott Ginder ∗, Seiro Omata , Karel Svadlenkaˇ 1) Grad. Sch. of Nat. Sci. and Tech., Kanazawa University 2) Institute of Science and Engineering, Kanazawa University 3) Institute of Science and Engineering, Kanazawa University

e-mail: [email protected] ∗ Abstract We introduce a method for computing multiphase motion by volume- preserving mean curvature flow. The method is variational and can be implemented in any dimension, with interfaces in the two dimensional setting described by closed curves, each of which encloses a prescribed area. The corresponding physical phenomena for curvature driven motions are numerous and include the grain boundary growth of crystals and ternary alloys, and applications also arise in image processing and biology. Frequently occurring in practical applications, curvature driven interfacial motions are a recurring subject of study. As such, analytical and numerical methods have been proposed for their treatment. Standard approaches include front tracking [8], level set [5], and phase field [1, 6] methods, but these techniques are limited to two phases whenever they are extended to handle the case of volume preservation (see, [2, 3]). Our approach to realizing multiphase volume-preserving mean curvature flow is based on an idea of Merriman, Bence and Osher (MBO)[4]. The MBO algorithm approximates mean curvature flow by diffusing, for a small time, the characteristic function of the set enclosed by the interface. In particular, the characteristic function of the set is taken as the initial condition for the heat equation, which is then solved for a short time under Neumann boundary conditions. The location of the evolved interface is then given by the 1/2 level set of the solution. The initial condition for the heat equation is renewed by truncation (that is, by taking the characteristic function of the 1/2 level set), and the process is repeated. The main benefit of this approach is that the diffusion process is able to automatically treat the occurrence of singularities, and there is no need directly compute curvatures. This method is given for the multiphase case in [4], and for a 2-phase volume-preserving method, see [7]. However, the idea in [7] does not apply to the multiphase case, and so we reformulate the idea of [4], and extend it to the case of multiphase motion by volume-preserving mean curvature flow.

The prescribed phase volumes are maintained by use of a variational technique known as the discrete Morse ﬂow. Time is discretized in this method and a functional is introduced for each time step. The functionals correspond to the discrete Morse ﬂow for a vector valued heat equation, with additional penalty terms for the prescribed phase volumes. Minimizing the functionals decreases their penalties, so that each phase maintains its prescribed volume.

Similar to the MBO algorithm, we then employ a truncation to obtain the evolution of the interfaces, and we repeat. We will explain the details of our algorithm and show results of its application in the two dimensional setting.

References

[1] L. Bronsard, F. Reitich: On three-phase boundary motion and the singular limit of a vector-valued Ginzburg-Landau equation, Arch. Rational Mech. Anal., 124 (1993), pp. 355-379.

[2] L. Bronsard, B. Soth: Volume preserving mean curvature ﬂow as a limit of a nonlocal Ginzburg-Landau equation, SIAM J. Math. Anal., 28 No. 4 (1997), pp. 769-807.

[3] S. Esedoglu, P. Smereka: A variational formulation for a level set representation of multiphase ﬂow and area preserving curvature ﬂow, Com- munications in Mathematical Sciences, 6 No. 1 (2008), pp. 125-148.

[4] B. Merriman, J. Bence, S. Osher: Motion of multiple junctions: a level set approach, J. Comp. Phys., 112 No. 2 (1994), pp. 334-363.

[5] S. Osher, J. A. Sethian: Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comp. Phys., 79 (1988), pp. 12-49.

[6] J. Rubinstein, P. Sternberg: Nonlocal reaction-diﬀusion equations and nucleation, IMA J. Appl. Math., 48 (1992), pp. 248-264.

[7] S. J. Ruuth, B T. R. Wetton: A simple scheme for volume-preserving motion by mean curvature, J. Scientiﬁc Computing, 19 (2003), pp. 373- 384.

[8] J. E. Taylor: The motion of multiple-phase junctions under prescribed phase-boundary velocities, J. Diﬀerential Equations, 119 (1995), pp. 109-136.

A High-Order Discontinuous Galerkin Method for Elliptic Interface Problems

1), 2) Min-Hung Chen ∗, Rong-Jhao Wu 1) Department of Mathematics, National Cheng Kung University, Taiwan 2) Department of Aeronautics and Astronautics, National Cheng Kung

University, Taiwan e-mail: [email protected] ∗ Abstract In this work, we implement the high-order local discontinuous Galerkin method [1] with the numerical flux proposed by Guyomarc’h et. al. [3], to solve the elliptic interface problem. Instead of the triangular elements and the P k-polynomial space used in [3], we use the quadrilateral elements and the Qk-polynomial space. To treat the curved interface and boundaries, we use the transfinite blending mappings [2] to construct curved elements. The discretization generates a symmetric system and the system can be solved directly with standard methods or reduced into smaller systems with matrix reduction techniques. The numerical experiments show the h and p convergence properties of the our high order scheme. Moreover, our numerical experiments show that the high-order method is more efficient than the low-order one. To achieve the same magnitude of accuracy, the degree of freedom of the generated system using the coarsest mesh is much smaller than the one using the finest mesh. As a result the CPU time required to solve the system using the coarsest mesh is much smaller, too.

References

[1] Cockburn, B.; Shu, C.-W.: The local discontinuous Galerkin method

for time-dependent convection-diﬀusion systems. SIAM J. Numer. Anal. vol. 35, no. 6, pp. 2440-2463 (1998). [2] Gordon, W. J.; Hall, C. A.: Transﬁnite element methods: blending- function interpolation over arbitrary curved element domains. Numer. Math. vol. 21, no. 2, pp. 109-129 (1972). [3] Guyomarc’h, G.; Lee, C.O.; Jeon, K.: A discontinuous Galerkin method for elliptic interface problems with application to electroporation, Com- mun. Numer. Methods Engrg. (2008).

A Coarse-Grain Parallel Scheme for Solving Poisson Equation by Chebyshev Pseudospectral Method

1); 2) 3) Teng-Yao Kuo , Hsin-Chu Chen , Tzyy-Leng Horng

1) Ph. D. Program in Mechanical and Aeronautical Engineering, Feng Chia University, Taichung, Taiwan 2) Department of Computer and Information Science, Clark Atlanta University, Atlanta, GA 30314, USA 3) Department of Applied Mathematics, Feng Chia University, Taichung, Taiwan e-mail: [email protected] Abstract Poisson equation is frequently encountered in mathematical modeling for scienti…c and engineering applications. Fast Poisson numerical solvers for 2D and 3D problems are, thus, highly requested for their simulations. In this paper, we consider solving the Poisson equation 2u = f(x; y) in the Cartesian r domain = [ 1; 1] [ 1; 1], subject to mixed Dirichlet / Neumann boundary conditions, discretized with the Chebyshev pseudospectral method. The main purpose of this paper is to propose a re‡exive decomposition scheme for orthogonally decoupling the linear system obtained from the discretization into independent subsystems via the exploration of a special re‡exivity property inherent in the second-order Chebyshev di¤erentiation matrix. The decomposition not only yields more e¢ cient algorithm but introduces coarse- grain parallelism. This approach can be applied to more general problems and to other partial di¤erential equations discretized with the Chebyshev pseudospectral method as well, so long as the discretized problems possess re‡exive symmetry. Numerical examples and the performance of numerical experiments are presented to demonstrate the validity and advantage of the proposed approach. Keywords Poisson equation, Chebyshev pseudospectral method, Cheby- shev di¤erentiation matrix, Coarse-grain parallelism, Re‡exivity property

References

[1] A. L. Andrew: Solution of equations involving centrosymmetric matrices, Technometrics, 15(2) (1973), 405–407.

[2] A. L. Andrew: Eigenvectors of certain matrices, Linear Algebra Appl., 7 (1973), 151–162.

[3] A. Cantoni, P. Butler: Eigenvalues and eigenvectors of symmetric centrosymmetric matrices, Linear Algebra Appl., 13 (1976), 275–288.

[4] C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang: Spectral Meth- ods in Fluid Dynamics, Springer-Verlag, Berlin, 1987.

[5] H-C. Chen: Increasing parallelism in the …nite strip formulation: static analysis, Neural, Parallel & Scienti…c Computations, 2 (1994), 273–298.

[6] H-C. Chen, A. Sameh: A matrix decomposition method for orthotropic elasticity problems, SIAM J. Matrix Anal. Appl., 10(1) (1989), 39–64.

[7] H-C. Chen, T-L Horng, Y-H. Yang: Re‡exive decompositions for solving Poisson equation by Chebyshev pseudospectral method, Proceedings of Neural, Parallel, and Scienti…c computations, 4 (2010), 98–103.

[8] R. Peyret: Spectral methods for incompressible viscous ‡ow, Applied Mathematical Sciences 148, Springer-Verlag, 2002.

[9] H. Dang-Vu, C. Delcarte: An accurate solution of the Poisson equation by the Chebyshev collocation method, J. Comput. Phys., 104 (1993), 211–220.

[10] U. Ehrenstein, R. Peyret: A Chebyshev collocation method for Navier- Stokes equations with applications to double-di¤usive convection, Int’l J. for Numerical Methods in Fluids, 9 (1989), 427–452.

[11] G.H. Golub, C.F. Van Loan: Matrix Computations, The Johns Hopkins University Press, Maryland, USA, 1983.

[12] D. B. Haidvogel: The solution of Poisson equation by expansion in Chebyshev polynomials, J. Comput. Phys., 30 (1979), 167–180.

[13] J.d.J. Martinez, P.d.T.T. Esperanca: A Chebyshev collocation spectral

method for numerical simulation of incompressible ‡ow problems, J. of the Braz. Soc. of mech. Sci. & Eng., XXIX(3) (2007), 317–328.

[14] Lloyd N. Trefethen: Spectral Methods in MATLAB, SIAM, Philadel- phia, 2000.

[15] J.R. Weaver: Centrosymmetric (cross-symmetric) matrices, their basic properties, eigenvalues, and eigenvectors, The American Mathematical Monthly, 92(10) (1985), 711–717.

100

On the stability of an oscillation-free ADI method in laser beam propagation computations Qin Sheng1), Hai-Wei Sun2),∗

1) Department of Mathematics, Center for Astrophysics, Space Physics and Engineering Research, Baylor University, USA 2) Department of Mathematics, University of Macau, Macao, China

∗ e-mail: [email protected]

Abstract If a slowly varying envelope approximation of the light beam, such as a laser, is concerned, then one of the most eﬀective mathematical models used can be the paraxial Helmholtz equation,

2iκuz = uxx + uyy + g(u), ∀(x, y, z) ∈ D, (1) √ where i ≡ −1 is the imaginary unit, κ is the wave number, z is the beam propagation direction, x, y are dimensional directions perpendicular to the light, u is the complex envelope of the wave to be investigated, g is the source, and D = {a < x < b, c < y < d, z > z0} . Equation (1) is often called the one-way Helmholtz equation and can be viewed as a dimensionless simpli- fication of Maxwell’s field equations. Moreover, (1) has been widely used in laser optical material researches, particularly in crystal or liquid crystal photorefractive and negative indexed (or left-handed) material developments. Since κ can be extremely large in optical applications, the complex envelope is highly oscillatory. Consequently, higher efficiencies in solution computations for (1) are difficult to achieve, since mesh steps cannot be unre- alistically small. Although for the spectrum, Gaussian beam methods and integral transformations may possess certain advantages in the situation, the main challenge pertains in balancing the algorithmic simplicity and accuracy. This motivates a recent study of the eikonal transformation based numerical methods. For simplicity in the following discussion, we remove the source g(u) from (1). Assume φ(x, y, z) and ψ(x, y, z) to be the amplitude and phase functions of the underlying highly oscillatory wave respectively. Based on the theory of geometrical optics, we claim that

u(x, y, z) = φ(x, y, z)ei·κψ(x,y,z), ∀(x, y, z) ∈ D. (2)

101

By substituting (2) into (1), we immediately acquire the following coupled eikonal diﬀerential equations in D, i.e.,

φz = α (ψxx + ψyy) + f1, (3)

ψz = β (φxx + φyy) + f2, (4) where φ 1 1 h i α = , β = − , f = φ ψ + φ ψ , f = (ψ )2 + (ψ )2 . (5) 2 2κ2φ 1 x x y y 2 2 x y

Note that φ 6= 0, solutions of (3) and (4) are real and non-oscillatory even when κ is extremely large. Therefore, mesh steps anticipated for solving them need not to be particularly small. Needless to say, this implies a terrific potential for fast and highly efficient computations of the numerical solution of (1). Discussions of continuing discretizations of (3) and (4) can be found in numerous recent publications, in particular, with important electro-optical and material engineering applications. Although various kinds of finite difference schemes based on (3) and (4) have been accomplished and used in practical laser computations, their numerical stability has been a serious concern in many applications. In fact, the algorithms comprised are often unstable, especially when a large beam propagation distance is considered. This proposes an extremely important question on the applicability of the eikonal transformation based schemes in modern laser beam computations. Therefore, it becomes more imminently for us to study the stability of the popular oscillation-free split step method. In this talk, we study the stability of an ADI finite difference method for solving highly oscillatory paraxial Helmholtz equations in laser optics applications. We show that the ADI method is asymptotically stable when wave numbers anticipated are large.

102

Flows in thoracic aorta with torsion

1),4) 2),4) 3), Hiroshi Suito ∗, Takuya Ueda , Daniel Sze

1) Graduate School of Environmental Science, Okayama University, Japan 2) Department of Radiology, Saint Luke’s Hospital, Japan 3) Department of Radiology, Stanford University School of Medicine, USA 4) CREST, JST

e-mail: [email protected] ∗ Abstract Thoracic endovascular aortic repair (TEVAR), or stent-graft treatment, has become widely accepted as an important option for treatment of thoracic aortic diseases. Many studies have proven the safety and efficacy of TEVAR with satisfactory short-term to mid-term outcomes. Nevertheless, even if the initial TEVAR treatment technically succeeds, some patients show recur- rence and progression of disease many years after treatment[5, 6]. Based on long-term follow-up examinations, such long-term morphological change and effects of hemodynamic flow apparently interact synergically. Constant pul- satile hemodynamic effects from blood flow apparently induce degeneration of the underlying aorta to cause its morphological change and to induce mi- nor morphological changes that alter the hemodynamic state. These changes ultimately engender long-term adverse events. For this study, which investigates the effect of vascular hemodynamics on long-term adverse events, we considered patient-specific models of the thoracic aorta as constructed from CT scans. As a procedure for the reconstruction of aorta morphology from medical images, median axis transform technique[4] is applied to extract the (x, y, z)- coordinate of the centerline and radius at position s from CT images, where s is a length along the centerline from the proximal end of aorta. Then a finite difference mesh system (ξ, η, ζ) is generated, one coordinate axis of which is nearly parallel to the aorta centerline. The ζ axis is set not to be strictly parallel but to be nearly parallel to the centerline.− This treatment avoids numerical instabilities arising from severe skewness of finite difference meshes. On each mesh point, the characteristic function is computed, which takes the value 0 in the aorta and value 1 out of the aorta. It varies gradually near the boundary with certain smoothness. As governing equations, incompressible viscous Navier-Stokes equations are used with the continuity equation. The governing equations are discretized on the finite difference mesh and SMAC method is applied for time

103 advances. At the proximal end, a pulsating velocity profile is given, whereas the constant pressure is given at the distal end. It is apparent from numerical results that the stress distributions are strongly dependent to the aorta morphology. From comparison of the flow fields of several examples of thoracic aorta morphologies, it has been inferred that torsions of the centerline are a candidate of the critical parameter of the existence of swirling flow in diastole phases. In order to examine the effect of torsion on the flows in curved tubes, numerical tests in simplified geometries are performed. Through computations of the flows in realistic aorta morphologies and in simplified spiral tubes, the importance of the torsion of aorta has been shown. The values of torsion can vary widely among patients. Our results suggest that the torsion is a more important parameter in considering the relation between aorta morphologies and the hemodynamics in them.

References

[1] C.Y. Wang: On the low-Reynolds-number ﬂow in a helical pipe, J. Fluid

Mech., 108 (1981), 185–194. [2] M. Germano: On the effect of torsion on a helical pipe flow, J. Fluid Mech., 125 (1982), 1–8. [3] H. Fujita, H. Kawahara, and H. Kawarada: Distribution theoretic approach to fictitious domain method for Neumann problems, East-West J. Numer. Math., 3, 2 (1995), 111–126. [4] G.D. Rubin, D.S. Paik, P.C. Johnson, and S. Napel: Measurement of the aorta and its branches with helical CT, Radiology, 206 (1998), 823–829. [5] M. Tillich, R.E. Bell, D.S. Paik, D. Fleischmann, M.C. Sofilos, L.J. Logan, and G.D. Rubin: Iliac arterial injuries after endovascular repair

of abdominal aortic aneurysms: correlation with iliac curvature and diameter, Radiology, 219 (2001), 129–136. [6] T. Ueda, D. Fleischmann, G.D. Rubin, M.D. Dake, and D.Y. Sze: Imag- ing of the thoracic aorta before and after stent-graft repair of aneurysms and dissections, Semin. Thorac. Cardiovasc. Surg., 20, 4 (2008), 348–357. [7] H. Suito, T. Ueda, M. Murakami, and G.D. Rubin: Vortex dynamics in thoracic aortic aneurysms, ECCOMAS CFD 2010, Laboratorio Nacional de Engenharia Civil, Lisbon, Portugal, (2010).

104

A Pore-scale Network Flow Model for Two Phase Flow: The Genesis

and Migration of Gas Phase

Koukung Alex Chang e-mail: [email protected]

Abstract We present a network flow model to compute the transport, through a pore network, of a compositional fluid consisting of water with a dissolved hydrocarbon gas. Interest in this research is to develop simulation capability at the pore scale for specific application in order to investigate reactive flow resulting from CO2 sequestration and capture pore-level flow detail. The model presented here contains two parts: one describes fluid transport equations which follow mass conservation law; the other is reactive equations which represent phase equilibrium conditions. Combining these two parts, this model forms a nonlinear system which contains six unknowns as well as six equations. The challenge of this model is that the mass conservation equations and the phase equilibrium conditions be solved simultaneously while the algorithm maintains enough flexible to allow for the spontaneous creation and extension of the gas phase. The numerical model employs sequential solution for the pore pressures, saturations, and concentrations each time-step. The challenge in computing saturations and concentrations is to satisfy both mass conservation and phase equilibrium. This is done sequentially, updating species concentrations first due to transport followed by a phase-change/equilibrium (flash) calculation for equilibrium concentration and saturations in each pore. Computational tests with constant injected flow rate were run on a “quarter 5-spot” flow scenario to model horizontal flow; the solution obeys the symmetry of the quarter-5-spot pattern.

Under constant temperature conditions, the concentration of CO2 dissolved in the liquid phase throughout the medium is linearly related to the liquid pressure. In addition, the saturation of the gas phase throughout the medium is also linearly related to the liquid pressure

105

Convective transport and stability in ﬁlms of binary mixtures Santiago Madruga1),∗, Fathi Bribesh2) Uwe Thiele3)

1) Universidad Polit´ecnica de Madrid, ETSI Aeron´auticos,Plaza Cardenal Cisneros 3, 28040 Madrid, Spain 2), 3) School of Mathematics, Loughborough University, Loughborough, Leicestershire, LE11 3TU, UK

∗ e-mail: [email protected]

Abstract Thin polymer films are increasingly used in advanced technological applications. The use of these films as coatings is often limited by their lack of stability due to their wettability properties on the substrates. The instabilities may be employed to create complex morphologies for polymeric functional layers and their development is relatively well understood for single component liquids. However, in many relevant applications the film consists of a binary mixture such as a polymer blend. For such systems the dynamics of the decomposition within the film and of the dewetting of the film itself may couple. This allows for new pathways of structuring like decomposition induced dewetting [1] that do not exist in single component films. We present a model for the evolution of films of isothermal binary liquid mixtures with a free evolving surface [2]. The model is based on model-H [3] supplemented by appropriate boundary conditions at the free surface and the solid substrate. The equations account for the coupled transport of the concentration of a component (convective Cahn-Hilliard equation) and the momentum (Korteweg-Navier-Stokes equation). The inclusion of convective motion makes surface deflections possible, i.e., the model allows to study couplings between the decomposition of the mixture and the evolving surface corrugations. After determining homogeneous and vertically stratified base states of free surface films of polymer mixtures we analyse their linear stability with respect to lateral perturbations [4]. For purely diffusive transport, an increase in film thickness either exponentially decreases the lateral instability or entirely stabilizes the film. The inclusion of convective transport leads to a further destabilization as compared to the purely diffusive case. In some cases the inclusion of convective transport and the related widening of the range of available film configurations (films are then able to change its surface profile) change the stability behavior qualitatively.

106

We show two dominant driving mechanisms for the convective motion in binary mixtures with diffuse interface: Marangoni driving for energetic biased surfaces, and Korteweg driving for neutral surfaces. In addition, we discuss the role of composition for off-critical mixtures on surface deflections, and the dependence of the instability on parameters such as the Reynolds number, the surface tension number and the ratio of velocities of convective and diffusive transport. S. Madruga acknowledges support by a Marie Curie European Reintegra- tion Grant (PERG04-GA-2008-234384) within the 7th European Community Framework Programme and by Universidad Politécnicade Madrid through grant AL11-PAC-01. U.Thiele acknowledges support by the EU via the ITN MULTIFLOW (PITN-GA-2008-214919).

References

[1] R. Yerushalmi-Rozen, T. Kerle, and J. Klein. Science. 285, 1254–1256 (1999).

[2] U. Thiele, S. Madruga, and L. Frastia. Phys. of Fluids. 19, 122106, 2007.

[3] D.M. Anderson, G.B. McFadden, and A.A. Wheeler. Ann. Rev. Fluid Mech. 30, 139-165 (1998).

[4] S. Madruga and U. Thiele. Phys. of Fluids. 21, 062104, 2009.

[5] S. Madruga and U. Thiele. Eur. Phys. J. Special Topics. 192, 101-108 (2011).

107

Longshore Wave Breaker with Reﬂected Beach

1), 2) 3) S.R. Pudjaprasetya ∗, F.P.H. van Beckum , Elis Khatizah

1) Industrial and Financial Mathematics Research Group, Fac. of Math. and Natural Sciences, Bandung Institute of Technology, Indonesia 2) Dept. of Applied Math., University of Twente, the Netherlands 3) Mathematics Dept., Bogor Agricultural University, Indonesia e-mail: sr [email protected] ∗ Abstract We consider waves pass over a seabed covered with parallel bars as wave reflector. In the previous research [1] an optimal dimension of the submerged parallel bars is obtained. The more bars installed the more incident wave amplitude is reduced. In that study we assume the beach is ideal and can absorb all transmitted wave energy. When there is a reflected beach, the situation is changed because there is a reflected wave from the beach. This reflected wave will interact with other waves and will change the long-term wave interaction behavior. Literatures suggest that submerged bars in front of beach can either provide shelter or worsen the hazard of incident wave, depending on the phase relation between the bars and the shoreline reflection. In this paper, we study the effectiveness of submerged bars as breakwater when there is a hard-wall beach on the right. We consider the shallow water equation in its Riemann invariant form with variables surface elevation and discharge. Applying the method of characteristic will give us exact formulas for surface elevation and discharge at any time. In this way we can then obtained the long-term wave interaction behavior. It turns out that the phase difference between right running waves will determined whether the wave will superpose destructively or constructively before they hit the hard-wall beach. It turns out that distance Lb between submerged bar and the beach plays an important role, because it determines the phase relation between interacting waves. We obtain formulas for the safest case

1 π 1 λ1 1 √gh1 Lb = (n + ) = (n + ) = (n + )π , (1) 2 k1 2 2 2 ω and the most dangerous case

π λ1 √gh1 Lb = n = n = nπ . (2) k1 2 ω Figure 1 shows results from computation for one-bar submerged bar that conﬁrm the formulas above. Basically, we can determine the safest distance Lb for any formation of submerged rectangular bars. Simulations for two-bar submerged bar and periodic rectangular bar are given.

108

1.5

0.5

−0.5

−1

−1.5

−2 0 20 40 60 80 100

Figure 1: The ampliﬁcation curve of surface elevation at the beach as functions of time for destructive superposition: Lb in (1) (dotted line), and for constructive superposition: Lb in (2)(solid line).

2.5

1.5 One bar

1 Two bar periodic, 3 0.5

0 0 20 40 60 80

Figure 2: Curves ampliﬁcation curve of surface elevation at the beach as functions of distace Lb for one-bar case (diamonds), two-bar (squares), periodic bar (triangles). All curves are periodically depending on Lb.

References

[1] Pudjaprasetya, S.R. & Chendra, H.D., 2009 An Optimal Dimension of Sub- merged Parallel Bars as a Wave Reflector, Bulletin of Malaysian Mathematical Soc., 2. [2] Mei, C.C., Stiassnie, M., Dick K.-P. Yue, 2004, Theory and Applications of Ocean Surface Waves, Advanced Series on Ocean Engineering Vol. 23, World Scientific. [3] Mattioli, F., 1990, Resonant Reflection of a Series of Submerged Breakwaters, Il Nuovo Comento Vol. 13C, No.109 5, 823-833. [4] Yu, J., Mei, C.C., 2000, Do longshore bars shelter the shore?, J. Fluid Mech. 404, 251-268, Wave Energy Dissipation in Porous Media Ikha Magdalena1),∗, S.R. Pudjaprasetya2)

1, 2) Industrial & Financial Mathematics, Fac. of Math. and Natural Sciences, Bandung Institute of Technology, Indonesia ∗ e-mail: [email protected]

Abstract An array of artificial reefs can be used as a submerged breakwater for offshore protection. This type of breakwater provides environmental enhancement and aesthetics that are not found in conventional breakwaters. An array of reefballs is assumed to be a submerged porous layer with certain dimension and diffusive parameters. This research study the effectiveness of a submerged porous media in dissipating wave energy of an incoming wave. The approach is one-dimensional and linear.

The full governing equation for this problem consists of two Laplace equations for a layer of ideal fluid, which is on top of a layer of fluid in the submerged porous media. Boundary conditions are kinematic and dynamic along the surface, continuity of pressure and fluid velocity along the interface, and impermeable bottom. Solving the full governing equation will yield the following dispersion relation

2 S S ω ε tanh k(h1 − h2) cosh kh1 − α sinh kh1 = S S (1) gk ε sinh kh1 tanh k(h1 − h2) − α cosh kh1

S S S with α = (C − if ). If we take parameter ω = 1, h1 = 1, h2 = 1.7, ε = 0.4, CS = 0.4, f S = 1.5, dispersion relation (1) will results a wave number k with negative imaginary part that results spatial dissipation of wave amplitude, see Fig. 1 (Left). Fig. 1 (Middle & Right) show how |=(k)| depends on other variables. From explicit formulas of velocity potential as solution of the full governing equation, we can derive an evolution equations in term of surface elevation η and discharge Q: 1 η + (nQ) = 0 (2) t n x 2 Qt + c ηx = 0 (3)

Coeﬃcients n and c are complex numbers depending on variable bottom, and characteristic of permeable breakwater. We solve the equation numerically using ﬁnite volume method in a staggered grid. Numerical simulation shows amplitude reduction of incoming waves, see Figure 2.

110

Figure 1: (Left) Solid line is the curve exp−i(kx−ωt) with k = 0.3117600956 − 0.02450305720i, dotted line is its envelope. (Middle & Right) The curve of |=(k)| 2 with respect to the non-dimensional variable d = (h2 − h1)/h2 and r = ω h2/g.

Figure 2: (Left) Amplitude of incoming monochromatic wave reduces after passing a submerged porous breakwater. (Right) Wave energy decreases due to submerged porous breakwater.

References

[1] Armono, H.D and Hall, K.R: Wave transmission on submerged breakwater made of hollow hemispherical shape artiﬁcial reefs, Proceeding of The 1st Coastal, Estuary and Oﬀshore Engineering Speciality Conference, (2003), CSC-185-1:10.

[2] Ching-Piao Tsai : Wave Transformation Over Submerged Permeable Break- water on Porous Bottom, Ocean Engineering, (2006), 1623-1643.

[3] Dean, R.G., Dalrymple, R.A., Water Wave Mechanics for Engineers and Sci- entist, World Scientiﬁc, (1991).

[4] Stelling, G.S., Duinmeijer, S.P.A., A staggered conservative scheme for every Froude number in rapidly varied shallow water ﬂows, Int. J. Numer. Meth. Fluids, (2003), 43, 1329-1354.

111

Multi-Compartment Modeling of Tumor Cells Interacting With Dynamic Chemotherapeutic Drug

Madhu Jain1)*, G. C. Sharma2)**

1) Indian Institute of Technology, Roorkee-247667(India) 2) Institute of Basic Science, Khandari, Agra-282002(India)

*[email protected], **[email protected]

Abstract

A tumor is an unusual growth of the tissues of the human body. Both benign and malignant types of tumors require some kind of therapy to remove or reduce their size. The growth of tumor is not regulated by normal body control mechanisms. This unchecked growth is not beneficial to the organs of the body. Chemotherapeutic drug has however been proved successful to great extent in the reduction of the size of tumor. Benign tumor is a well-defined growth with smooth boundaries and is not dangerous in nature; it grows slowly in diameter and compresses the nearby tissues. Malignant tumor usually has irregular boundaries and attacks the surrounding area; it is poisonous in nature and takes the form of cancer later on. Chemotherapy is a combination of two or more drugs which administrated to patients at different time and is very much useful in elimination or reduction in the size of tumors. The effect of drug on tumor growth can be mathematically modeled from different perspective. A three compartment mathematical model has been developed to study the dynamics of tumor cells interacting with chemotherapeutic drug. A system of differential equations that describe the dynamics of tumor cells, is constructed. The model has been analyzed at equilibrium as well as transient state. The steady state analysis has been done by using successive over relaxation method whereas Runge Kutta method is employed for facilitating transition results. The sensitivity analysis has also been performed to explore the effects of various parameters on tumor cell population. The model analyzed demonstrate that the drug with decaying factor has vital role in the decrease of proliferating cell population in all proliferating compartment, however the drug has no effect on non-proliferating cell population. It can be predicted through our model that population of tumor cell can be controlled by adjusting the parameters during the course of chemotherapeutic drug administration. The outcomes of the investigation may provide an insight for medical practitioners as well as pharmacists to get understanding the pharmacokinetic dynamics of the drug. Our model also reveals that how equilibrium points and stability analysis can be employed in handling the tumor cell population. 112

On LaSalle’s Invariance Principle and Its Application to Synchronize Hyperchaotic Systems Matthew Min-Hsiung Lin

Dept. of Mathematics, National Chung Cheng University

∗ e-mail: [email protected]

Abstract The notion of hyperchaotic synchronization is of fundamental importance because its wide range of important applications in secure communications, information processing, chemical reactions, and biological systems [1, 2, 3, 4]. In this talk, we want to focus on the adaptive control law to solve the synchronization problem. To synchronize two hyperchaotic systems, traditional approaches often require the number of nonzero outputs of the controller equal to the dimension of the state variables. However, in practice, it would be desirable to reduce the number of outputs but also synchronize two hyperchaotic systems. Compared to traditional approaches, we show that that two Rössslerhyperchaotic systems with known and unknown parameters can be synchronized by only using three control functions. Theoretical proof is based on the LaSalle’s invariant principle [5, 7], which is one of the very useful theorems in dynamical systems and control theory. Plenty of applications are referred to [6]. We believe that our idea of employing LaSalle’s invariant principle leads to a general strategy for claiming the synchronization between two hyperchaotic systems. There is plenty of room for future research, including a refinement of the control functions for more efficient synchronization and a generalization to more complex structures. We hope that our discussion in this investigation offers a unified and an effectual avenue of synchronizing between two given systems.

References

[1] P. Badola, S. S. Tambe, and B. D. Kulkarni, Driving systems with chaotic signals, Phys. Rev. A, 46 (1992), pp. 6735–6737.

[2] S. Bowong, Stability analysis for the synchronization of chaotic systems with diﬀerent order: application to secure communications, Physics Letters A, 326 (2004), pp. 102 – 113.

113

[3] G. Chen and X. Dong, From chaos to order: methodologies, per- spectives, and applications, World scientiﬁc series on non-linear science: Monographs and treatises, World Scientiﬁc, 1998.

[4] D. Ghosh, A. R. Chowdhury, and P. Saha, On the various kinds of synchronization in delayed duﬃng-van der pol system, Communications in Nonlinear Science and Numerical Simulation, 13 (2008), pp. 790 – 803.

[5] M. W. Hirsch, S. Smale, and R. Devaney, Diﬀerential Equations, Dynamical Systems, and an Introduction to Chaos - Second edition, 2nd edition, Academic Press, 2003.

[6] H. K. Khalil, Nonlinear Systems, 3rd edition, Prentice Hall, 2002.

[7] J. P. LaSalle, The Stability of Dynamical Systems, SIAM Press, 1976.

114

An unbiased estimator of noise correlations under signal drift

1,2), Keiji Miura ∗ 1) PRESTO, Japan Science and Technology Agency 2) Department of Molecular and Cellular Biology, Harvard University

e-mail: [email protected] ∗ Abstract Correlations in noises can play an important role in large-scale modeling and data analysis. In neuroscience, how reliably sensory information is represented by neural responses has been characterized by applying information theoretic approach in a stochastic stimulus-response framework. It has been theoretically shown that the correlations in response noises could be a major determinant for coding capacities of sensory information by neurons [1] and, even in a very simple homogeneous network with tiny correlations, having more neurons does not help at all [2]. Therefore, it is very important to estimate noise correlations accurately. While significant noise correlations has been reported from almost all cortical areas, it was pointed out that non- stationarity such as a drift in 0.1 mean responses could lead to artificial correlations even if there is no effective correla- 0.05 tion [3, 4, 5].

Although attempts to esti- Cross correlations 0 mate noise correlations under changing environments have −20 0 20 been made, they worked only Time shift for speciﬁc cases [5]. For some statistical mod- Figure 1: Cross correlation function esti- els, an unbiased estimator un- mated by the proposed method (full line) and der arbitrarily changing en- the conventional correlation coeﬃcient (dot- vironments has been success- ted line) for the time shifted data. Σ11 = fully obtained in a simple, an- Σ22 = 1 and Σ12 = 0.1. Time dependent alytically closed form by using signals ξ were generate by AR model. the information geometry of semiparametric models [6, 7].

115

In this paper I propose a semiparametric method for estimating noise correlations unbiasedly whatever the time course of mean drift is. I consider a bivariate normal distribution for activities of two neurons:

1 1 1 1 q(X; ξ, Σ− ) = exp (X ξ)′Σ− (X ξ) , (1) 2π Σ 1/2 {−2 − − } | | where X = x, y and ξ = ξ1, ξ2 are vectors. Here, I assume that the co- variance matrix{ Σ} is constant{ while} the signals ξ can change over time. Espe- cially, when the signals are randomly distributed, but, two consecutive signals are the same, the distribution of activities X = X1,X2 = x1, y1, x2, y2 can be described as a mixture model: { } { } { }

1 1 1 p( X ;Σ− , k) = k(ξ)q(X1; ξ, Σ− )q(X2; ξ, Σ− )dξ (2) { } ∫ where k(ξ) denotes an unknown distribution of signals. I derive optimal 1 1 1 1 1 estimators of the three constant parameters Σ− = Σ11− , Σ12− (= Σ21− ), Σ22− which work whatever k(ξ) is. { }

References

[1] H. Sompolinsky, H. Yoon, K. Kang, M. Shamir: Population coding in neuronal systems with correlated noise, Phys. Rev. E, 64 (2001), 051904. [2] E. Zohary, M. N. Shadlen, W. T. Newsome: Correlated neuronal discharge rate and its implications for psychophysical performance, Nature, 370 (1994), 140–35. [3] A. S. Ecker, P. Berens, G. A. Keliris, M. Bethge, N. K. Logothetis, A. S. Tolias: Decorrelated neuronal ﬁring in cortical microcircuits, Science, 327 (2010), 584–7. [4] A. Renart, J. de la Rocha, P. Bartho, L. Hollender, N. Parga, A. Reyes, K. D. Harris: The asynchronous state in cortical circuits, Science, 327 (2010),

587–90. [5] W. Bair, E. Zohary, W. T. Newsome: Correlated ﬁring in macaque visual area MT: time scales and relationship to behavior, J Neurosci., 21 (2001), 1676–97. [6] S. Amari, M. Kawanabe: Information geometry of estimating functions in semi-parametric statistical models, Bernoulli, 3 (1997), 29–54. [7] K. Miura, M. Okada, S. Amari: Estimating spiking irregularities under changing environments, Neural Comput., 18 (2006), 2359–86.

116

Construction of Probabilistic Boolean Networks: A Maximum Entropy Rate Approach

1), Wai-Ki Ching ∗,

1) Advanced Modeling and Applied Computing Laboratory Department of Mathematics The University of Hong Kong Pokfulam Road, Hong Kong.

e-mail: [email protected] ∗ Abstract Modeling genetic regulatory networks is an important problem in ge- nomic research. Boolean Networks (BNs) and their extensions Probabilistic Boolean Networks (PBNs) have been proposed for modeling genetic regulatory interactions. In a PBN, its steady-state distribution gives very important information about the long-run behavior of the whole network. However, one is also interested in system synthesis which requires the construction of networks. The inverse problem is ill-posed and challenging, because there may be many networks or no network having the given properties, and the size of the problem is huge. The construction of PBNs from a given transition-probability matrix and a given set of BNs is an inverse problem of huge size. Here we propose a maximum entropy approach for the above problem. Newton’s method in conjunction with the Conjugate Gradient (CG) method is then applied to solving the inverse problem. We investigate the convergence rate of the proposed method. Numerical examples are also given to demonstrate the eﬀectiveness of our proposed method.

This is a joint work with X. Chen, X.S. Chen, Y. Cong and N. Tsing [3].

References

[1] W. Ching, X. Chen and N. Tsing, Generating Probabilistic Boolean Net- works from a Prescribed Transition Probability Matrix, IET on Systems Biology, 6 (2009) 453-464.

[2] S. Zhang, W. Ching, X. Chen and N. Tsing, Generating Probabilistic Boolean Networks from a Prescribed Stationary Distribution, Informa- tion Sciences, 180 (2010) 25602570.

117

[3] X. Chen, W. Ching, X.S. Chen, Y. Cong and N. Tsing, Construction of Probabilistic Boolean Networks from a Prescribed Transition Probability Matrix: A Maximum Entropy Rate Approach, to appear in East Asian Journal of Applied Mathematics, (2011).

118

Compensation Technique for Transmission Delay and Packet Loss in Networked Control System based on Lagrange Interpolation

1),* 2), 2) Inseok Yang , Donggil Kim , Dongik Lee

1) Department of Industrial Applied Mathematics, Kyungpook National University, Korea 2) School of Electronics Engineering, Kyungpook National University, Korea * e-mail: [email protected]

Abstract In this paper, the compensation technique for transmission delay/packet loss in networked control system (NCS) is proposed. NCS is a feedback control system wherein the control loops are closed through a communication network thanks to recent advances in microelectronics [1]. NCS has many advantages compared to traditional control systems, including ease to maintenance, reduced system space and weight by replacing mechanical linkages to network nodes, etc. However, by inserting communication network in feedback control loop, new problems including transmission delay/packet loss among network nodes occur as shown in Fig. 1. These problems degrade the system performance and even make systems unstable. To overcome these problems, this paper proposes a compensation technique based on Lagrange polynomial interpolation [2]. The proposed Lagrange polynomial compensation method can compensate the lost data by updating the past received data. For this reason, the compensation module can be implemented as an independent network device, as shown in Fig. 2. The equation of the proposed Lagrange polynomial compensation method can be expressed as follow:  fn((− 3) T ) 3 3 nT−−() n j T fnTˆ ( )=  fniT ((− ))= 1,3,3−− fn (( 2)) T avT (1) 2 ∑ ∏ []n i=1 j=1 ()()niTn−−− jT ji≠ fn((− 1) T ) where, T is a sampling period. In (1), the lost data can be compensated by updating vn in real time. The performance of the proposed technique is evaluated by numerical simulation results applied to an “innovative control effectors (ICE)” aircraft [3]. The aircraft maneuvers with maximally 3.5 deg of error between the normal model and the uncompensated model as shown in Fig. 3 and 4. However, if the aircraft adopts the proposed compensation module, then the error is reduced significantly, to less than 0.5 deg. Hence, the simulation results show that the proposed method can compensate lost data fast and effectively.

119

References

[1] Y. Tipsuwan, M.-Y. Chow: Control Methodologies in Networked Control Systems, Control Engineering Practice, 11 (2003), 1099--1111. [2] S. C. Chapra, R. P. Canale: Numerical Methods for Engineers, McGrawHill, (2006).

[3] S. R. Wells: Application of Sliding Mode Methods to the Design of Reconfigurable Flight Control Systems, Ph. D. Dissertation, University of California, (2002).

System coordinate

Plant yt() Plant yt()

Actuator Sensor TDC Actuator Sensor

ykT() ukTˆ() ukT()−τ ca ykT() ukT()−τ ca

τ Network τ sc Network ca τ ca τ sc

ukT() ykT( −τ s ukT() Control coordinate ykT()−τ sc Controller ykTˆ() Controller TDC

Fig. 1. The structure of networked control Fig. 2. The proposed architecture of networked systems. control system.

Fig. 3. The results of turn reversal maneuver of the ICE aircraft with and without compensation.

Fig. 4. The errors of turn reversal maneuver of the ICE aircraft with and without compensation. 120

Mathematics of Crime

Andrea BERTOZZI University of California Los Angeles * e-mail: [email protected]

Abstract

There is an extensive applied mathematics literature developed for problems in the biological and physical sciences. Our understanding of social science problems from a mathematical standpoint is less developed, but also presents some very interesting problems, especially for young researchers. This lecture uses crime as a case study for using applied mathematical techniques in a social science application and covers a variety of mathematical methods that are applicable to such problems. We will review recent work on agent based models, methods in linear and nonlinear partial differential equations, variational methods for inverse problems and statistical point process models. From an application standpoint we will look at problems in residential burglaries and gang crimes. Examples will consider both ``bottom up'' and ``top down'' approaches to understanding the mathematics of crime, and how the two approaches could converge to a unifying theory.

121

Numerical Studies of Lorentz Transformation

* David C. Ni

Direxion Technology 9F, No. 177-1, Ho-Ping East Rd., Section 1, Taipei, Taiwan * e-mail: [email protected]

Abstract The invariance of Lorentz transformation, which the relativity theories are based upon, has been under extensive tests and elaborated extensions in several areas recently. In the area of astrophysics, the origin of magnetic fields in conjunction with generalized vorticity is connected to the space-time distortion [4]. In the area of quantum gravity, the speed of light is predicted to vary at Planck scale [1 and references therein]. At the high-energy or ultraviolet limit, new developments, such as lattice gauge theory, which evolving from Yang-Mills gauge theory [8] and renormalization [2, 3, 7], still have debated issues as those in quantum gravity. We present numerical studies of Lorentz transformation with focus on the nonlinear extensions of the equations. We find interesting results such as fractal patterns, chaos, nonlinear-to-linear degeneracy, continuum-to-discreteness transition, vorticity and symmetry broken.

The efforts in testing Lorentz invariance are proposed with the new theoretical models based on phenomenological modifications of the usual Lorentz invariant dispersion law. One of the models modifies the dispersion law E2 = p2c2 + m2c4 by adding nonlinear terms in powers of particle momentum as follows:

2 2 2 2 4 2 3 E = p c + m c + h1p + h2p + h3p + … (1)

Where c is the light velocity, E and p are the energy and momentum respectively, m is the mass of particle, and hj is the coefficient. Since these proposals are still conjectural, we are motivated to examine the dispersion law and related phenomenological modifications by numerical analysis [5, 6].

Given two inertial frames with different velocities, u and v, the observed velocity, u’ , from v-frame is as follows:

2 u’ = ( u - v ) / ( 1 - vu/c ) (2)

We set c2 = 1 and the multiply a phase connection, exp(iχ(u)), to the normalized complex form of the equation (2) based on the concept of gauge transformation [3] as follows:

122

f = (u’/u) = exp(iχ(u))(1/u)[(u-v)/( 1-uv)] (3)

We adopt the appraoch by iterating the function f for all the points on the complex plane and examine the convergent subsets. We find several interesting observations as follows:

(1) The convergent subsets show fractal patterns (2) For particular v values, chaotic phenomena occurs

(3) Nonlinear-to-linear degeneracy occurs when v value, which corresponding to the normalized velocity of observing frame, is approaching to unity. (4) Abrupt continuum-to-discreteness transition when v value further approaching to unity in the range of 10-16. (5) After the transition occurs, the convergent subsets are symmetrical to both real and imaginary axes.

References

[1] A. A. Abdo et al, A limit on the variation of the speed of light arising from quantum gravity effects, Nature 462 (Nov., 2009) 331-334 [2] G. 't Hooft and M. Veltman, Regulation and renormalization of gauge fields, Nuclear Physics B 44, (1972) 189–219 [3] G. 't Hooft, ed., 50 Years of Yang-Mills theory, (2005) World Scientific [4] S. M. Mahajan and Z. Yoshida, Twisting Space-Time: Relativistic Origin of Seed Magnetic Field and Vorticity, Phys. Rev. Lett. 105 (2010) 095005 [5] David C. Ni and C. H. Chin, Classification on Herman Rings of Extended Blaschke Equations, Differential Equations and Control processes, #2, Article 1 (2010) ( http://www.neva.ru/journal/j/EN/numbers/2010.2/issue.html) [6] D. C. Ni, N. B. Ampilova, B. Alina, and C. H. Chin, Chaotic Behavior and Bifurcation in N-Body Systems, Proceedings for 8th AIMS International Conference on Dynamical Systems, Differential Equations, and Applications (2011) [7] Kenneth G. Wilson, The renormalization group: critical phenomena and the Kondo problem, Rev. Mod. Phys., 47 (1975), 773-839 [8] C. N. Yang and R. Mills, Conservation of Isotopic Spin and Isotopic Gauge Invariance, Physical Review 96 (1954) 191–195

123

Numerical studies on the Monge-Amp´ere equation arising from free-form design of geometric optics Yu-Lin Tsai1), Ming-Chiang Jiang2) and Chin-Tien Wu2),∗

1) Department of Photonics and Institute of Electro-Optical Engineering, Chiao-Tung University, Taiwan. 2) Department of Applied Mathematics, Chiao-Tung University, Taiwan.

∗ e-mail: [email protected]

Abstract The Monge-Ampere (MA) equation is a fully nonlinear PDE arising in various research fields including differential geometry, mass transportation, geostrophic fluid and optical design. Numerical methods such as finite difference methods proposed by Oberman etc [7], Dean and Glowinshi [3, 4] and finite element methods proposed by Feng and Neilan [5, 6] and Awanou [8] have been successfully applied to solve the standard MA equation. In this talk, we introduce some of known theoretical results obtained by Caffarelli [9], Oliker [1], Mader [12] and Wang [11] etc., and the above mentioned numerical methods. Particularly, we employee the vanished-moment method to solve the MA equation arising from the free form surface (FFS) design problem in geometric optics[2] [13]. To ensure the FFS obtained from numerical computation is continuous and to be able to compute the curvature of the FFS easily, we solve the linearized MA problem using BCIZ finite element method[10]. The discrete linear system is solved by the conjugate gradient method (CG) in which multigrid approximation of the discrete biharmonic operator is used as the preconditioner of the CG method. Accuracy and robustness of our approach are demonstrated on several benchmark examples. We compare our numerical results with the results obtained by Feng and Neilan, Froese and Oberman, and Dean and Glowinski.

References

[1] V. Oliker, Mathematical aspects of design of beam shaping surface in geometrical optics., In: Trends in Nonlinear Analysis, Springer-Verlag, pp. 191–222, 2002.

[2] J. S. Schruben, Formulation of a reﬂector-design problem for a light- ing ﬁxture., J. Opt. Soc. Am. 62, pp. 1498– 1501, 1972.

124

[3] E. J. Dean and R. Glowinshi, Numerical solution of the two- dimensional elliptic Monge-Ampére equation with Dirichlet boundary conditions: an augmented Lagrangian approach., C. R. Acad. Sci. Paris, Ser. I 336, pp. 779–784, 2003. [4] E. J. Dean and R. Glowinshi, Numerical methods for fully nonlinear elliptic equations of the Monge-Ampére type., Comput. Methods Appl. Mech. Engrg. 195, pp. 1344–1386, 2006. [5] X. Feng and M. Neilan, Mixed finite element methods for the fully nonlinear Monge-Ampere equation based on the vanishing moment method., SIAM J. Numer. Anal., 47, pp. 1226-1250, 2009. [6] X. Feng and M. Neilan, Vanishing moment method and moment solutions for fully nonlinear second order partial differential equations., J. Sci. Comput., 38, pp. 74-98, 2009. [7] J. D. Benamou, B. D. Froese and A. D. Oberman, Two numerical methods for the elliptic Monge-Ampere equation., ESAIM: Mathe. Modell. Numer. Anal., pp. 737–758, 2009.

[8] G. Awanou, Numerical methods for fully nonlinear elliptic equations., Inter. Conf. App. Thm., 2010. [9] L. A. Caffarelli and X. Cabre, Fully nonlinear elliptic equations., American Mathematical Society Colloquium Publications. 43, American Mathematical Society, Providence, RI, 1995. [10] G. P. Bazeley, Y. K. Cheung, B. M. Irons and O. C. Zienkiewicz, Triangular elements in plate bendingconforming and nonconforming solutions, Proc. Conf. on Matrix Methods in Structural Me- chanics, WPAFB, Ohio, pp. 547-576, 1965. [11] P.Guan and X. J. Wang, On a Monge-Ampere equation arising in

geometric optics J. Diﬀerential Geometry, 48, pp. 205-223, 1998. [12] L. Marder,Uniquessness in reﬂector mappings and the Monge-Ampere equation Proceeding of Royal Society of London. Series A, Mathematical and Physical Sciences, 378, No. 1775, pp.529-538, 1981. [13] P. Benitez, J. C. Minano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernandez andW. Falicoff Simultaneous multiple surface optical design methof in three dimensions., Opt. Eng. 43(7). pp. 1489-1502, 2004.

125

Renormalization approach to solve Langevin’s equations Takashi Ichinomiya

School of Medicine, Gifu University

e-mail: [email protected]

Abstract The importance of molecular dynamics(MD) simulations in physics, chem- istry and biology has been increased in recent years. Unfortunately, when we want to investigate very rare events, such as chemical reactions, the computational costs of the MD can be extremely large. Several methods have been proposed to overcome this diﬃculty[1, 2, 3]. In this talk, we propose a new method to accelerate MD simulation, using the idea of renormalization. The application of renormalization to MD simulation has been proposed by Faccioli[4]. However our method use the different renormalization technique, inspired by the work of Chen, Goldenfeld, and Oono[5]. In the following, we consider the following Langevin’s equation dx i = f(x) + √Dξ (t), (1) dt i where x = (x1, x2, , xn), and ξi(t) is Gaussian white noise, which satisﬁes ··· ξi(t)ξj(t0) = δ(t t0)δij. If we use the Euler-Maruyama scheme to solve this equationh numerically,i − x(∆t) is given by

xi(∆t) = ∆tf(x(0)) + √D∆tηi(t), (2) where ηi is Gaussian random noise. Therefore the distribution of x(∆t) is given as

T P (x(∆t) x0) exp[ (x ∆tf(x0)) A0(x ∆tf(x0))], (3) | ∝ − − − 1 where (A0)ij = δij(2D∆t)− . The Euler-Maruyama scheme is valid only when ∆t is small enough. Our basic idea is to assume that P (x(t) x0) can be approximated | T P (x(t) x0) exp[ (x tf(x˜(t))) At(x tf(x˜(t)))], (4) | ∝ − − −

126 for larger t, where x˜(t) and At is renormalization parameters. To calculate x˜(t) and At, we use the equation

P (x(2t) x0) = dx0P (x0(t) x0)P (x(t) x0). (5) | ∫ | |

By the Taylor expansion of f(x) around x = x˜(t), we get the approximated equation for x˜(2t) and A2t as

2 x˜(2t) = x˜(t) + t f1(x˜(t))f(x˜(t)) + (x˜(t) x0) (6) − and A t A2t = (Atf1(x˜(t)) + f1(x˜(t))At), (7) 2 − 4 ∂f n where f1 = ∂x . By applying these equations n-times,we can obtain P (x(2 ∆t) x0). Eqs.(6) and (7) serves us a new simulation algorithm. In my talk, we will | present our new method in detail, and discuss on the advantages and disadvantages of this method, with some results of numerical simulations.

References

[1] C. D. Dellago, P. G. Bolhuls, F. S. Csajka and D. Chandler, Transition path sampling and the calculation of rate constants, The Journal of Chemical Physics, 108(1998) 1964-1977.

[2] A. F. Voter, A method for accelerating the molecular dynamics simulation of infrequent events, Journal of Chemical Physics, 106(1997), 4665-4677.

[3] G. Hummer and I. G. Keverkidis, Coarse molecular dynamics of a pep- tide fragment: Free energy, kinetics, and long-time dynamics computations, Journal of Chemical Physic, 118(2003) 10762-10773.

[4] P. Faccioli, Molecular dynamics at low time resolution, The Journal of Chemical Physics, 133(2010), 164106.

[5] L-Y. Chen, N. Goldenfeld, Y. Oono, Renormalization group and singular perturbations: Multiple scales, boundary layers, and reductive perturbation theory, Physical Review E, 54(1996) 376-394.

127

Consistent approximation-sampling with multi pre and post ﬁltering

1), 1) Kil Hyun Kwon ∗, Jaekyu Lee

1) Department of Mathematical Sciences, KAIST, S. Korea.

e-mail: [email protected] ∗ Abstract Sampling is the process of representing signals of a continuous variable t by sequence of numbers, that is, discrete signal representation. If a signal f(t) is band-limited, that is, f(t) PWπ, then f(t) is com- ∈ pletely characterized by its samples f(n) n Z as f(t) = f(n)sinc(t n) = ∈ { } n Z − ∈ (c φ)(t), where c = f(n) n Z l2 and φ(t) = sinc(t)∑ is the interpolation kernel(or∗ interpolator).{ } ∈ ∈ If a signal f(t) L2(R) is not band-limited, then the standard procedure ∈ is to apply an additional prefiltering with the ideal low pass filter χ[ π,π](ξ) = − φ(ξ), φ(t) = sinc(t) before sampling. Then the corresponding band-limited reconstruction f(t) = f(t), φ(t k) φ(t k) is the orthogonal projection b k Zh − i − of f(t) onto PW . ∈ eπ ∑ More generally for any Riesz generator φ(t) L2(R), ∈ f(t) = f(t), φ(t k) φ(t k) (0.1) h − i − k Z ∑∈ e e is the orthogonal projection of f(t) L2(R) onto V (φ), where φ(t) is the dual Riesz generator of φ(t). ∈ Note that f(t) in (0.1) is the L2-least square approximation of ef L2(R) ∈ in V (φ). We call h(t) := φ( t) the analysis function and f f as in (0.1) the ”approximation-sampling”e − [1]. 7−→ However, in most applications,e the analysis function is givene a priori as the impulse response h(t) of the acquisition device. So, [6] introduces the consistent approximation-sampling procedure, where the consistency means that the original signal and its approximation are essentially equivalent to observers(through sensors or acquisition devices). Many authors[2, 3, 4, 5, 7, 8] have extended it several ways. In this work, we consider the consistent approximation-sampling procedure satisfying m-shift-invariance when the samples f(t), ψi(t qk) (1 i 2 h − i ≤ ≤ M, k Z) of f(t) L (R) and the reconstruction filters φj(t) 1 j N ∈ ∈ { | ≤ ≤ } 128 are given, where ψi(t qk) 1 i M, k Z is a Riesz sequence , { − | ≤ ≤ ∈ } φj(t) 1 j N is a Riesz generator and q is a rational number. Then we {find some| ≤ equivalent≤ } conditions that the approximation exists. Also, we discuss the properties of the approximation and which approximation is closest to the least square solution (0.1).

References

[1] A. Aldroubi and M. Unser: Sampling procedure in function spaces and asymptotic equivalence with Shannon’s sampling theory, Numer. Func. Anal. Opt., 15 (1994), 1–21.

[2] Y. Eldar: Sampling with arbitary sampling and reconstruction spaces and oblique dual frame vectors, J. Fourier Anal. Appl., 9 (2003), 77–96.

[3] Y. Eldar and T. G. Dvorkind: A minimum squared-error framework for generalized sampling, IEEE T. Signal Proces., 54 (2006), 2155–2167.

[4] Y. Eldar and T. Werther: Generalized framework for consistent sampling in Hilbert spaces, Int. J. Wavelets. Multi., 3 (2005), 347–359.

[5] A. Hirabayashi and M. Unser: Consistent sampling and signal recovery, IEEE T. Signal Proces., 55 (2007), 4104–4115.

[6] M. Unser and A. Aldroubi: A general sampling theory for nonideal acquisition devices, IEEE T. Signal Proces., 42 (1994), 2915–2925.

[7] M. Unser and J. Zerubia: Generalized sampling: Stability and performance analysis, IEEE T. Signal Proces., 45 (1997), 2941–2950.

[8] M. Unser, J. Zerubia: A generalized sampling theory without band- limiting constraints, IEEE T. Circuits-II., 45 (1998), 959–969.

129

A Tailored Finite Point Method for Convection Diﬀusion Reaction Problems with Variable Coeﬃcients

1, 2 Yintzer Shih ∗, R. Bruce Kellogg

1. Department of Applied Mathematics, National Chung Hsing University Taichung 402, Taiwan 2. Department of Mathematics, University of South Carolina, SC 29208, USA

e-mail: yintzer [email protected] ∗ Abstract

We present a tailored finite point method (TFPM) [1, 2, 3] for solving the convection-diffusion-reaction equation with variable coefficients. We develop an algorithm to construct a streamline-aligned grid for the TFPM. Our numerical tests show for small diffusion coefficient the TFPM solution resolves the internal and boundary layers regardless the mesh size, and depicts that TFPM method with a streamline grid has excellent performance compared with the tailored finite point method and a streamline upwind finite element method when ε is small.

References

[1] H. Han, Z. Huang and R.B. Kellogg: The tailored ﬁnite point method and a problem of P. Hemker, Proceedings of the International Confer- ence on Boundary and Interior Layers - Computational and Asymptotic Methods, Limerick, July 2008.

[2] Y. Shih, R.B. Kellogg and P. Tsai: A Tailored Finite Point Method for Convection-Diﬀusion-Reaction Problems, J. of Sci. Compu., 43 (2010), pp. 239-260.

[3] Y.T. Shih, R.B. Kellogg and Y. Chang: Characteristic Tailored Finite Point Method for Convection-Dominated Convection-Diﬀusion-Reaction Problems, J. of Sci. Compu., DOI: 10.1007/s10915-010-9433-7.

130

On the interpolation constants over triangular elements Kenta Kobayashi Institute of Science and Engineering, Kanazawa University

e-mail: Kenta.k@staﬀ.kanazawa-u.ac.jp

Abstract We present new formulas which give sharp upper bounds for the interpolation constants on the triangles. Our formulas provide better upper bounds than the former ones. Furthermore, from the point of practical use, our formulas are convenient and easy to calculate. In the proof of the formulas, we employ numerical veriﬁcation method.

Let T be the triangle in R2 and V 1,1(T ),V 1,2(T ),V 2(T ) be the function spaces deﬁned by

V 1,1(T ) = φ H1(T ) φ dxdy = 0 , ∈ T ½ ¯ Z ¾ ¯ V 1,2(T ) = φ H1(T ) ¯ φ ds = 0, k = 1, 2, 3 , ∈ γ ½ ¯ Z k ¾ 2 2 ¯ V (T ) = φ H (T ) φ(pk) = 0, k = 1, 2, 3 , ∈ ¯ n ¯ o where p , p , p and γ , γ , γ are the¯ vertices and sides of T respectively. 1 2 3 1 2 3 ¯ Then, it is known that the following constants C1(T ),C2(T ),C3(T ),C4(T ) exist:

φ L2(T ) φ L2(T ) C1(T ) = sup ∥ ∥ ,C2(T ) = sup ∥ ∥ , φ V 1,1(T ) 0 φ L2(T ) φ V 1,2(T ) 0 φ L2(T ) ∈ \ ∥∇ ∥ ∈ \ ∥∇ ∥ φ L2(T ) φ L2(T ) C3(T ) = sup ∥ ∥ , C4(T ) = sup ∥∇ ∥ , φ V 2(T ) 0 φ H2(T ) φ V 2(T ) 0 φ H2(T ) ∈ \ | | ∈ \ | | 2 where φ 2 is a H semi-norm of φ deﬁned by | |H (T ) 2 2 2 2 φ 2 = φxx 2 + 2 φxy 2 + φyy 2 . | |H (T ) ∥ ∥L (T ) ∥ ∥L (T ) ∥ ∥L (T ) For these constants, we have obtained the formulas which give sharp upper bound of Cj(T ) as

Cj(T ) < Kj(T ), j = 1, 2, 3, 4.

Concrete form of the formulas Kj(T ) will be shown in the talk.

131

A posteriori estimates of inverse linear ordinary diﬀerential operators

1), 2) 2) T. Kinoshita ∗, T. Kimura and M.T. Nakao

1) Research Institute for Mathematical Sciences, Kyoto University 2) Sasebo National College of Technology

[email protected] ∗ Abstract We present constructive a posteriori estimates of inverse operators for initial value problems in linear ordinary diﬀerential equations on a bounded interval. Here, “constructive” indicates that we can obtain the bounds of the operator norm in which all constants are explicitly given or are represented in a numerically computable form. In general, it is diﬃcult for us to obtain this estimates of this inverse operators in the meaning of a priori. We propose the technique for obtaining the a posteriori estimates by using Galerkin approximation of inverse operators.

References

[1] M.T. Nakao, N. Yamamoto and S. Kimura, On the Best Constant in the 1 Error Bound for the H0 -Projection into Piecewise Polynomial Spaces, Journal of Approximation Theory, 93 (1998), 491–500.

[2] M.T. Nakao, K. Hashimoto, Y. Watanabe, A numerical method to verify the invertibility of linear elliptic operators with applications to nonlinear problems, Computing, 75 (2005), 1–14.

[3] M.T. Nakao and K. Hashimoto, Guaranteed error bounds for ﬁnite element approximations of noncoercive elliptic problems and their applications, Journal of Computational and Applied Mathematics, 218 (2008), no. 1, 106–115.

132

Numerical veriﬁcation for solution existence of semilinear elliptic partial diﬀerential equation on arbitrary polygonal domain

1) 2), 2) Akitoshi Takayasu , Xuefeng Liu ∗, Shin’ichi Oishi 1) Graduate School of Fundamental Science and Engineering, Waseda University 2) Faculty of Science & Engineering, Waseda University; CREST/JST

e-mail: xﬂ[email protected] ∗ Abstract We consider the semilinear elliptic partial diﬀerential equation on polygonal domain of arbitrary shape.

∆u = f(u) on Ω, associated with boundary condition . (1) − To verify the existence of solutions to equation (1), there have appeared several methods adopting invariant point theories, c.f. [2, 3, 4], which declare the solution existence in a small range nearby approximate solution. In case that the domain has non-convex corner, the solution u of equation (1) can be singular one, that is, the L2 integration of second order derivative of u is not bounded, which brings diﬃculty for constructing error estimation. By using the Prager-Synge’s theorem, we successfully developed computable a priori error estimation as below [1]:

(u uh) Ch f 2 , k∇ − kL2(Ω) ≤ k k where uh is approximate solution from linear conforming FEM method and Ch is computable quantity depending only on domain and mesh size. By further applying the Newton-Kantorovich theorem, see also [4], we can verify the solution existence of equation (1) even on polygonal domain of arbitrary shape. In the presentation, we will display numerical computation results to demonstrate the eﬀciency of proposed method.

References

[1] Xuefeng Liu and Shin’ichi Oishi, Veriﬁed eigenvalue evaluation for the laplacian on arbitrary polygonal domain, submitted to Numerische Mathematik, Dec,2010

[2] Mitsuhiro T. Nakao and Yoshitaka Watanabe, Numerical veriﬁcation methods for solutions of semilinear elliptic boundary value problems, Nonlinear Theory and Its Applications, IEICE, vol.2, no.1, pp.2-31., 2011

133

[3] Plum.M, Computer-assisted enclosure methods for elliptic diﬀerential equations, Linear Algebra and its Applications, Vol.324, pp.147-187., 2001

[4] Shin’ichi Oishi, Akitoshi Takayasu and Takayuki Kubo, Numerical ver- iﬁcation of existence for solutions to Dirichelt boundary value problems of semilinear elliptic equations, Preprint submited to Journal of Com-

putational & Applied Mathematics.

134

Reconstruction of the density of tree-shaped networks from boundary measurements of waves

1), WADA Naoki ∗

1) Graduate School of Informatics, Kyoto University

e-mail: [email protected] ∗ Abstract On finite graphs whose edges are identified with open intervals, we consider the inverse problem for wave equation to determine the unknown coefficient from boundary measurements. In general case this inverse problem is not uniquely solvable. We restrict ourselves in the case of trees and give a reconstruction by reducing the inverse problem to an infinite family of Fredholm integral equations of the second kind.

1 Waves on Graphs

K Let Ω be a finite connected tree, which is consisting of edges E = ej j=1 L { } and vertices V = vj j=1. Each edge ej is identified with an open interval and parameterized{ through} this identification. The vertices with more than one incident edge are called the interior vertices and the others are called the boundary vertices. The set of the interior vertices and the boundary vertices are denoted by N and Γ, respectively. Let ρ( ) be a positive C2-smooth function on Ω V and let T be a positive number. We· consider the wave equation \

ρ(x)utt(t, x) uxx(t, x) = 0, t (0,T ), x Ω V (1) − ∈ ∈ \ u(t, ) C(Ω), t [0,T ] (2) · ∈ ∈ ∂νu e (t, v) = 0, t [0,T ], v N (3) | j ∈ ∈ ej v ∑≻ u(0, x) = ut(0, x) = 0, x Ω (4) ∈ u(t, γ) = f(t, γ), t [0,T ], γ Γ, (5) ∈ ∈ where u ej is the restriction of u on the edge ei, ∂ν is the outward normal at endpoints| of intervals and the sum in (3) is taken all over the edges incident to the interior vertex v N, which condition is called Kirchhoﬀ condition. ∈

135

For this initial boundary value problem, we can diﬁne the operator RT which f f maps Dirichlet data u [0,T ] Γ = f to the Neumann data ∂νu [0,T ] Γ. We diﬁne the operator| ×CT : L2 (0,T ) Γ L2 (0,T|) ×Γ by the relation × → × T ( f g) ( ) C f, g)L2((0,T ) Γ) = u (T, x) u (T, x) ρ(x) dx. × Ω ∫ This operator(CT is expressed through the response operator RT [2].

2 Reconstruction

We give a local reconstruction of ρ from the boundary using the boundary measurements RT . On an edge e which connected to a boundary vertex T γ0, we introduce some formulae [1]. Let α, β be functions whose supports T are contained in [0,T ] γ0 and α(t, γ0) = t, β (t, γ0) = T t. For these functions, we have the× relation { } lim RT α (t) = ρ(γ) and equations− t +0 → − T f ( T) T C u = (R )∗β (6) T g T C u = β . (7)

Functions ft , gt that satisfies supp ft , supp gt [T t0,T ] γ0 and (6), 0 0 0 0 ⊂ − × { } (7) in [T t0,T ] γ0 exist uniquely respectively. The equations (6),(7) are expressed− throgh× { the} boundary measurements RT and have the form of Fredholm integral equations of the second kind. These solution satisfies 1 1 ρ(ˆx(t0)) 4 ρ(ˆx(t0)) 4 that limt t0+0 ft0 (t) = , limt t0+0 gt0 (t) = xˆ(t0), where → ρ(γ0) → ρ(γ0) xˆ(t) is the inverse function( of)τ(x) = x ρ(x) dx.( τ(x) is) the so-called γ0 optical distance, which is the travel time for waves from γ to x. Using these √ 0 relations, by solving the equations (6)(7),∫ we can reconstruct the unknown coefficients ρ on the edge e. By generalizing these process mentioned above, we show a reconstruction of the unknown coefficients on trees.

References

[1] M. Belishev: Boundary control and inverse problems: The one- dimansional variant of the BC-method, J. Math. Science, Vol.155, No.3, (2008) pp.343–378 [2] M. Belishev, N. Wada: On revealing graph cycles via boundary measurements, Inverse Problems, 25 (2009)

136

Cubic helical splines with Frenet-frame continuity Chang Yong Han1),∗ Song-Hwa Kwon2)

1) Department of Applied Mathematics, Kyung Hee University 2) Department of Mathematics, The Catholic University of Korea

∗ e-mail: [email protected]

Abstract

A parametric curve r(t) in R3 is called a helix if its tangent vector maintains a constant angle against a fixed axis [7]. A helix admits polynomial parametrization only if it satisfies Pythagorean-hodograph (PH) condition [4], i.e., its speed is a polynomial in the curve parameter [1]. Although helical PH curves comprise a proper subset of spatial PH curves, all spatial PH cubics are helical [2, 3]. As the simplest class of PH curves and polynomial helices, spatial PH cubics have been extensively studied in the context of Hermite interpolation [9, 5, 8, 6]. Our focus in this presentation is on the geometric spline interpolating a 3 sequence of points p0,..., pn in R . This subject has also been discussed in the aforementioned papers as a straightforward extension of G1 Hermite interpolation problem: at each pk a unit tangent tk is either derived from the original curve [5], or assigned according to a shape-preserving interpolation scheme [8] to generate a sequence of PH cubic segments r1(t),..., rn(t) that constitute a G1 spline for the given data points. Instead of directly specifying the unit tangents, we determine each tk in such a way that the Frenet frames of the adjacent PH cubic segments are coincident at their juncture point pk, therefore improving the continuity of the resulting spline to be not only tangent-continuous but also Frenet-frame continuous. It turns out that to ensure a finite number of solutions, an additional condition is needed for each segment. We require each rk(t) to satisfy a certain symmetry property which is geometrically attractive and facilitates the subsequent analysis. The construction of rk(t) involves a new kind of geometric Hermite data consisting of an initial Frenet frame and the displacement vector ∆pk = pk −pk−1. The existence condition of a PH cubic interpolant is formulated in terms of the position of ∆pk relative to the Frenet frame. Once an initial Frenet frame at p0 is provided, the entire collection of PH cubic segments can be sequentially generated using the Frenet frame of rk−1(t) at pk−1 as the initial Hermite data for rk(t).

137

References

[1] R.T. Farouki: Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable, Springer, Berlin, 2008.

[2] R.T. Farouki, C. Giannelli, A. Sestini: Helical polynomial curves and double Pythagorean hodographs I. Quaternion and Hopf map represen- tations. Journal of Symbolic Computation, 44 (2009), 161–179.

[3] R.T. Farouki, C. Giannelli, A. Sestini: Helical polynomial curves and double Pythagorean hodographs II. Enumeration of low-degree curves. Journal of Symbolic Computation, 44 (2009), 307–332.

[4] R.T. Farouki, C.Y. Han, C. Manni, A. Sestini: Characterization and construction of helical polynomial space curves. Journal of Computa- tional and Applied Mathematics, 162 (2004), 365–392.

[5] B. J¨uttler,C. M¨aurer:Cubic Pythagorean-hodograph spline curves and applications to sweep surface modeling. Computer-Aided Design, 31 (1999), 73–83.

[6] S-H. Kwon: Solvability of G1 Hermite interpolation by spatial Pythagorean-hodograph cubics and its selection scheme. Computer Aided Geometric Design, 27 (2010), 138–149.

[7] B. O’Neill: Elementary Diﬀerential Geometry (2nd ed.), Academic Press, 1997.

[8] F. Pelosi, R.T. Farouki, C. Manni, A. Sestini: Geometric Hermite interpolation by spatial Pythagorean-hodograph cubics. Advances in Com- putational Mathematics, 22 (2005), 325–352.

[9] M.G. Wagner, B. Ravani: Curves with rational Frenet-Serret motion. Computer Aided Geometric Design, 15 (1997), 79–101.

138

Free vibration of symmetric angle-ply laminated cylindrical shells of variable thickness including shear deformation theory: spline method

K.K.Viswanathan*, Zainal Abdul Aziz, Saira Javed

Department of Mathematics, Faculty of Science, Universiti Teknologi Malaysia, 81310 Johor Bahru, Johor, Malaysia * e-mail: [email protected]

Abstract The vibrational study of structural elements is of persistent interest in the areas of missiles, aviation, shipping, surface transport and a large number of industries related to the cement and chemicals. Shells of layered construction possess more desirable characteristics over those of homogeneous structure, in terms of higher specific stiffness and better damping and shock absorbing characteristics.Variation in thickness of the layers and inclusion of shear deformation and rotary inertia provides further facilities for the designer in realizing the desired vibrational characteristics of the structure. Transverse effects become more pronounced as the shell become thicker relative to its in-plane dimension and radius of curvature. Free vibrational study of symmetric angle-ply laminated cylindrical shells of variable thickness including first order shear deformation theory using spline function approximation is studied. The researchers [1-3] worked on cross-ply and angle-ply plates with constant thickness and conical shells with variable thickness neglecting the shear deformation and rotary inertia. The available works which consider also these complexities are very few and, to the authors’ knowledge, do not use the versatile spline function method of solutions and variable thickness [4-5]. In the present work, the results are expected to be more accurate and interesting, and more suitable for immediate application in the areas already explained elsewhere.

The equations of motion for the cylindrical shells are derived using first order shear deformation theory. The solutions of displacement functions are assumed in a separable form to obtain a system of coupled differential equations in terms

139 displacement and rotational functions and these functions are approximated by Bickley-type splines of order three. The vibrations of three and five layered shells, made up of several types of layered materials and subjected to two types of boundary conditions are considered. A generalized eigenvalue problem is obtained and solved numerically for an eigenfrequency parameter and an associated eigenvector of spline coefficients. Parametric studies are made for the frequency parameters with respect to the coefficients of thickness variations, length-to-radius ratio, length-to-thickness ratio, various types of materials and ply angles under different boundary conditions.

References

[1] K.K.Viswanathan, S.K. Lee: Int. J. Mechanical Science, 49 (2007), 352—363. [2] K.K.Viswanathan, K.S. Kim: Int. J. Mechanical Sciences, 50 (2008),

1476—1485. [3] K.K.Viswanathan, P.V. Navaneethakrishnan: Int. J. Solids & Structures, 42 (2005), 1129--1150. [4] U. Topal, U. Uzman: Materials & Design, Vol.30 (2009), 532--536.

[5] T.Timarci, K.P. Soldatos: J. Engineering Mathematics, Vol.37 (2000),

211--230.

140

Point vortex equilibria enhancing forces over two parallel plates

1), Takashi Sakajo ∗

1) Department of mathematics, Hokkaido University, CREST, Japan Science and Technology Agency

e-mail: [email protected] ∗ Abstract In ’70s, Kasper designed a high aspect ratio wing with a flap, which is known as “Kasper airfoil”. He suggested that the lift of the airfoil was enhanced by placing vortex structures around its wing. The concept has been examined by some wind-tunnel experiments[3, 7]. Rossow[4] demonstrated that it was possible to keep a spanwise vortex line coupled with an axial flow along its core over the wing with a vertical flap. Saffman and Sheffield[5] considered a potential flow model describing a free point vortex over a flat plate in a uniform flow. They found that some of stationary point vortices can stay stably above the plate and they enhance the lift without any flap. In the present talk, we consider a flow over two parallel flat plates in the presence of a uniform flow whose schematic picture is shown in Figure 1. This is an extension of the analysis due to Saffman and Sheffield into a doubly connected flow domain to find vortex equilibria enhancing the lift on the double wing. It is also motivated by a study of an design for vertical axis wind turbines, which are wind-powered electronic generators with vertical multiple blades[8]. With keeping the same concept as the Kasper airfoil in mind, we expect that stationary vertical vortex lines could enhance the rotational force when they were attached stably in the neighborhood of the blades. As a simple two-dimensional model, we consider two stationary point vortices around the two parallel plates and observe how the stationary point vortices enhance the forces acting on the two plates. The analysis is based on a calculus for potential flows in multiply connected domains introduced by Crowdy[1]. The equation of motion for two point vortices is derived from a mathematical formulation of the N-vortex problem in multiply connected domains given in [6]. We find that there exist loci of stationary two point vortices that enhance the upward lift and the clockwise rotation of the two parallel plates.

141

w２ ʣ ʣ u１

Figure 1: We consider fixed configurations of two point vortices behind two parallel plates in a uniform flow and observe how the fixed equilibria enhance the force on the plates.

References

[1] D. G. Crowdy, A new calculus for two dimensional vortex dynamics, Theor. Comput. Fluid Dyn., 24 9–24 (2010) doi:10.1007/s00162-009- 0098-5

[2] D. Hummel, On the vortex formation over a slender wing at large angles of incidence, AGARD CP-247, paper no. 15.

[3] E. W. Kruppa, A wind tunnel investigation of the Kasper vortex concept, AIAA paper, 77-310, Jan. (1977)

[4] V.G. Rossow, Lift enhancement by an externally trapped vortex, AIAA paper 77-672, June (1977)

[5] P. G. Saﬀman and J. S. Sheﬃeld, Flow over a wing with an attached

free vortex, Stud. Appl. Math., 57 107–117 (1977)

[6] T. Sakajo, The N-vortex problem in multiply connected domains, Proc. Roy. Soc. A, 465 2589–2611 (2009)

[7] D. Walton, A brief wind tunnel test of the Kasper aerofoil, Soaring, 38 No.11 26–27 (1974)

[8] See the webpage at http://en.wikipedia.org/wiki/Vertical- axis wind turbine for a description of vertical axis wind turbines.

142

Bifurcation analysis for the Lugiato-Lefever equation in a square

1), 2) 3) Tomoyuki Miyaji ∗, Isamu Ohnishi , Yoshio Tsutsumi 1) Research Institute for Mathematical Science, Kyoto University 2) Department of Mathematical and Life Sciences, Hiroshima University

3) Department of Mathematics, Kyoto University

e-mail: [email protected] ∗ Abstract We show some results on bifurcation analysis and numerical simulation for the Lugiato-Lefever equation (LLE) on a square domain. LLE is a cubic nonlinear Schr¨odingerequation (NLS) with damping, detuning and driving force. It is a model for describing the evolution of transversal patterns in an optical cavity with a Kerr medium [1]. It is given by

2 2 ∂tE = (1 + iθ) E + ib ∆E + i E E + Ein, x Ω, t > 0, (1) − | | ∈ T 2 where x = (x1, x2) , Ω = ( 1/2, 1/2) R is a unit square, and ∆ = 2 2 − ⊂ ∂x1 +∂x2 is the Laplacian. We impose the periodic boundary conditions. The parameters b2, θ R are diffraction and detuning parameters, respectively. ∈ E = E(x, t) C denotes a slowly varying envelope of electric field. Ein 0 denotes the intensity∈ of spatially homogeneous driving field, and it is a main≥ control parameter. Numerical simulations suggest that LLE in 1- or 2-dimensional space has solitary wave solutions in a certain range of parameters[2]. In contrast to NLS, LLE does not satisfy any conservation law of NLS. Typically, such a solution of LLE appears as an equilibrium point. Moreover, for 2D LLE, a localized spot can undergo a Hopf bifurcation, and it results in a spatially localized and temporally oscillating solution called oscillon [3]. Such solutions are understood as dissipative structures resulting from the balance between gain and loss of energy. However, no explicit analytical solutions are known for LLE. The authors have studied steady-state bifurcations for 1D LLE [4]. They have proved that a small localized roll can bifurcate from homogeneous state in R, and that two mixed-mode solutions can bifurcate as a secondary bifurcation in T1, which are considered to be “germs” of localized structures. We are interested in oscillons for 2D LLE. As a first step, we study steady- state bifurcation of spatially homogeneous steady state in a mathematically rigorous sense, because it is responsible for the occurrence of stationary localized patterns. For this purpose, we apply the center manifold reduction and group theoretic bifurcation theory [5].

143

We study (1) with periodic boundary conditions. Taking into account of the symmetry of the system, we can obtain a catalog of model-independent bifurcation behaviors. Then we apply the center manifold reduction near the modulational instability of homogeneous state. The symmetry of LLE restricts the function form of the vector field on the center manifold. It helps to study model-specific behaviors. The symmetry of LLE is described by 2 Γ = D4 n T , where n means semidirect product of groups. D4 is generated 2 by the reflection across x2 = x1 and π/2-rotation about the origin. T is a group of translations modulo 1. Suppose that k = (l, n) Z2 is a critical wave vector of modulational instability. We may assume ∈l n 0 without loss of generality. Since we consider single-mode bifurcations,≥ we≥ assume that the solution (l, n) Z2 to l2 + n2 = k for a given k N is unique in the above sense, There∈ are three cases to be distinguished:∈ i) l = n > 0; ii) l > n = 0; iii) l > n > 0. In the first two cases, the center manifold is 4-dimensional, while that is 8-dimensional in the third case. We classify the possible bifurcations for (1) in a small neighborhood of the bifurcation point in a mathematically rigorous sense. Especially, it turns out that all bifurcating solutions are unstable near the bifurcation point. However, numerical simulations suggest that there exist stable patterns. In order to capture them, we have to study bifurcations with higher codimension.

References

[1] L. A. Lugiato and R. Lefever, Spatial Dissipative Structures in Passive Optical Systems, Phys. Rev. Lett., 58 (1987), 2209–2211. [2] A. J. Scroggie, W. J. Firth, G. S. McDonald, M. Tlidi, R. Lefever and L. A. Lugiato, Pattern formation in a passive kerr cavity, Chaos, Solitons & Fractals, 4 (1994), 1323–1354. [3] P. Colet, D. Gomila, A. Jacobo and M. A. Mat´ıa, Excitability medi- ated by dissipative solitons in nonlinear optical cavities, Lect. Notes in Physics, 751 (2008), 113–135. [4] T. Miyaji, I. Ohnishi and Y. Tsutsumi, Bifurcation analysis to the Lugiato-Lefever equation in one space dimension, Physica D, 239 (2010), 2066–2083. [5] M. Golubitsky, I. Stewart and D. G. Schaeﬀer, Singularities and Groups in Bifurcation Theory II, Applied Mathematical Sciences, 69, Springer- Verlag, New York (1988).

144

A theorem on S-shaped bifurcation curve for a positone problem with convex-concave nonlinearity and its applications to the perturbed Gelfand problem Kuo-Chih Hung*, Shin-Hwa Wang

Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan

e-mail: [email protected] ∗ Abstract We study the bifurcation curve and exact multiplicity of positive solutions of the positone problem

00()+()=0, 1 1, ( 1) = (1) = 0, − ½ − where 0 is a bifurcation parameter,  2[0 ) satisﬁes (0)  0 and ()  0 for 0,and is convex-concave∈ on ∞(0 ) and is asymptotic sublinear. Under a mild condition, we prove that the∞ bifurcation curve is S-shaped on the (  )-plane. We give an application to the perturbed Gelfand problem k k∞

 00()+ exp =0, 1 1, + − ( 1) = (1) = 0, ½ − ¡ ¢ where 0 is the Frank—Kamenetskii parameter and 0 is the activation energy parameter. We show that, if  ∗ 4166 for some constant ∗,the bifurcation curve is S-shaped on the ≥( ≈ )-plane. Our results improve those in Korman and Li [1] and Wang [2].k k∞

References

[1] P. Korman, Y. Li, On the exactness of an S-shaped bifurcation curve, Proc. Amer. Math. Soc. 127 (1999) 1011—1020.

[2] S.-H. Wang, On -shaped bifurcation curves, Nonlinear Anal. 22 (1994) 1475—1485.

145

On the existence of a double S-shaped bifurcation curve with six solutions for a combustion problem

Kuo-Chih Hung, Shin-Hwa Wang∗, Chien-Shang Yu

Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan

e-mail: [email protected] ∗ Abstract We study the combustion problem with nonlinear boundary conditions given by βu u00(x) = λ exp( ), 0 < x < 1, − β+u u(0) = 0, (1)  u(1) u(1)  u0(1) + [1 ]u(1) = 0,  u(1)+1 − u(1)+1 where λ > 0 is the Frank—Kamenetskii parameter or ignition parameter,  a > 0 is the activation energy parameter, u(x) is the dimensionless tempera- βu ture, and the reaction term exp( β+u ) is the temperature dependence obeying the simple Arrhenius reaction-rate law in irreversible chemical reaction kinetics. We mainly prove that, for β > β1 6.459 for some constant β1, the bifurcation curve of the problem is double≈ S-shaped on the (λ, u )-plane and the problem has at least six positive solutions for a certaink rangek∞ of λ. We give rigorous proofs of some computational results of Goddard II, Shivaji and Lee [1].

Fig. 1. Double S-shaped bifurcation curve of (1) for β > β 6.459. 1 ≈

146

Theorem 1. (See Fig. 1.) Consider problem (1). Suppose β >

β1 6.459 for some constant β1, then the bifurcation curve of problem (1) is double≈ S-shaped on the (λ, u )-plane. Moreover, there exist positive k k∞ numbers λ3 < λ1 < λ0 < λ4 < λ2 such that

(i) If λ0 < λ < λ4, problem (1) has at least 6 positive solutions,

(ii) If λ1 < λ λ0 or λ = λ4, problem (1) has at least 5 positive solutions, ≤

(iii) If λ4 < λ < λ2 or λ = λ1, problem (1) has at least 4 positive solutions,

(iv) If λ3 < λ < λ1 or λ = λ2, problem (1) has at least 3 positive solutions,

(v) If λ > λ2 or λ = λ3, problem (1) has at least 2 positive solutions,

(v) If 0 < λ < λ3, problem (1) has a unique positive solutions.

References

[1] J. Goddard II, R. Shivaji, E. K. Lee: A double S-shaped bifurcation curve for a reaction-diﬀusion model with nonlinear boundary conditions, Boundary Value Problems, 2010, Article ID 357542, 23 pages.

147

Appearance and Disappearance of Non-symmetric Progressive Waves of Deep Water

1),* 2) Mayumi SHŌJI , Chika SHIMIZU

1),2) Department of Mathematical and Physical Sciences Japan Women’s University

* e-mail: [email protected]

Abstract We consider a problem of progressive water waves with a constant speed. It is supposed to be two-dimensional and irrotational flow of incompressible inviscid fluid. Under the effect of gravity and surface tension, it becomes the bifurcation problem which has two bifurcation parameters concerned with these two forces respectively. It is a classical problem and there have been a lot of researches mathematically and numerically. However most of them are restricted to symmetric waves and we know a few about non-symmetric waves. Numerically there are only the works of Zufiria for gravity waves on water of finite depth [2], infinite depth [3] and capillary-gravity waves of finite depth [4]. All of them are the waves with six peaks in one wave length. We made a thorough investigation in case of capillary-gravity waves of infinite depth. We confirmed the existence of similar non-symmetric waves to [4] and saw the mechanism of symmetry breaking. And furthermore we were able to obtain some other new types of non-symmetric waves.

We studied the problem using the formulation with Levi-Civita’s equation. And numerically we applied the spectral-collocation method and Keller’s method (see [1] for details). The formulation and the numerical methods we used here are quite

(A) (B)

Figure 1. p=1.5. (A) The dotted line is the branch of non-symmetric waves. The branches of symmetric waves are just on the q-axis. (B) The wave profile of a solution at (b) on the branch of non-symmetric waves. 148 different from [2-4]. Figure 1 shows an example of our numerical results of non-symmetric waves with six peaks. The left figure is the bifurcation diagram, where the vertical axis q is the bifurcation parameter related to surface tension. The horizontal axis b2 represents the asymmetric degree of non-symmetric waves, namely b2=0 if wave is symmetric. Another bifurcation parameter p related to gravity is fixed here. The branch of non-symmetric waves constitutes a closed loop (dotted line in (A)) and the branches of symmetric waves are just on the q-axis. The symmetry-breaking occurs in the following mechanism: a trivial solution a simple bifurcation of mode 6 symmetric waves a secondary bifurcation of mode 2 a tertiary bifurcation of mode 1 symmetry-breaking. We can see the looped branch of non-symmetric waves becomes smaller as p decreases and disappears when p 1.3.

Other than the above type of waves with six peaks, we were able to obtain some new types of non-symmetric waves. One is a type of non-symmetric wave with three peaks in one wave length and another one is a type with seven peaks. Maybe, if any other non-symmetric solution exist, we predict that might be the bifurcation solution of a higher mode than six at first. Hence it is unexpected and interesting for us to obtain non-symmetric waves of mode 3.

Figure 2. Non-symmetric waves of mode 3 and mode 7.

References

[1] H. Okamoto and M Shōji, The Mathematical Theory of Permanent Progressive Water-Waves, World Scientific (2001). [2] J. A. Zufiria, Weakly nonlinear non-symmetric gravity waves on water of finite depth, J. Fluid Mech., 180 (1987), pp. 371--385. [3] J. A. Zufiria, Non-symmetric gravity waves on water of infinite depth, J. Fluid Mech., 181 (1987), pp. 17--39. [4] J. A. Zufiria, Symmetry breaking in periodic and solitary gravity-capillary waves on water of finite depth, J. Fluid Mech. 191 (1987), pp. 341--372.

149

Bifurcation structure and spontaneous pattern formation in binary ﬂuid convection

1), 1),2) 1) Takeshi WATANABE ∗, Makoto IIMA , Yasumasa NISHIURA

1) Research Institute for Electronic Science, Hokkaido University 2) JST, CREST

e-mail: [email protected] ∗ Abstract We study dynamical behavior of localized pattern in a binary fluid convection. In a binary fluid mixture, thermal convection exhibits localized patterns such as steady pulse (SP), a stationary spatially localized convection cell; and periodic traveling pulse (PTP), a moving spatially localized convection cell whose shape periodically changes with time, as well as ordinary Rayleigh- Bénardtype convection. So far, PTP solution has not been obtained even numerically because the solution has two unknown variables: group velocity and temporal period in the comoving frame with the group velocity. It is important to obtain such solutions including their stability in both views of determining mathematical structure of this system and controlling numerical experiments. We succeeded in obtaining bifurcation structure of PTP, SP, and SOC solutions[2]. The method to obtain PTP solutions is established by setting an appropriate dimensional Poincarésection. For N-dimensional dynamical system, PTP solution can be determined uniquely by setting (N 2)-dimensional Poincarésection because PTP solution have time and spatial− translational invariance. In other words, a point on a torus spanned by PTP solution vector is identified uniquely by two values. Therefore we search for a solution to the N-dimensional dynamical system in (N 2)-dimensional space and thus the number of unknowns does not equal the− number of equations, however, this lack is compensated by two additional unknowns, i.e., group velocity and period, and then a closed set of equations is obtained. The Newton-Raphson method is used to find the fixed point on the Poincarésection together with the GMRes method[2]. The obtained global bifurcation diagram for Γ = 32 revealed that bifurcation branch of SP forms a snake-like structure which is frequently observed in many dissipative systems including a plane Couette flow[4] and a reaction- diffusion system[5]. Thus, various SP solutions with different numbers of convection cells coexist in a parameter range. Further, PTP solutions also coexists in the same parameter range. This fact is consistent with the result

150 of collision analysis of two counter-propagating PTPs reported by Toyabe[3], in which various states are obtained as the result of the collision depending on the phase of each PTPs. However, he obtained PTP by calculating asymptotic state of time evolution, that is, the accuracy was not enough to investigate the detailed input-output relationships of the collision. In the present study, we investigate detailed phase dependency of the PTP-PTP collision by controlling obtained solutions. Surprisingly, almost all cases result in a single PTP in the asymptotic state and SP states are observed for a few cases where the initial value is close to mirror-symmetric. This result supports the robustness of PTP. We also study how PTP is generated. Since there exist stable SP and PTP branches even though the conductive state is unstable, it is suggested that existence and stability of these solutions depend on the size of the system. We performed time evolution in a very large domain (Γ = 500) to clarify the size eﬀect and found that PTP and SP lose their stability even if they are stable when Γ = 32. The convective instability contribute to the perturbation growth, which is consistent with the result of Batiste et al[1]. After complex interactions, we observed several PTPs propagating in the same direction with non-uniform intervals. A part of this study is supported by Grant-in-Aid for Scientiﬁc Research (KAKENHI) No. 21340019 and Core Research for Evolutional Science and Technology (CREST) No. PJ74100011.

References

[1] O. Batiste, E. Knobloch, A. Alonso, and I. Mercader, Spatially localized binary-fluid convection, J. Fluid Mech. 560, 149–158 (2006). [2] T. Watanabe, K. Toyabe, M. Iima, and Y. Nishiura, Time-periodic traveling solutions of localized convection cells in binary fluid mixture, The- oretical and Applied Mechanics Japan, 59, 211–219 (2010). [3] Kazutaka Toyabe, Collision dynamics of localized convection cells in binary fluid mixture: network structure of collision orbit, Master Thesis, Graduate School of Science, Hokkaido University (2009) [in Japanese]. [4] T. M. Schneider, J. F. Gibson, and John Burke, Snakes and ladders: localized solutions of plane Couette flow, Phys. Rev. Lett. 104, 104501 (2010). [5] Y. Nishiura, D. Ueyama, Spatio-temporal chaos for the Gray-Scott model, Physica D, 150, 137–162 (2001).

151

Single-machine group scheduling problems with time-dependent learning effect and position-based setup

time learning effect Wen-Hung Kuo

Department of Information Management, National Formosa University e-mail: [email protected]

Abstract Learning and its effect on productivity is well recognized in industry. In general, the worker learns from experience in performing repetitive tasks and the learning effect manifests itself in labor productivity improvement. Therefore, scheduling research has increasingly taken the concept of learning into consideration. There are two main classes of learning effect models studied in the scheduling literature. One is position-based learning. For example, Biskup [1] introduced a scheduling model with a job-independent learning effect in which the actual processing time of a job Ji is pir = pir if it is scheduled in position r of a sequence, where pi and a ( a < 0 ) are the normal processing time and the learning rate of job Ji , respectively. He showed that the single-machine scheduling problems with the learning effect to minimize the deviation from a common due date and the total completion time are polynomially solvable. Mosheiov and Sidney [2] considered a more general model of learning effect in which the learning rates of the jobs are

ai different, i.e., the learning effects are job-dependent so that pir = pir . They showed that some scheduling problems with the job-dependent learning effect are polynomially solvable. The other is sum-of-processing-time-based learning. For example, Kuo and Yang [3] introduced a scheduling model with a time-dependent learning effect in a r−1 which the actual processing time of job J is pp1 p if it is i ir=+ i∑ [] k k =1 scheduled in position r of a sequence, where pi and p[]k are the normal processing times of job Ji and the job scheduled in position k, respectively, and a is the learning rate. They showed that the problem to minimize total completion time is polynomially solvable. Koulamas and Kyparisis [4] proposed another sum-of-processing-time-based learning model in which the actual processing time r 1 α − p ∑ k =1 []k of job Ji is ppir= i 1− n if it is scheduled in position r of a sequence, p ∑ k =1 []k 152 where pi and p[]k are the normal processing times of job Ji and the job scheduled in position k, respectively, and α ≥1. They also provided polynomial solutions for some scheduling problems. For a comprehensive survey on this stream of scheduling research, the reader may refer to Biskup [5]. Biskup [5] stated that it may be more appropriate to assume that position-based learning takes place during machine setups only, while sum-of-processing-time-based learning occurs in considering the experience that workers have gained from producing jobs. To the best of our knowledge, there is still no study focusing on the viewpoint in the survey paper [5]. Therefore, in this paper, we will consider sum-of-processing-time-based learning on job processing time and position-based learning on setup time in some single-machine group scheduling problems. The objective is to minimize the makespan. We provide two polynomial time algorithms to solve the makespan minimization problems.

References

[1] D. Biskup. Single-machine scheduling with learning considerations, European Journal of Operational Research, 115 (1999), 173-178. [2] G. Mosheiov, J.B. Sidney. Scheduling with general job-dependent learning curves, European Journal of Operational Research, 147 (2003), 665-670. [3] W.H. Kuo, D.L. Yang. Minimizing the total completion time in a single-machine scheduling problem with a time-dependent learning effect, European Journal of Operational Research, 174 (2006), 1184-1190. [4] C. Koulamas, G.J. Kyparisis. Single-machine and two-machine flowshop scheduling with general learning functions, European Journal of Operational Research, 178 (2007), 402-407. [5] D. Biskup. A state-of-the-art review on scheduling with learning effects. European Journal of Operational Research, 188 (2008), 315-329.

153

Optimal Retirement with Unemployment Risks and Market Completion Bong-Gyu Jang∗, Seyoung Park, Yuna Rhee Department of Industrial and Management Engineering, POSTECH

∗ e-mail: [email protected]

Abstract This paper investigates individual’s optimal retirement in the presence of involuntary unemployment risks and positive wealth constraint. Most existing literature on optimal retirement problem, e.g., the paper of Farhi and Panageas [1], tries to find individual’s optimal consumption, investment, and voluntary retirement time simultaneously. We add the possibility of her involuntary unemployment to the conventional set-up, and assume that the involuntary unemployment readily breaks out when a crucial exogenous shock such as the bankruptcy of the company for which she is working comes into being. To reach the goal, we introduce a complete market with frictions. In this regard we consider a complete market with a personalized unemployment insurance, and accordingly, the individual can hedge the involuntary unemployment risks by purchasing the insurance policy. However, we assume that the price of the insurance policy is costly compared with its actuarially fair price to reflect the current situation that the unemployment insurance markets are quite thin. We utilize the martingale approach to take into account an optimal stopping for voluntary retirement under the conditions above. The technical contribution of our paper is in this point of view; to find an analytic solution of the optimal retirement problem in the presence of the uncertainties of retirement time by using the martingale approach. In our model, since the stopping time of the optimal retirement should be determined endogenously under the condition that the individual might lose her job accidentally, one can say that our paper is an extension of the work of Farhi and Panageas [1]. We indeed do such job even with individual’s positive wealth constraint, which makes our model go near the real world and precludes arbitrage op- portunities. Even though a plethora of studies explored individual’s optimal retirement, as far as we know, this is the first study examined not only with the voluntary retirement but also with the involuntary unemployment. More importantly, it allows one to capture more various economic features than those of Farhi and Panageas [1]. Basically, in our model, a wealth process equipped with an unemployment insurance is considered. When πt, ct, I, k, τ, τU , and W t present the dollar

154 amount of investment in the stock, consumption rate, income rate, fixed rate of the difference between the market price and actuarially fair price of the unemployment insurance rate, voluntary retirement time, involuntary unemployment time which is governed by exponential distribution with intensity λU , and potential voluntary retirement wealth level, respectively, the wealth process W follows by t h i dWt = rWt + πt(µ − r) − ct + I1{t<τ∧τU } − k − λU {W t − Wt} dt + πtσdBt, where r and σ denotes the mean rate of return and volatility of a stock, respectively. Note that, to reflect the current unemployment insurance markets, we introduce the additional rate k representing market frictions in trad- ing the unemployment insurance policy which can leads to the effect of reduction in income I. The insurance policy helps to achieve a complete market in that involuntary unemployment risks can be hedged through the instrument. Besides, an individual gets unemployment coverage purchased for a premium of λU (W t − Wt) per unit time and a receipt of W t − Wt at unemployment. We find some interesting properties are related to λU and k in the following view points: (1) the optimal retirement wealth level, (2) optimal consumption, (3) optimal risky portfolio, (4) the expected time to retirement, and (5) the probability that the individual enters voluntary retirement before involuntary unemployment. The high intensity of involuntary unemployment and high cost of the insurance policy tend to lead the individual tries to enter voluntary retirement earlier, which causes the lower retirement wealth level and less amount of consumption but the larger proportion of stockholdings. This is because she gets lower income due to the cost and wants to hedge against the involuntary unemployment risks. Moreover, the expected time to retirement decreases as the wealth level increases since the wealth level increasingly close to threshold wealth level. Thus, the higher intensity of involuntary unemployment and cost of the insurance policy, the shorter expected time to retirement. Finally, growing wealth level makes the probability that the individual enters voluntary retirement before involuntary unemployment high, and the probability rapidly increases as the high intensity and cost.

References

[1] E. Farhi, S. Panageas: Saving and Investing for Early Retirement: A Theretical Analysis, Journal of Financial Economics, 83 (2007), 87–121.

155

Rescaled Gradient-based Methods with Fixed Stepsize for Large-scale Optimization

1,2), 2) 1) Wah June Leong ∗, Mahboubeh Farid , Malik Abu Hassan

1) Department of Mathematics, Faculty of Science, University Putra Malaysia 2) Institute for Mathematical Research, University Putra Malaysia

e-mail: [email protected] ∗ Abstract A new family of gradient-type method with improved Hessian approximation for large-scale unconstrained optimization problems is introduced. The new method resembles the Barzilai-Borwein (BB) [1] method, except that the Hessian matrix is approximated by a diagonal matrix rather than the multiple of the identity matrix in the BB method. Then the diagonal Hes- sian approximation is derived based upon the commonly used least change updating strategy for quasi-Newton methods with the added restriction that full matrices are replaced by diagonal matrices [2]. Using some appropriate matrix norms, some members of this family are introduced. Numerical experiments with some particular members of the family are performed and the results show that the proposed method yields desirable improvement.

References

[1] J. Barzilai and J.M. Borwein: Two point step size gradient methods, IMA J. Numer. Anal. 8 (1988) 141-148.

[2] M.A. Hassan, W.J. Leong and M. Farid, A new gradient method via quasi-Cauchy relation which guarantees descent, J. Comput. Appl.

Math. 230 (2009) 300-305.

156

Behavior of polygonal curves by crystalline curvature ﬂow with a driving force

1), Tetsuya Ishiwata ∗ 1) Department of Mathematical Science, Faculty of System Science and Technology, Shibaura Institute of Technology

e-mail: [email protected] ∗ Abstract In this presentation we consider the motion of planar polygonal curves governed by generalized crystalline curvature ﬂow with a driving force:

β(Nj)Vj = U g(Hj), (1) − where Vj,Nj and Hj denote an outward velocity, an outward normal vector and a crystalline curvature of the j-th facet of solution curve, respectively. Here, “facet” means a lines segment of solution curve Γ(t) and j denotes the j-th facet of Γ(t). The positive function β and U describe anF anisotropy of the mobility and a driving force, respectively. In this paper we consider the case that U > 0 and the function g = g(λ) is odd, locally Lipschitz continuous, monotone increasing in λ and has linear or superlinear growth for λ . The∼ ±∞equation (1) with g(λ) = λ is a simple model of an interface motion of crystals (See [1] and [6]). In the case when the solution region is bounded, many authors discuss the behavior of the solutions for several types of crystalline motions (See [2, 7, 3] and their references). On the other hands, in the case when the solution region is unbounded, there are a few results. Maru- tani et al. [5] shown the existence of traveling “V-shaped” solutions for the case g(λ) = λ as the limit of the smooth solution for weighted curvature flow. In [4], we consider the simple case β 1 and g(λ) = λ and show that the solution with non-V-shaped initial region≡ eventually become V-shaped under some conditions. This kind of monotonicity phenomena are already known for the case when the solutions are bounded. The most famous result is shown by M. Grayson in 1987 for smooth solution curves to curve-shortening flow. He showed that the solution curves from non-convex initial curve becomes convex in finite time. For crystalline case, it is shown in [3] for the case U = 0 that crystalline curvature of all facets of the solution polygons from non-convex initial polygon becomes non-negative in finite time. In this presentation we show the extended results in [3, 4] for generalized motion (1).

157

References

[1] S. Angenent and M.E. Gurtin: Multiphase thermomechanics with interfacial structure, 2. Evolution of an isothermal interface, Arch. Rational Mech. Anal. 108 (1989), 323–391.

[2] Y. Giga and M. E. Gurtin: A comparison theorem for crystalline evolution in the plane, Quart. J. Appl. Math. LIV (1996), 727–737.

[3] T. Ishiwata: Motion of non-convex polygons by crystalline curvature and almost convexity phenomena, Japan Journal of Industrial and Applied Mathematics, 25 (2008), no. 2, 233–253.

[4] T. Ishiwata: On the motion of polygonal curves with asymptotic lines by crystalline curvature ﬂow with bulk eﬀect, to appear in Discrete and Continuous Dynamical Systems, Series S.

[5] Y. Marutani, H. Ninomiya and R. Weidenfeld: Traveling curved fronts of anisotropic curvature ﬂows, Japan Journal of Industrial and Applied Mathematics, 23 (2006), no. 1, 83–104.

[6] J. E. Taylor: Constructions and conjectures in crystalline nondiﬀerential geometry, Proceedings of the Conference on Diﬀerential Geometry, Rio de Janeiro, Pitman Monographs Surveys Pure Appl. Math. 52 (1991), 321–336, Pitman London.

[7] S. Yazaki: Point-extinction and geometric expansion of solutions to a crystalline motion, Hokkaido Math. J. 30 (2001), 327–357.

158

Some remarks on the ﬁctitious domain method with penalty for elliptic problems

1), 1) Guanyu ZHOU ∗, Norikazu SAITO

1) Graduate School of Mathematical Sciences, The University of Tokyo

e-mail: [email protected] ∗ Abstract The principle of the fictitious domain method is to solve the problem in a larger domain (the fictitious domain) containing the domain of interest with a very simple shape. Then, the fictitious domain is discretized by a uniform mesh, independent of the original boundary. The advantage of this approach is that we can avoid the time-consuming construction of a boundary-fitted mesh. One of these approaches is the penalty fictitious domain method which is based on a reformulation of the original problem in the fictitious domain by using penalty parameter. Obviously, this approach is of use to treat time-dependent moving-boundary problems. However, it seems that penalty fictitious domain method is not studied very well at present. The main purpose of this work is to study the penalty fictitious domain method applied to these time-dependent moving-boundary problems. As a primary step towards this final end, herein we examine some new methods of error analysis for elliptic problems that can be applied to time-dependent problems with no much difficulties. We consider the original elliptic problem (Q),

1 Find u H0 (Ω) such that ∈ 1 (1) ( u, v)Ω =(f,v)Ω, v H (Ω), ∇ ∇ ∀ ∈ 0 2 where Ω R denotes a smooth bounded domain, ( , )Ω is the inner product 2 ⊂ 2 · · of L (Ω) and f L (Ω). We can ﬁnd a rectangular domain D Ω, Ω1 = ∈ ⊃ D Ω, and turn to solve the approach problem (Qǫ) with penalty coeﬃcient 0 <ǫ\ 1, ≪ 1 Find uǫ H0 (D) such that ∈ 1 1 (2) ( uǫ, v) + ( uǫ, v) =(f,v˜ )D, v H (D), ∇ ∇ Ω ǫ ∇ ∇ Ω1 ∀ ∈ 0 where f˜ is the zero extension of f into D. The error uǫ u 1,Ω is analyzed by many authors, where 1,Ω denotes the H1(Ω) norm.k − Ink [1], it is bounded by C√ǫ (C is some constant,k · k so as in the following), and in [2, 3] the sharper estimate Cǫ is achieved. In this

159 paper, we give a new way to derive the sharper estimates, as well as some other estimates, such as uǫ 1,Ω1 Cǫ. We ﬁnd our method of analysis is easy to be applied to parabolick k problems.≤ Moreover, we present some other 1 2 analysis and estimates of uǫ, which is useful for studying the H and L error between the solutions of approach problem (Qǫ) and its discretization problem, which is denoted as (Q ). ǫ,h A Cartesian mesh can be introduced to the rectangular domain D to get a uniformed triangulation h, h is the maximum diameter of the triangles of T h. Vh(D) is a subspaces of piecewise linear continuous functions subordinate T to h. Then (Qǫ,h) reads as: T

Find uǫ,h Vh(D) such that ∈ 1 (3) ( uǫ,h, vh) + ( uǫ,h, vh) =(f,v˜ h)D, vh Vh(D). ∇ ∇ Ω ǫ ∇ ∇ Ω1 ∀ ∈ In the literature, there are numerous works devoted to the study of the H1 1 error between uǫ,h and uǫ in Ω. For example, in [2], it is proved that the H error is bounded by C(ǫ + ǫ√h + h√ǫ + √h). In our work, we prove a similar result. Moreover, we show the L2 error is bounded by C(ǫ + h + √ǫh). In above, we have restricted our attention to Dirichlet boundary problem. For Neumann and mixed boundary problems we also study the approximation of approach problems( they are a little diﬀerent from the Dirichlet case), as well as the discretization problem for Neumann case. Although the main results we obtain are much similar to those of [2], we omit to present the detail here. However, as we mentioned in the beginning, our work is focus on solving parabolic problem with time-dependent domain, and all our analyzing methods are applicable well.

References

[1] A. L. Mignot: Méthodes d’approximation des solutions de certains problèmes aux limites linéaires. I, Rend. Sem. Mat. Univ. Padova, 40 (1968) 1-138.

[2] S. Zhang: Analysis of ﬁnite element domain embedding methods for curved domains using uniform grids, SIAM J. Numer. Anal. 46 (2008), no. 6, 2843-2866.

[3] B. Maury: Numerical analysis of a ﬁnite element/volume penalty method, SIAM J. Numer. Anal. 47 (2009), no. 2, 1126-1148.

160

Analysis of the ﬁnite volume approximation for a degenerate parabolic equation

1), Norikazu SAITO ∗ 1) Graduate School of Mathematical Sciences, The University of Tokyo

e-mail: [email protected] ∗ Abstract We consider the initial-boundary value problem for a degenerate parabolic equation, ut ∆f(u) = 0 in Ω (0,T ), − × u = 0 on ∂Ω (0,T ), (1)  ×  u t=0 = u0(x) on Ω, | where Ω R2 denotes a polygonal domain, T an arbitrary positive constant, ⊂  and f a non-decreasing continuous function defined on R satisfying f(0) = 0. We also consider the boundary value problem for a degenerate elliptic equation, u λ∆f(u) = g in Ω, − (2) { u = 0 on ∂Ω, where λ is a positive constant and g a function defined on R. As is well-known, Problem (1) describes, for instance, the flow of homogeneous fluid in porous media if

γ 1 f(u) = u u − (3) | | with γ > 1, the fast diﬀusion if (3) with 0 < γ < 1, and the two phase Stefan problem in enthalpy formulation if

α(u + 1) (u 1) ≤ − f(u) = 0 ( 1 < u < 1)  − β(u 1) (u 1) − ≥ with α, β > 0.  The purpose of this paper is to report some operator theoretical properties of the ﬁnite volume method (FVM) in the sense of [3] applied to (1) and (2). As is well-known, L1 theory of Brezis and Strauss [1] is of great use to deal with this problem. In this paper, we shall see that FVM is a suitable discretization method for (1) and (2) in the sense that the discrete version

161 of [1] can be applied. Consequently, we immediately deduce the generation of the nonlinear semigroup, namely, the unique existence of a solution to a semidiscrete (in space) FVM for (1). Then, we readily obtain stability results 1 in L and L∞, and order-preserving property for numerical solutions via the nonlinear semigroup theory. This is totally new approach to study FVM for degenerate elliptic and parabolic problems.

FVM is a discretization method based on local conservation properties of equations so that it is well suited for PDEs of conservation laws. Although the range of application seems to be smaller than that of the finite element method (FEM), FVM has its own advantages. For example, FVM naturally satisfies the discrete maximum principle, if a linear diffusion is considered. We recall that the discrete maximum principle in FEM holds only when some shape conditions on the triangulation are satisfied, and such a restriction often causes some difficulties. In this paper, we shall reveal another advantage of FVM through the degenerate diffusion problems and the nonlinear semigroup theory. Though we shall restrict our consideration to the two dimensional polygon in this paper, it is not difficult to extend those results to smooth domains and the three dimensional cases. This is a joint work with T. Suzuki and A. Mizutani, and a sequel of our previous work [4].

References

[1] H. Brezis and W. Strauss: Semi-linear second-order elliptic equations in L1, J. Math. Soc. Japan, 25 (1973), 565–590.

[2] M. G. Crandall and T. Liggett, Generation of semi-groups of nonlinear transformations on general Banach spaces, Amer. J. Math. 93 (1971), 265–293.

[3] R. Eymard, T. Gallou¨etand R. Herbin: Finite Volume Methods, Hand- book Numer. Anal. VII, Elsevier, 2000, 713–1020.

[4] A. Mizutani, N. Saito and T. Suzuki: Finite element approximation for degenerate parabolic equations. An application of nonlinear semigroup theory, ESAIM: Math. Modell. Numer. Anal. 39 (2005), 755–780.

162

FEM analysis of the Stokes equations under boundary conditions of friction type Takahito Kashiwabara 1) Graduate School of Mathematical Sciences, The University of Tokyo

e-mail: [email protected] ∗ Abstract

1 Boundary conditions of friction type

We are concerned with the stationary Stokes equation in a two-dimensional domain Ω R2, that is, ⊂ ν∆u + p = f in Ω, div u = 0 in Ω, (1) − ∇ where ν, u, p, and f stand for a viscosity constant, velocity ﬁeld, pressure, and external force, respectively. As for the boundary, we assume that Γ := ∂Ω is polygonal, that Γ is a union of two non-overlapping parts Γ0, Γ1, namely,

Γ = Γ0 Γ1, Γ0 Γ1 = , ∪ ∩ ∅ and that Γ1 coincide with whole one side of the polygon Ω. We associate with (1) the adhesive boundary condition u = 0 on Γ0, while on Γ1 we impose one of the following nonlinear boundary conditions:

un = 0, στ g, στ uτ + g uτ = 0, (SBCF) | | ≤ | | which is called the slip boundary condition of friction type, and

uτ = 0, σn g, σnun + g un = 0, (LBCF) | | ≤ | | which is called the leak boundary condition of friction type. Here, un (resp. u ) is the normal (resp. tangential) component of u, σ (resp. σ ) is the τ τ n tangential (resp. normal) components of the stress vector, and g is a given positive number. In (SBCF) (resp. (LBCF)), if the magnitude of the tangential (resp. normal) stress is strictly less than g, then u = 0; if it reaches the threshold g, then it can happen that uτ = 0 (resp. un = 0). This is why the above boundary conditions are called of̸ “friction type”.̸ In this study, ﬁnite element approximation of (1) under (SBCF) or (LBCF) is examined. We propose approximate problems formulated by a variational inequality, prove the unique existence of an approximate solution, establish error analysis, and consider an Uzawa-type iteration method for numerical implementation.

163

2 Continuous and approximate problems

Due to space limitations, we restrict ourselves to (SBCF). Let Th be a { } regular family of triangulations of the polygon Ω. Let Pk(T ) denote the set of the polynomial functions of degree k, deﬁned on T Th. We introduce: ≤ ∈ 1 2 Vn = v H (Ω) v = 0 on Γ0, vn = 0 on Γ1 , ∈ ◦ 2 Q = {q L (Ω) ( q, 1)0 = 0 , } ∈ 2 2 Vnh = {vh C(Ω) vh T P2(}T ) ( T Th), vh = 0 on Γ0, vhn = 0 on Γ1 , ∈ | ∈ ∀ ∈ ◦ Qh = {qh C(Ω) qh T P1(T )( T Th), (qh, 1)0 = 0 , } ∈ | ∈ ∀ ∈ ◦ Now, the{ continuous and approximate problems, which seek} (u, p) Vn Q ∈ × ◦ and (uh, ph) Vnh Qh respectively, are formulated by the following variational inequalities:∈ ×

a(u, v u) + b(v u, p) + j(v) j(u) (f, v u)0 ( v Vn), (VI) − − − ≥ − ∀ ∈ { b(u, q) = 0 ( q Q◦ ). ∀ ∈ a(uh, vh uh)+b(vh uh, ph)+jh(vh) jh(uh) (f, vh uh)0 ( vh Vnh), (VIh) − − − ≥ − ∀ ∈ ◦ {b(uh, qh) = 0 ( qh Q ). ∀ ∈ h ∂u Here, we deﬁne the bilinear forms a, b by a(u, v) = ν 2 ∂ui + j 2 i,j=1 Ω ∂xj ∂xi ∂v ∂vi + j dx, b(v, q) = divv q dx. The nonlinear∑ term j∫, and( its Simp-) ∂xj ∂xi − Ω son approximation j , are deﬁned as ( ) h ∫

j(v)=g vτ ds (v Vn), Γ1 | | ∈ ∫m MiMi+1 jh(vh)=g | | vhτ (Mi) + 4 vhτ (M ) + vhτ (Mi+1) (vh Vnh), 6 i+1/2 i=1 | | | | | | ∈ ∑ ( ) where Mi’s denote the vertices and the midpoints of two adjacent vertices, contained in Γ1 and numbered in ascending order. We will give details of our results, especially error analyses and numerical examples, in the present talk. See also our preprint [1], where all of the theoretical and numerical results are described in detail.

References

[1] T. Kashiwabara: On a ﬁnite element approximation of the Stokes problem under leak or slip boundary conditions of friction type, arXiv:1012.4982.

164

A mesh generator using a self-replicating system

1), 2) 3) Hirofumi NOTSU ∗, Masahiro YAMAGUCHI Daishin UEYAMA 1) Meiji Institute for Advanced Study of Mathematical Sciences 2) Graduate School of Science and Technology, Meiji University 3) School of Science and Technology, Meiji University e-mail: [email protected] ∗ Abstract In this paper, a new triangular-mesh generator for numerical simulations, especially for ﬁnite element analyses, is presented. For numerical simulations, meshes are necessary. However, it is not easy to generate good meshes, because we need to add nodes, move the nodes and change connections of triangles according to the shape of domain and required mesh size, where the details are not simple. Now, complex patterns in nature have been modeled by nonlinear systems, which have been mathematically and computationally analyzed, e.g., see [1]. We focus on so called self-replicating system. One of the simplest self-replicating systems is the following Gray-Scott model [2], which is a nonlinearly-coupled system of two partial diﬀerential equations and represents a chemical reaction.

∂u 2 = Du∆u uv + F (1 u), ∂t − − (1)  ∂v  = D ∆v + uv2 + (F + k)v,  ∂t v where Du and Dv are diffusion constants and F and k are parameters. For a parameter region, the system (1) give spot patterns. The properties of the patterns to be created are as follows. The domain is filled with spots. • The distances between spots are almost equal. • The spots fit the domain shape. • It is important that these properties are automatically realized in the system. Using the system (1) and considering the spots as nodes, we generate a mesh. Let Ω be a given domain and Γ be the boundary of Ω. For the given domain Ω, what we need for computations is a mesh h = Kl l, where Kl is a triangle (2D) or a tetrahedra (3D) and h is a representativeT { } mesh size. Let Pi i be nodes of the mesh h. We propose a triangular-mesh generator using{ the} system (1), whose procedureT is as follows.

165

1. The domain Ω is given. 2. Artificial spots are put on the boundary Γ as an initial condition of the system (1). 3. The system (1) is numerically solved in an artificial rectangle domain Ω˜ ( Ω) by a finite difference method (FDM). ⊃ 4. A set of nodes Pi i is made by the created spots. { } 5. The Delaunay triangulation [3] is applied for the set of nodes and a set of triangles h = Kl l is made. T { } We note that a rough computation is okay in the third process, and that the mesh size h can be controlled by the ratio of diffusion constants Du and Dv. Figure 1 shows snapshots of a mesh generation. It is observed that a good mesh has been generated for a complex domain. Since the procedure is simple, and complex processes, e.g., smoothing, increasing and decreasing nodes and so on, are solved by the self-replicating system, the mesh generator is useful.

Figure 1: Snapshots of a mesh generation, a given domain (left), created spots (center) and a generated mesh (right).

References

[1] Y. Nishiura, Far-from-Equilibrium Dynamics, American Mathematical Society, Providence, 2002. [2] J. Pearson: Complex patterns in a simple system, Science, 261 (1993) 189–192. [3] T. Taniguchi, Automatic mesh generation for FEM -Application of De- launay triangulation, Morikita, Tokyo, 1992 (in Japanese).

166

A Remark on Numerical Instability of Complex Inverse Laplace Transforms using Multiple-Precision Arithmetic

1), Hiroshi FUJIWARA ∗

1) Graduate School of Informatics, Kyoto University

∗ e-mail: [email protected]

Abstract We give a remark on numerical computation of the inverse Laplace transform based on the Bromwich integral. In particular numerical instability are discussed using multiple-precision arithmetic in Hosono’s inversion method [1]. ∞ st The Laplace transform Lf(s) = F (s) = 0 e− f(t) dt appears as a fundamental method in mechanics, engineering, mathematical science, and R1 mathematical ﬁnance. And its inversion f = L− F also plays an important role. Under some assumptions, the Bromwich integral

1 c+iT f(t)= lim etzF (z) dz, 2πi T c iT →∞ Z − gives the inverse formula, and several numerical treatments have been proposed [2, 3]. Hosono has proposed a numerical scheme

k 1 µ eσ − 1 f(t) ≈ f (t)= F + A F , σ,N t n 2k+1 µ,ν k+ν n=1 ν=0 ! X X where σ > 0 is an approximation parameter of the integral kernel etz, N = n 1 1 (k,µ) is a truncation parameter, Fn =(−1) Im F (zn), zn = t {σ+ n − 2 πi}, µ +1 Aµ,µ = 1, and Aµ,ν 1 = Aµ,ν + . − ν In the numerical treatments of the Bromwich integral, the numerical instability prevents reliable and accurate numerical computations. In Hosono’s 1 method the numerical solution is given by fσ,N = Lσ,N− FN . The actual numerical computations are processed with ﬂoating-point arithmetic with precision p, and the actual numerical result is

1 fσ,N,p = Lσ,N,p− FN,p.

The diﬀerence |fσ,N − fσ,N,p| is caused by ﬂoating-point arithmetic, and it is called a rounding error in the precision p.

167

We propose the use of multiple-precision arithmetic [4] for quantitative analysis of rounding errors. Figure 1 shows quantity of rounding errors by f (t) − f (t) σ,N,p σ,N , ulp fσ,N,p(t)

where ulp(x) is the unit in the last place of a ﬂoating-point value x. From the ﬁgure, the magnitude of rounding errors in the numerical solution is approximately 1016 for σ = 40, and the standard double precision arithmetic, which has 16 decimal digits precision, is not enough for the reliable computations.

1e+25 σ = 40 σ = 20 σ = 10 1e+20

1e+15

1e+10

100000

rounding error [ULP] (logarithmic scale) 1 0 5 10 15 20 25 30 35 40 t

Figure 1: Rounding Errors in Numerical Solution by Hosono’s Method

The truncation error |f − fσ,N | tends to zero as σ tends to inﬁnity in Hosono’s method. We need large σ to reduce truncation errors, which leads the numerical instability. Multiple-precision arithmetic is eﬀective for accurate and reliable numerical computations.

References

[1] T. Hosono: Numerical Inversion of Laplace Transform (in Japanese), IEEJ Trans. FM, 99 (1979), 494–50. [2] A. M. Cohen: Numerical methods for Laplace transform inversion, Springer, 2007. [3] D. Sheen, I. H. Sloan, and V. Thom´ee: A parallel method for time discretization of parabolic equations based on Laplace transformation and quadrature, IMA J. Num. Anal. 23 (2002), 269–299.

[4] exﬂib: http://www-an.acs.i.kyoto-u.ac.jp/~fujiwara/exflib

168

Numerical Laplace inversion using multi-precision

Jiwoon Kim∗ Dongwoo Sheen†

Abstract Numerical inversion of Laplace transform has attracted intensive attention from scientists and engineers as well as mathematicians. Various numerical methods have been devised. In particular, we review several efficient methods based on deformed contour that have been developed for about ten years. A comparative study using multi-precision will be exposed. We examine the effects of roundoff errors, computation is carried out both in double precision and a multi-precision, the latter which provides better understanding of the numerical Laplace inversion algorithms. In this talk we also review and compare the most efficient methods for numerical Laplace inversion, in particular, using multi-precision. In this study, several methods have been classified and interpreted as the contents related to three issues: (i) the choice of contour, (ii) its parameterization, and (iii) numerical quadrature rule.

∗Department of Mathematics, Seoul National University, Seoul 151-747, Korea; E-mail: ji- [email protected] †the same address; also, Interdisciplinary Program in Computational Science & Technology, Seoul Na- tional University, Seoul 151–747, Korea; E-mail: [email protected]

169

CPU-GPU Hybrid Approaches in Multifrontal Methods for Large and Sparse Linear System

1), 1) 2) Chenhan D. Yu ∗, Weichung Wang , Dan’l Pierce

1) Department of Mathematics, National Taiwan University, 10617, Taiwan 2) Access Analytics Intl, Redmond, Washington 98073, USA

E-mail: [email protected] ∗ Abstract Solving large-scale sparse linear systems Ax = b is at the heart of various scientiﬁc and engineering computations. To solve these problems, direct methods relying on symbolic and numerical factorizations have been used for decades and remain popular choices, as evidenced by the many well-designed algorithms and well-developed software packages that are available. Most recently, a growing number of computing programs have tried to use graphic processing units (GPU) to accelerate the computations, due to its highly parallel computational capability. We intend to investigate how we can inte- grate a GPU with a central processing unit (CPU) system to accelerate the direct solving process for sparse linear systems. Among various direct methods, we focus on the multifrontal method particularly [1, 2]. The choice of multifrontal method is mainly motivated by the following observation. A multifrontal method uses a factorization tree to transform a large sparse linear system problem into many operations involving smaller dense matrices. These dense operations are promising candidates that can be performed on a GPU to gain possible time savings without down- grading the accuracy. As all of the frontal matrices are dense and the main computational tasks involving these dense frontal matrices are actually level 2 and level3 Basic Linear Algebra Subprograms (BLAS) computations, we can accelerate the whole multifrontal process if we can accelerate the BLAS computations via a GPU. A GPU contains a large number of computational cores on a single board. These cores can work simultaneously and thus justify the ideas to accelerate the BLAS and other operations in a multifrontal method via a GPU. How- ever, it is important to realize that a GPU is not a standalone platform, but a device driven by a CPU. Consequently, the data must be transferred between the host CPU and the device GPU. The communication cost between the CPU and GPU thus plays a critical role in the overall performance. For unsymmetric linear systems, we analyze the multifrontal method from both an algorithmic and implementation perspective to see how a GPU, in

170 particular the NVIDIA Fermi series, can be used to accelerate the computations. Our main accelerating strategies include (i) performing BLAS on both CPU and GPU, (ii) improving the communication efficiency between the CPU and GPU by using page-locked memory and asynchronous memory copy, and (iii) a modified algorithm that reuses the memory between different GPU tasks and sets thresholds to determine whether a certain tasks be performed on GPU. For symmetric positive definite systems, the step-by-step operations of the numerical factorization can be exactly predicted after the symbolic factorization is done. However, data communication between CPU and GPU remains a critical issue that can affect the overall performance significantly. We analyze the symmetric multifrontal method to find the minimal communication requirements between CPU and GPU for symbolic factorization, coefficient matrix, numerical factorization and triangular solver. Besides, we propose a timing-optimized algorithm that further improve the overall performance. In the algorithm, some data communications are hidden by overlapping the data transfers with other operations. In addition, we show how shorter overall computational time can be achieved by setting thresholds to let small frontals be factored in CPU, even extra data communication traffics are introduced. The proposed acceleration strategies are implemented by modifying UMF- PACK (http : //www.cise.ufl.edu/research/sparse/umfpack/) and TAUCS (http : //www.tau.ac.il/ stoledo/taucs/) which is an unsymmetric and symmetric multifrontal linear system solver, respectively. Numerical results for a series of three-dimensional Poisson problems and standard large test problems will be presented. The results show that the CPU-GPU hybrid approach can accelerate the unsymmetric and symmetric multifrontal solver, especially for computationally expensive problems.

References

[1] T. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method, ACM Transactions on Mathematical Software (TOMS) 30 (2) (2004) 195.

[2] J.Liu,The multifrontal method for sparse matrix solution: Theory and practice, SiamReview 34 (1) (1992) 82109.

171

Design and implementation of a multiple-precision system on GPU

1), Shugo Manabe ∗

1) Graduate School of Informatics, Kyoto University

∗ e-mail: [email protected]

Abstract In the present research, we design and implement a multiple-precision system which is useful for parallel computation. In scientific computation, there are some hard problems to obtain desired accuracy of the results due to accumulation of rounding errors. We propose adopting multiple-precision system, but arithmetic speed is slow on such systems. In scientific computation, vector operations usually occupy most of the computation time. GPU (Graphics Programming Unit) has highly parallel structure. In the current research, we design and implement the multiple-precision system on GPU. GPU is a specialized microprocessor that offloads and accelerates 3D or 2D grahics rendering from CPU (Central Processing Unit). Modern GPU is not only a powerful graphics engine but also a highly parallel programmable processor. Nowadays application of GPU to scientific computation develops. GPU has a large number of cores. For example, Intel Core i7-950 (CPU) has only 4 cores, on the other hand Geforce GTX 480 (GPU) has 480 cores. Now we have implemented 1) vector operations, 2) decimal input and output, 3) type conversion, and 4) summation and inner product on the proposed system. In particular, 1) and 4) are optimized for GPU. The following are typical vector operations.

Z[i]= X[i]+ Y [i] (i =0, 1,...,N − 1) (1)

Z[i]= X[i] − Y [i] (i =0, 1,...,N − 1) (2) Z[i]= X[i] × Y [i] (i =0, 1,...,N − 1) (3) Z[i]= X[i] ÷ Y [i] (i =0, 1,...,N − 1) (4) where X[i],Y [i], and Z[i] are multiple-precision numbers.

We do not have to care how to use GPU. For example, if we compute (1), we only write

vectZ = vectX + vectY;

172 on the C++ programming language. We can use the proposed system without knowledge of GPU. Figure 1 and Figure 2 compare the proposed system with exﬂib (multiple- precision arithmetic library)[1, 2]. On the proposed system, as the number of elements of vector gets larger, you can obtain more performance.

×106 operations per sec

N=106 3833.5(×130.3) 5 × proposed system Geforce GTX 480 N=10 3332.8( 113.3) N=104 1875.3(×63.8) N=103 218.3(×7.4) exﬂib Athlon64 2.20GHz 29.4(×1.00) Core i7 2.66GHz 14.8(×0.50)

- Fast Figure 1: comparison of speed of addition with 100 decimal precision

×106 operations per sec

N=106 307.7(×40.9) 5 × proposed system Geforce GTX 480 N=10 303.7( 40.4) N=104 250.1(×33.3) N=103 54.4(×16.9) exﬂib Athlon64 2.20GHz 7.52 (×1.00) Core i7 2.66GHz 3.22 (×0.50)

- Fast Figure 2: comparison of speed of multiplication with 100 decimal precision

References

[1] exﬂib home page; http://www-an.acs.i.kyoto-u.ac.jp/∼fujiwara/ exflib/index.html [2] Hiroshi Fujiwara : Design and implementation of a fast multiple- precision environment for scientiﬁc computation (in Japanese), Jourm- nal of the Society of Instrument and Control Engineers , vol. 49, No. 5 (2010) pp. 285–290.

173

On Optimality Theorems for Robust Multiobjective Optimization Problems Gue Myung Lee

Department of Applied Mathematics, Pukyong National University, 608-737, Korea

e-mail: [email protected]

Abstract Recently, many authors have studied roust (scalar) optimization problems ([1-12]). In this talk, the robust approach (the worst-case approach) for the standard multibjective optimization problem (MP) is considered. We define three kinds of robust efficient solutions for the robust counterpart (RMP) of the uncertain multiobjective optimization problem (UMP) which consists of more than two objective functions with uncertainty data and constraint functions with uncertainty data. We give necessary optimality theorems for weakly and properly robust efficient solutions for (RMP). This talk is based upon the paper ”Robust Multiobjective Optimization” written by Daishi Kuroiwa and Gue Myung Lee.

References

[1] A. Beck and A. Ben-Tal, Duality in robust optimization: primal worst equals dual best, Oper. Res. Lett., 37 (2009), 1-6.

[2] A. Ben-Tal and A. Nemirovski, Robust optimization—methodology and applications, Math. Program., Ser B, 92 (2002), 453-480.

[3] A. Ben-Tal, L.E. Ghaoui and A. Nemirovski, Robust Optimization, Princeton Series in Applied Mathematics, 2009.

[4] A. Ben-Tal and A. Nemirovski, A selected topics in robust convex optimization, Math. Progr. B, 112 (2008), 125-158.

[5] D. Bertsimas and D. Brown, Constructing uncertainty sets for robust linear optimization, Oper. Res., 57 (2009), 1483-1495.

[6] D. Bertsimas, D. Pachamanova and M. Sim, Robust linear optimization under general norms, Oper. Res. Lett., 32 (2004), 510-516.

174

[7] V. Jeyakumar and G. Li, Characterizing robust set containments and solutions of uncertain linear programs without qualiﬁcations, Oper. Res. Lett., 38(2010), 188-194.

[8] V. Jeyakumar and G. Y. Li, Robust Farkas’ lemma for uncertain linear systems with applications, Positivity, DOI 10. 1007/s11117-010-0078-4.

[9] V. Jeyakumar and G. Y. Li, Strong duality in robust convex programming: complete characterizations, SIAM J. Optim., 20(2010), 3384- 3407.

[10] V. Jeyakumar, G. Y. Li and G. M. Lee, A robust von Neumann minimax theorem for zero-sum games under bounded payoﬀ uncertainty, Oper. Res. Lett., 39(2011), 109-114.

[11] V. Jeyakumar, G. Y. Li and G. M. Lee, Robust duality for generalized convex programming problems under data uncertainty, to appear in Nonlinear Anal..

[12] G. Y. Li, V. Jeyakumar and G. M. Lee, Robust conjugate duality for convex optimization under uncertainty with application to data classi- ﬁcation, Nonlinear Anal., 74(2011), 2327-2341.

175

The Resolvent average on symmetric cones Sangho Kum1),∗, Yongdo Lim2)

1) Department of Mathematics Education, Chungbuk National University, 2) Department of Mathematics, Kyungpook National University

∗ e-mail: [email protected]

Abstract Recently Bauschke et al. introduced a very interesting and new notion of proximal average in the context of convex analysis, and studied this subject systemically in [3–7] from various viewpoints. In addition, this new concept was applied to positive semidefinite matrices under the name of resolvent average, and basic properties of the resolvent average are successfully established by themselves from a totally different view and techniques of convex analysis rather than the classical matrix analysis [8]. Inspired by their works and the well-known fact that the convex cone of positive definite matrices is a typical example of a symmetric cone (self-dual homogeneous convex cone), we study the resolvent average on symmetric cones, and derive corresponding results in a different manner based on a purely Jordan algebraic technique. By the way, the functional f(x) = − ln det(x) on a symmetric cone is a self-scaled barrier. Indeed, this logarithmic barrier function f(x) lies at the heart of interior point methods in optimization over symmetric cones [1, 12, 13, 14]. In regard to this fact, we provide a variational characterization of the resolvent average as a unique minimizer of a Bregman distance induced by the parameterized logarithmic barrier functional fµ(x) = −ln det(x + µe).

References

[1] M. Baes, Convexity and diﬀerentiability properties of spectral functions and spectral mappings on Euclidean Jordan algebras, Linear Algebra Appl. 422 (2007), 664–700.

[2] H. H. Bauschke, E. Matouskova and S. Reich, Projection and proximal point methods: convergence results and counterexamples, Nonlinear Anal. TMA 56 (2004), 715–738.

[3] H. H. Bauschke, Y. Lucet and X. Wang, Primal-dual symmetric intrinsic methods for ﬁnding antiderivatives of cyclically monotone operators, SIAM J. Control Optim. 46 (2007), 2031–2051.

176

[4] H. H. Bauschke, Y. Lucet, and M. Trienis, How to transform one convex function continuously into another, SIAM Rev. 50 (2008), 115–132.

[5] H. H. Bauschke, R. Goebel, Y. Lucet, and X. Wang, The proximal average: basic theory, SIAM J. Optim. 19 (2008), 766–785.

[6] H. H. Bauschke and X. Wang, The kernel average for two convex functions and its applications to the extension and representation of monotone operators, Trans. Amer. Math. Soc. 361 (2009), 5947–5965.

[7] H. H. Bauschke, Y. Lucet and M. Trienis, The piecewise linear-quadratic model for computational convex analysis, Comput. Optim. Appl. 43 (2009), 95–118.

[8] H. H. Bauschke, S. M. Moﬀat and X. Wang, The Resolvent average for positive semideﬁnite matrices, Linear Algebra Appl. 432 (2010), 1757– 1771.

[9] S. Kim, J. Lawson and Y. Lim, The matrix geometric mean of parameterized, weighted arithmetic and harmonic means, submitted.

[10] J. Faraut and A. Koranyi, Analysis on Symmetric Cones, Clarendon Press, Oxford, 1994.

[11] Y. Lim, Birkhoﬀ formula for conformal compressions of symmetric cones, Amer. J. Math. 125 (2003), 167-182.

[12] Y. E. Nesterov and M. J. Todd, Self-scaled barriers and interior-point methods for convex programming, Math. Oper. Res. 22 (1997), 1-42.

[13] Y. E. Nesterov and M. J. Todd, On the Riemannian geometry deﬁned by self-concordant barriers and interior-point methods, Found. Comput. Math. 2 (2002), 333-361.

[14] Y. E. Nesterov and A. Nemirovski, Primal central paths and Riemannian distances for convex sets, Found. Comput. Math. 8 (2008), 533–560.

177

Optimizing Latin Hypercube Designs by Particle Swarm with GPU Acceleration Dai-Ni Hsieh1),∗, Ray-Bing Chen2), Ying Hung3), Weichung Wang1)

1) Department of Mathematics, National Taiwan University, Taiwan 2) Department of Statistics, National Cheng Kung University, Taiwan 3) Department of Statistics and Biostatistics, Rutgers University, USA

∗ E-mail: [email protected]

Abstract Computer and physical experiments of many kinds are important tools in scientific researches. As many of these experiments are time consuming, it is important to design the experiment carefully with small number of experimental points. In order to explore an experimental region efficiently, the experimental design we used in a computer experiment is required to satisfy certain properties like space-filling and non-collapsing. Based on these two properties, a space-filling Latin hypercube design (LHD) is an appropriate and popular choice. Formally, an n-run and k-factor LHD is an n by k matrix. Each row of the matrix corresponds to a design point, and each column is a permutation of {1, 2, . . . , n}. Note that the total number of n-run and k-factor LHDs is (n!)k−1, which grows fast as n or k increase. Although a LHD is guaranteed to be non-collapsing, a randomly generated LHD is usually not space-filling. To satisfy the space-filling property, Morris and Mitchell [1] suggested applying the maximin criterion to LHDs, i.e. maximizes the minimal pairwise distance of design points. Furthermore, they proposed the extended maximin criterion to break ties among optimal maximin LHDs . In order to use the extended maximin criterion in a easier way, they introduced the φp criterion, 1 n−1 n  p X X −p φp =  dij  , (1) i=1 j=i+1 where p is a positive integer and dij is the distance between the i-th and j-th design points. When p is large enough, φp ranks LHDs in the same way as the extended maximin criterion does. In this paper, we search the optimal LHD by minimizing φp with respect to all possible LHDs. The main challenge of this work arises from the huge number of feasible points in the discrete optimization problem min φp. To tackle the problem, we modify the particle swarm optimization (PSO) [2] framework to fit it into the LHD optimization problem. PSO is a population based stochastic

178 heuristic methodology and is capable of solving high-dimensional optimization problems with multiple optima. The key idea of PSO is that all particles cooperate to ﬁnd the optimum in the search domain through social interaction. However, the original PSO is to solve continuous optimization problems. Following the structure of PSO, we alter the step of position update such that the movement of a particle is from one LHD to another LHD. Moreover, we use graphic processing unit (GPU) to accelerate the proposed PSO. To assess the quality of LHDs we found, the squared minimum distance of optimal LHDs generated by PSO and other approaches are presented in Table 1. In the table, the column UB shows the upper bounds obtained by van Dam et al. [3]. For the column Rand, we show the best values of randomly generated LHDs whose total number is 100 times the number of LHDs visited in PSO. The column Web are the results from http://www.spaceﬁllingdesigns.nl, which contains optimal maximin LHDs of several papers. Comparing with random LHDs, PSO always gets better results. If we compare with the results from the website, PSO is capable of achieving the same value and even making some improvements. For others that are not available from the website, the results obtained by PSO are near to the upper bounds. In addition to the better squared minimum distance, we usually get more than 20X speedup on a NVIDIA Telsa C2070 GPU comparing to the CPU version that is run on a HP BL460cG6 workstation equipped with Intel Xeon X5570 2.93GHz CPU. These results together with the acceleration of GPU suggest that the proposed PSO is promising to solve even larger problems.

Table 1: Optimal results comparison. (A higher value is better.) n k UB Rand Web PSO n k UB Rand Web PSO 6 10 70 65 68 68 7 10 93 83 90 90 6 11 77 72 74 75 7 11 102 91 N/A 100 6 12 84 79 82 84 7 12 112 101 N/A 109 6 13 91 85 89 89 7 13 121 109 N/A 119 6 14 98 92 N/A 96 7 14 130 118 N/A 128 6 15 105 99 N/A 104 7 15 140 128 N/A 138

References [1] M.D. Morris, T.J. Mitchell: Exploratory designs for computational experiments, J. Statist. Plann. Inference, 43 (1995), 381–402. [2] J. Kennedy, R.C. Eberhart: Particle swarm optimization, Proc. IEEE international conference on neural networks, 4 (1995), 1942–1948. [3] E.R. van Dam, G. Rennen, B. Husslage: Bounds for maximin Latin hypercube designs, Oper. Res., 57 (2009), 595–608. 179

Maximum Entropy Approach For Optimal Repairable Mx/G/1 Queue With Bernoulli Feedback And Setup M. Jain

Indian Institute of Technology, Roorkee-247667 (India)

Abstract Queueing theory was developed to predict the behavior of various service systems wherein the server may experience unpredictable breakdowns. Therefore queueing systems with unreliable servers reflect more practical situations that we face in real time systems. Bernoulli feedback queueing models have been developed to analyze many stochastic service systems such as telecommunication systems, manufacturing and production systems, packet transmission, etc. In such type of systems, the customers are fed back again to the head of the queue in case if he is not satisfied with the service provided by the server in the first attempt. This process is continued until the customer is fully satisfied and leaves the system after the completion of the service. The model developed in this paper has been designed to analyze the behavior of Mx/G/1 queueing system with balking. The server initiates the service only if there are N customers accumulated in the system. To make the model more versatile, the assumptions of unreliable server, general phase repair, Bernoulli feedback, startup and setup times are taken into consideration. Using the supplementary variable technique and the generating function method, the stationary distribution of the queue length and various other queueing characteristics are established. Using principle of maximum entropy (PME), one can propose a feasible method for approximately analyzing many complex queueing problems. The idea behind this approach is to develop a maximum entropy analysis which in turn provides the widest probability distribution subject to the known mean constraints. The PME used for estimating the probabilistic information measures has been employed to obtain queue size distribution of various queueing systems in different frameworks. Using the maximum entropy approach, we intend to determine the approximate results for the steady-state probability distributions of the queue length. The method of Lagrange’s undetermined multipliers to maximize the entropy function subject to the relevant constraints is used. Moreover, we derive the approximate formulae for the expected waiting time of an arbitrary customer in the queue. A comparative study between the exact results obtained using supplementary variable technique and the approximate results obtained using maximum entropy approach has been facilitated by taking numerical illustration. The comparative analysis facilitate an insight to the practitioners and system engineers to make appropriate choice of the service time distribution and repair time distribution, which give the minimum expected waiting time. The numerical illustration provided indicate that the PME is accurate enough for practical purposes and provides a useful method to cope up with many complex queueing congestion situations which are not easily explored by using classical analytical techniques.

180

Application of the Bees Algorithm to Multi-Objective Optimization Engineering Problems

A.Ghanbarzadeh1),* , A.Moradi2) , A.Mirzakhani Nafchi3)

1) School of Mechanical Engineering, Shahid Chamran University, Ahvaz, IRAN

2) Graduate School of Mechanical Engineering, Shahid Chamran University, Ahvaz, IRAN

3) Graduate School of Mechanical Engineering, Shahid Chamran University, Ahvaz, IRAN

*[email protected], [email protected], [email protected]

Abstract

This paper describes application of the Bees Algorithm [1] to multi-objective optimization problems. The Bees Algorithm is a search procedure inspired by the foraging behavior of honey bees [2]. In the present paper, two standard mechanical design problems, the design of a gear train design and the design of coil springs, were used to benchmark the Bees Algorithm to compare with other optimization techniques. Multi-objective optimization using the Bees Algorithm is developed in order to obtain a set of geometric design parameters, lead to find optimum solution. Multi- objective optimization procedure yields a set of non-dominated solutions, called Pareto optimal set, each of which is a trade-off between objectives and can be selected by the user, regarding the application and the project’s limits. The presented work takes care of numerous geometric parameters in the presence of logical constraints. Study results showed that the approach using the Bees Algorithm, found better solutions than traditional calculus-based algorithm, BFGS. The coil spring design problem consists of minimizing the weight of a tension/compression spring subject to constraints on shear stress, surge frequency and minimum deflection. The design variables are the mean coil diameter D ; the wire diameter d, and the number of active coils N [3]. The objectives of the gear train design is to find the number of teeth in each of the four gears so as to minimize (i) the error between the obtained gear ratio and a required gear ratio of 1/6.931 and (ii) the maximum size of any of the four gears. Since the number of teeth must be integers, all four variables are strictly integers [3].

181

The Bees Algorithm has been successfully applied to various benchmark and standard engineering optimization problems. Numerical results reveal that the proposed algorithm can find better solutions when compared to other methods and is a powerful search algorithm for various engineering optimization problems.

References

[1] Pham D.T., Ghanbarzadeh A., Koc E., Otri S., Rahim, S., and Zaidi M., 2005. “Technical Note: The Bees Algorithm”. Technical Report No MEC 0501, Manufacturing Engineering Centre, Cardiff University: Cardiff. [2] Von Frisch, K. Bees: their vision, chemical senses and language, revised edition, 1976 (Cornell University Press,Ithaca, NY). [3] Deb, K., Pratap, A., and Moitra, S., 2000. “Mechanical Component Design for Multiple Objectives Using Elitist Non-Dominated Sorting GA”. Technical Report No 200002, Indian Institute of Technology: Kanpur, India. pp.10.

182

A quadrilateral Morley element for biharmonic equations Chunjae Park1),∗, Dongwoo Sheen2)

1) Department of Mathematics, Konkuk University 2) Department of Mathematics, Seoul National University

∗ e-mail: [email protected]

Abstract

We consider the biharmonic problem in a polygonal domain Ω ⊂ R2, ∆2u = f in Ω, u = 0 on ∂Ω, (1) ∂u ∂ν = 0 on ∂Ω.

2 The variational formulation of (1) is to ﬁnd a solution u ∈ H00(Ω) such that

2 a(u, v) = (f, v), ∀v ∈ H00(Ω), (2) where ∂u H2 (Ω) = {u ∈ H2(Ω) | u = = 0 on ∂Ω}, 00 ∂ν 2 2 and the bilinear form a : H00(Ω) × H00(Ω) → R which is deﬁned by Z 2 a(u, v) = uxxvxx + 2uxyvxy + uyyvyy dxdy, ∀u, v ∈ H00(Ω). Ω

2 There are several conforming finite element spaces in H00(Ω) such as Ar- gyris, Bell, Hsieh-Clough-Tocher, Singular Zienkiewicz triangles and Fraeijs 2 1 de Veubeke-Sander quadrilateral [2]. To be in H00(Ω) is to be of class C for a finite element space in practice. In order to solve (2) with a finite element space of lower regularity, some mixed methods and nonconforming finite element methods have been developed [1]. For given triangulation T of Ω, define a space W (T ) as Z ∂v W (T ) = {v ∈ L2(Ω) v(P ), dσ are continuous E ∂ν at all vertices P and edges E in T}.

The Morley ﬁnite element space is one of the simplest nonconforming ﬁnite element spaces with the simplicial triangulation T S of Ω. It is a subspace

183 of W (T S) and its element is piecewise quadratic. Some rectangular Mor- ley finite element spaces, that is, subspaces of W (T R) to solve (2) with a rectangular triangulation T R have been designed [3]. Let T Q be a triangulation of Ω into convex quadrilaterals. In this talk, we will introduce a quadrilateral Morley finite element space in W (T Q). For each quadrilateral Q ∈ T Q, our finite element space belongs to

2 M P (Q) < L1L3L13,L2L4L24 >, where Li is a linear function which vanishes at edge Ei of Q for i = 1, ··· , 4 and Li i+2 is also linear which vanishes at the bimedian between Ei,Ei+2 for i = 1, 2. Ei, i = 1, ··· , 4 are numbered counterclockwisely as in Figure 1.

L 2 =0

E1 L 24=0

L 4 =0 E4

L =0 L 1 =0 3 L 13 =0

Figure 1: Name convention in Q

We will prove the values at vertices and integrals of normal derivatives over edges are the degrees of freedom of the proposed element. An optimal order of convergence is analyzed and several numerical tests conﬁrm it.

References [1] D. N. Arnold, F. Brezzi : Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates, RAIRO Modél.Math. Anal. Numér.,19 (1985), 7–32. [2] P. G. Ciarlet, J. L. Lions: Handbook of Numerical Analysis, Vol. II, North Holland, 1991. [3] M. Wang, Z. Shi, J. Xu : Some n-Rectangle Nonconforming Elements for Fourth Order Elliptic Equations, Journal of Computational Mathe- matics, 25 (2007), 408–420.

184

Applying Snapback Repellers in Ecology Shu-Ming Chang Department of Applied Mathematics, National Chiao Tung University e-mail: [email protected]

Abstract In ecology, Satake’s generalized resource budget model [6] that modified from Isagi’s resource budget model [2], Satake and Iwasa illustrated by computing the positive Lyapunov exponent that if the depletion coefficient is greater than one, then the system is chaotic. However, a positive Lyapunov exponent implies only sensitivity in Devaney’s chaos [1, 5]. Therefore, this work presents mathematical viewpoints and numerical analysis on Satake’s generalized resource budget model, to rigorously prove that the generalized resource budget model is chaotic in Devaney’s sense by using the snapback repeller theory [4, 7] and the topological entropy theory [3]. Moreover, this work also investigates that the behaviors are different between positive odd depletion coefficients and positive even depletion coefficients under numerical computations.

References

[1] R. L. Devaney. An Introduction to Chaotic Dynamical Systems, Second Edition. Addison-Wesley, Redwood City, Canada, 1989. [2] Y. Isagi, K. Sugimura, A. Sumida, and H. Ito. How does masting happen and synchronize? J. theor. Biol., 187:231–239, 1997. [3] D. Kwietniak and M. Misiurewicz. Exact devaney chaos and entropy. Qualitative Theory Dynamical Systems, 6(95):169–179, 2005. [4] F. R. Marotto. On redeﬁning a snap-back repeller. Chaos, Solitons and Fractals, 25:25–28, 2005. [5] C. Robinson. Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, Second Edition. CRC, Boca Raton, Florida, 1998. [6] A. Satake and Y. Iwasa. Pollen coupling of forest trees: Forming synchronized and periodic reproduction out of chaos. J. theor. Biol., 203:63– 84, 2000. [7] Y. Shi and P. Yu. Chaos induced by regular snap-back repellers. J. Math. Anal. Appl., 337:1480–1494, 2008.

185

Veriﬁed solutions of linear systems on GPU

1), 2) 3) Yusuke Morikura ∗, Katsuhisa Ozaki , Shin’ichi Oishi

1) Graduate School of Fundamental Science and Engineering, Waseda University 2) Department of Mathematical Sciences, Shibaura Institute of Technology & JST, CREST 3) Faculty of Science and Engineering, Waseda University & JST, CREST

e-mail: [email protected] ∗ Abstract This talk is concerned with accuracy of numerical solutions of liner systems, where numerical solutions mean computed results by floating-point arithmetic defined by the IEEE 754 standard [1]. Due to accumulation of rounding errors, the numerical solution may be inaccurate. The aim of our research is to obtain an error bound for a numerical solution of a linear system. This topic is frequently discussed in the field of verified numerical computations. Letx ˜ be an approximate solution of a linear system Ax = b where A is a real n n matrix and b is a real n-vector. Let R be an approximate inverse matrix× of A and I be the n n identity matrix. If RA I < 1, then the matrix A is nonsingular and× k − k∞

1 R(Ax˜ b) x˜ A− b k − k∞ . (1) k − k∞ ≤ 1 RA I − k − k∞ Based on (1), several algorithms obtaining an upper bound of the error have been developed [2, 3, 4]. Nowadays, GPU (Graphics Processing Unit) is used for not only acceleration of building of images but also for numerical computations since the performance is very high. For example, Fermi from Nvidia Corporation has

448 cores and peak performance is 515 Gflops for double precision arithmetic. Basically, it was difficult for a beginner for high performance computing to exploit GPU efficiently. Recently, useful toolboxes for GPU has been developed, for example, CUDA, JACKET and so forth. Therefore, GPU will be widely used. However, there are two problems of implementation of algorithms for verified numerical computations on GPU. Rounding modes defined by the IEEE 754 standard (rounding to nearest, rounding upward, rounding downward) are useful for verified numerical computations. For example, they are used for interval arithmetic. If we use routines supported in such libraries, then the rounding mode cannot be switched dynamically.

186

An algorithm on GPU must output the error bounds without switches of the rounding modes, namely, only rounding to nearest is used. Next, amount of working memory is little comparing to CPU’s space, for example, C2050 and C2070 by Nvidia Corporation have 3 GBytes and 6 Gbytes working memory, respectively. Therefore, computable size of the matrix is limited. We develop an algorithm for the verification of linear systems which is specialized for GPU. Our algorithm only requires floating-point computations with the rounding to nearest mode. Moreover, applying block algorithms, we overcome these problems. Finally numerical result will be shown to illustrate the efficiency of the proposed algorithm.

References

[1] ANSI/IEEE: IEEE Standard for Binary Floating-Point Arithmetic: ANSI/IEEE Std 754-1985, New York, IEEE, 1985 .

[2] S. Oishi, S. M. Rump: Fast Veriﬁcation Solutions of Matrix Eqiations, Numer.Math.,90:4(2002),755-773.

[3] T. Ogita, S. M. Rump, S. Oishi: Veriﬁed Solution of Liners Systems without Directed Rounding,11th GAMM-IMACS International Sympo- sium on Scientiﬁc Computing. Computer Arithmetic, and Validated Nu- merics, Fukuoka,Japan,2004

[4] K. Ozaki, T. Ogita, S. Miyajima, S. Oishi, S. M. Rump: A Method of Obtaining Veriﬁed Solutions for Linear Systems suited for Java, Journal of Computational and Applied Mathematics, 199:2(2007), pp. 337-344.

187

Speaker Index

Name Page no. Session id Afshin Ghanbarzadeh 21 181 [29-4-C]

Akitoshi Takayasu 12 33 [27-3-A] Akiyasu Tomoeda 15 78 [28-3-C] Alex Chang 17 105 [28-4-C] Andrea Bertozzi 18 121 [29-1-A] Buyung Sarita 16 91 [28-3-D] Carolina Mendoza 15 80 [28-3-C] Chang Yong Han 19 137 [29-2-D] Changtong Luo 13 60 [27-4-D] Chenhan Yu 21 170 [29-4-B]

Chien-Hong Cho 13 49 [27-4-C] Chin-Tien Wu 18 124 [29-2-B] Ching-An Wang 13 44 [27-4-C] Chiun-Chang Lee 12 32 [27-3-A] Chunjae Park 22 183 [29-4-D] Dai-Ni Hsieh 21 178 [29-4-C] David Ni 18 122 [29-2-B] Dongwoo Sheen 21 169 [29-4-B]

Edi Cahyono 16 90 [28-3-D] Elliott Ginder 16 96 [28-4-B] Eun-Jae Park 14 66 [28-2-A(1)] G Sharma 17 112 [28-4-C] Guanyu Zhou 20 159 [29-3-D] Gue Myung Lee 21 174 [29-4-C] ( to be continued )

188

Speaker Index

Name Page no. Session id HUI JI 11 27 [27-2-A(2)]

Haiwei Sun 16 101 [28-4-B] Hendra Gunawan 13 51 [27-4-C] Hidenori Yasuda 15 70 [28-3-B] Hirofumi Notsu 20 165 [29-3-D] Hiroshi Fujiwara 21 167 [29-4-B] Hiroshi Kokubu 16 85 [28-3-D] Hiroshi Suito 17 103 [28-4-C] Hisashi Okamoto 15 84 [28-3-C] Ikha Magdalena 17 110 [28-4-C]

Inseok Yang 17 119 [28-4-D] Kaname Matsue 16 86 [28-3-D] Katsuhisa Ozaki 12 42 [27-4-B] Keiji Miura 17 115 [28-4-D] Keita Iida 15 82 [28-3-C] Kenta Kobayashi 18 131 [29-2-C] Kil H Kwon 18 128 [29-2-B] Kuo-Chih Hung 19 145 [29-3-B]

Kwok Kin Wong 13 53 [27-4-C] Lei Du 12 30 [27-3-A] Leo Hari Wiryanto 13 55 [27-4-D] Madhu Jain 21 180 [29-4-C] Masataka Kuwamura 16 92 [28-3-D] Masato Kimura 16 94 [28-4-B] ( to be continued )

189

Speaker Index

Name Page no. Session id Matthew Min-Hsiung Lin 17 113 [28-4-D]

Mayumi Shoji 19 148 [29-3-B] Min-Hung Chen 16 98 [28-4-B] Mitchell Luskin 11 24 [27-1-A] Mohd Omar 15 75 [28-3-C] Naoki Wada 19 135 [29-2-D] Naoya Yamanaka 15 73 [28-3-B] Norikazu Saito 20 161 [29-3-D] Roberd Saragih 12 41 [27-4-B] Roslinda Mohd Nazar 14 68 [28-2-A(2)]

Sangho Kum 21 176 [29-4-C] Santiago Madruga 17 106 [28-4-C] Seyoung Park 20 154 [29-3-C] Shin-Hwa Wang 19 146 [29-3-B] Shu-Ming Chang 22 185 [29-4-D] Shugo Manabe 21 172 [29-4-B] Sri Redjeki Pudjaprasetya 17 108 [28-4-C] Swaroop Nandan Bora 13 62 [27-4-D]

Syakila Binti Ahmad 13 56 [27-4-D] Takafumi Miyata 12 34 [27-4-B] Takahito Kashiwabara 20 163 [29-3-D] Takanori Katsura 15 76 [28-3-C] Takashi Ichinomiya 18 126 [29-2-B] Takashi Sakajo 19 141 [29-3-B] ( to be continued )

190

Speaker Index

Name Page no. Session id Takehiko Kinoshita 18 132 [29-2-C]

Takeshi Fukaya 12 36 [27-4-B] Takeshi WATANABE 19 150 [29-3-B] Tanay Deshpande 15 72 [28-3-B] Teng-Yao Kuo 16 99 [28-4-B] Tetsuya Ishiwata 20 157 [29-3-D] Tomoyuki Miyaji 19 143 [29-3-B] Tsuyoshi Mizuguchi 16 88 [28-3-D] Victor Didenko 13 58 [27-4-D] Viswanathan Kodakkal Kannan 19 139 [29-2-D]

Wah June Leong 20 156 [29-3-C] Wai-Ki CHING 17 117 [28-4-D] Wen-Hann Sheu 11 26 [27-2-A(1)] Wen-Hung Kuo 20 152 [29-3-C] Xiao-Qing Jin 12 38 [27-4-B] Xuefeng LIU 18 133 [29-2-C] Yasumasa Nishiura 14 64 [28-1-A] Yintzer Shih 18 130 [29-2-C]

Yongyong Cai 12 28 [27-3-A] Yukihito Suzuki 13 47 [27-4-C] Yung-Ta Li 12 39 [27-4-B] Yusuke Morikura 22 186 [29-4-D]

191