SCAN 2016 17th International Symposium on Scientific Computing, Computer Arithmetic and Verified Numerics
Book of Abstracts Department of Mathematics Uppsala University Sweden September 26 – 29, 2016
17th International Symposium on Scientific Computing, Computer Arithmetic and Verified Numerics
SCAN 2016
Book of Abstracts
Department of Mathematics Uppsala University Sweden
September 26–29, 2016
SCAN 2016
Scientific Committee
• G. Alefeld (Karlsruhe, Germany) • A. Bauer (Ljubljana, Slovenia) • G.F. Corliss (Milwaukee, USA) • T. Csendes (Szeged, Hungary) • R.B. Kearfott (Lafayette, USA) • V. Kreinovich (El Paso, USA) • W. Luther (Duisburg, Germany) • G. Mayer (Rostock, Germany) • S. Markov (Sofia, Bulgaria) • J.-M. Muller (Lyon, France) • M. Nakao (Fukuoka, Japan) • T. Ogita (Tokyo, Japan) • S. Oishi (Waseda, Japan) • K. Ozaki (Tokyo, Japan) • M. Plum (Karlsruhe, Germany) • A. Rauh (Rostock, Germany) • N. Revol (Lyon, France) • J. Rohn (Prague, Czech Republic) • S. Rump (Hamburg, Germany/Tokyo, Japan) • S. Shary (Novosibirsk, Russia) • W. Tucker (Uppsala, Sweden) • W. Walter (Dresden, Germany) • J. Wolff von Gudenberg (Wuerzburg, Germany) • N. Yamamoto (Tokyo, Japan) Organzing Committee
• Warwick Tucker (Chair) • Anna Belova
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SCAN 2016
Preface
SCAN 2016 will continue the long tradition of promoting the area of rigorous computing, a field that has reached a great level of maturity. This conference will cover the research areas of
• reliable computer arithmetic • enclosure methods • self validating algorithms and will bring many new ideas for future applications and research. The booklet you have before you contains the abstracts of the invited and contributed talks. The conference starts with awarding the R. E. Moore Prize for Ap- plications of Interval Analysis to Arnold Neumaier, Bal´azsB´anhelyi, Tibor Krisztin, and Tibor Csendes for their paper Global attractivity of the zero solution for Wright’s equation. Each morning and after- noon begins with a plenary talk, and I am very proud that the fol- lowing distinguished experts have accepted to present their research:
• Maciej Capinski (Poland) • Martine Ceberio (USA) • Mioara Joldes (France) • Hiroshi Kokubu (Japan) • Jean-Philippe Lessard (Canada) • Weldon Lodwick (USA) • Kaori Nagatou (Germany) • Mark Stadtherr (USA)
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Organizing SCAN 2016 in Uppsala
Planning and organizing this conference has been a great honour and pleasure for me. With the help of the international scientific committee many practical issues have been greatly simplified – a huge thanks to all of you! Locally, I have enjoyed the fine assistance of Inga-Lena Assarsson, Anna Belova, and Susanne Gauffin – I thank you all for supporting me in all aspects. I also thank Akademikonferens for taking care of many of the practical matters. Special thanks to Anders K¨allstr¨om , for typesetting this book of Abstracts. With regards to financing, the Swedish Research Council, as well as Uppsala University have been very generous. Without their con- tributions, it would have been very difficult to organise an event of this magnitude. But most importantly it is all of you who have graciously agreed to participate, and to disseminate your knowledge to us that have made the greatest contribution to SCAN2016. And for this I thank you, and wish you all a very reliable SCAN2016!
Uppsala, September 14 Warwick Tucker
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Schedule – SCAN 2016 As of September 12, 2016.
Sunday, September 25, 2016
18:00 – 20:00 Get-together and registration (Norrlands Nation)
Monday, September 26, 2016
08:00 – 09:00 Registration (Norrlands Nation) 09:00 – 09:30 Opening 09:30 – 10:30 R. E. Moore Prize Awarding Ceremony (Gamla salen; Chair: Baker Kearfott) Tibor Csendes, University of Szeged Interval based checking algorithm fit to the parallel architecture of GPUs with an application to circle covering problems 10:30 – 11:00 Coffee break 11:00 – 12:00 Plenary talk (Gamla salen) Hiroshi Kokubu, Kyoto University Computer-assisted methods for detecting global structure of dynamics 12:00 – 13:20 Lunch (Norrlands Nation) 13:20 – 14:20 Plenary talk (Gamla salen) Mioara Joldes, LAAS-CNRS Validated numerics for robust space mission design or Beyond Gravity (2013) 14:20 – 15:10 Parallel Sessions Session A1: Fuzzy computations (Inre l¨as,Chair: Vladik Kreinovich)
14:20 – 14:45 Weldon A. Lodwick (University of Colorado Denver), Interval Methods in the Calculation of Solutions to Fuzzy Interval Lin- ear Systems
14:45 – 15:10 K.K. Semenov (Peter the Great St. Petersburg Poly- technic University), Interval computations in the metrology
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Session A2: ODEs (Gamla salen, Chair: Michael Plum)
14:20 – 14:45 Masahide Kashiwagi (Waseda University), A study on verified ODE solver from the standpoint of stiffness 14:45 – 15:10 Akitoshi Takayasu (University of Tsukuba), Verified numerical computations for blow-up solutions of ODEs
15:10 – 15:40 Coffee break 15:40 – 17:45 Parallel Sessions Session B1: Linear algebra (Inre l¨as,Chair: Denis Gaidashev)
15:40 – 16:05 Roman Iakymchuk (KTH Royal Institute of Technol- ogy), Towards Fast, Accurate and Reproducible LU Factorization 16:05 – 16:30 Katsuhisa Ozaki (Shibaura Institute of Technology), Linear Systems with the Exact Solution for Numerical Tests 16:30 – 16:55 Yuka Yanagisawa (Waseda University), Verification method for system of linear equations by QR factorization 16:55 – 17:20 Yuka Kobayashi (Tokyo Woman’s Christian University), An Accurate and Efficient Solution of Ill-conditioned Linear Systems by Preconditioning Methods 17:20 – 17:45 Xuefeng Liu (Niigata University), A framework for high- precision verified eigenvalue bounds by using finite element methods
Session B2: Control (Gamla salen, Chair: Vladik Kreinovich)
15:40 – 16:05 Luc Jaulin (ENSTA Bretagne), Secure a zone with robots 16:05 – 16:30 Simon Rohou (ENSTA Bretagne), Tube Programming Applied to State Estimation 16:30 – 16:55 Andreas Rauh (University of Rostock), An Interval- Based Algorithm for Feature Extraction from Speech Signals 16:55 – 17:20 Andreas Rauh (University of Rostock), Interval-Based Identification of Friction and Hysteresis Models 17:20 – 17:45 Andreas Rauh (University of Rostock), Toward the Optimal Parameterization of Interval-Based Variable-Structure State Estimation Procedures
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Tuesday, September 27, 2016
09:00 – 10:00 Plenary talk (Gamla salen) Mark A. Stadtherr, University of Notre Dame Rigorous Method for Robust Optimization and Design of Nonlinear Dy- namic Systems 10:00 – 10:30 Coffee break 10:30 – 12:10 Parallel Sessions Session C1: Optimization (Inre l¨as,Chair: Vladik Kreinovich)
10:30 – 10:55 Arnold Neumaier (University of Vienna), Generalized intervals in global optimization
10:55 – 11:20 Ryo Kobayashi (Waseda University), A method of ver- ified computation for convex programming
11:20 – 11:45 J¨urgenGarloff (University of Konstanz, HTWG Kon- stanz), Fast determination of the tensorial and simplicial Bernstein enclosure
11:45 – 12:10 Ralph Baker Kearfott (University of Louisiana at Lafayette), Simplicial Branch and Bound in Interval Global Optimization
Session C2: Arithmetic (Gamla salen, Chair: Denis Gaidashev)
10:30 – 10:55 Nathalie Revol (ENS de Lyon), HPC and interval com- putations
10:55 – 11:20 Yusuke Morikura (Waseda University), Fast enclosure for matrix multiplication on a GPU
11:20 – 11:45 Siegfried M. Rump (Hamburg University of Technol- ogy), The origin of interval arithmetic
11:45 – 12:10 Siegfried M. Rump (Hamburg University of Technol- ogy), Sharp error bounds for the Gamma function over the whole floating- point range
12:10 – 13:30 Lunch (Norrlands Nation) 13:30 – 14:30 Plenary talk (Gamla salen) Kaori Nagatou, Karlsruhe Institute of Technology
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Orbital stability investigation for travelling waves in a nonlinearly sup- ported beam 14:30 – 17:50 Viking excursion (Gamla Uppsala)
Wednesday, September 28, 2016
09:00 – 10:00 Plenary talk (Gamla salen) Maciej Capi´nski,AGH University of Science and Technology Geometric methods and computer assisted proofs for invariant mani- folds in dynamical systems 10:00 – 10:30 Coffee break 10:30 – 12:10 Parallel Sessions Session D1: Software (Inre l¨as,Chair: Nathalie Revol)
10:30 – 10:55 Matthias H¨usken (University of Wuppertal), IeeeCC754++ – an advanced tool to check IEEE 754-2008 conformity
10:55 – 11:20 Fran¸coisF´evotte (EDF R&D), VERROU: CESTAC without recompilation
11:20 – 11:45 Romain Picot (Sorbonne Universit´es,EDF R&D), PROMISE: floating-point precision tuning with stochastic arithmetic
11:45 – 12:10 David P. Sanders (Universidad Nacional Aut´onomade M´exico(UNAM)), The Julia package ValidatedNumerics.jl and its application to the rigorous characterization of open billiard models
Session D2: General (Gamla salen, Chair: Luc Jaulin)
10:30 – 10:55 Peter Franek (IST Austria), Zero Verification in Sys- tems of Equations: Interval-based Implementation of a Topological Test
10:55 – 11:20 David Romero i S`anchez (Universitat Aut`onomade Barcelona), Numerical computation of invariant objects with wavelets
11:20 – 11:45 Denis Gaidashev (Uppsala University), Golden-mean universality for Siegel disks
11:45 – 12:10 Pedro Barragan (University of Texas at El Paso), Why superellipsoids: an explanation
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12:10 – 13:30 Lunch (Norrlands Nation) 13:30 – 14:30 Plenary talk (Gamla salen) Martine Ceberio, University of Texas at El Paso Using Interval Methods to handle Large Numerical Simulations 14:30 – 15:20 Parallel Sessions Session E1: PDEs (Inre l¨as,Chair: Michael Plum)
14:30 – 14:55 Jonathan Wunderlich (Karlsruhe Institute of Technol- ogy), Computer-assisted existence proofs for one-dimensional Schr¨odinger- Poisson systems
14:55 – 15:20 Hussein Awala (Temple University), Validated Numer- ics Methods for the Mixed Boundary Value Problem for the System of Elastostatics
Session E2: General (Gamla salen, Chair: Denis Gaidashev)
14:30 – 14:55 Ivo List (University of Ljubljana), Efficient Dedekind reals in Haskell
14:55 – 15:20 Anastasia Volkova (Sorbonne Universit´es, UPMC), Computing the Worst-Case Peak Gain of Digital Filter in Interval Arithmetic
15:20 – 15:50 Coffee break 15:50 – 17:55 Parallel Sessions Session F1: ODEs (Inre l¨as,Chair: Michael Plum)
15:50 – 16:15 Nobito Yamamoto (The University of Electro-Communications), Numerical verification of existence of homoclinic orbits in dynamical systems
16:15 – 16:40 Kaname Matsue (The Institute of Statistical Mathe- matics), Rigorous numerics of global trajectories for fast-slow systems with an explicit range of multi-scale parameter
16:40 – 17:05 Alexandre Chapoutot (ENSTA ParisTech, Universit´e Paris-Saclay), Runge-Kutta Theory and Constraint Programming
17:05 – 17:30 Irmina Walawska (Jagiellonian University), An implicit algorithm for validated enclosures of the solutions to variational equa- tions for ODEs
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17:30 – 17:55 Florent Br´ehard(LAAS-CNRS), A New Efficient Al- gorithm for Computing Validated Chebyshev Approximations Solutions of Linear Differential Equations
Session F2: PDEs (Gamla salen, Chair: Vladik Kreinovich)
15:50 – 16:15 Takuma Kimura (Saga University), Optimal order con- structive a priori error estimates for a full discrete approximation of the heat equation
16:15 – 16:40 Kouta Sekine (Waseda university), A norm estimation for an inverse of linear operator using a minimal eigenvalue
16:40 – 17:05 Akitoshi Takayasu (University of Tsukuba), On ver- ification methods for parabolic partial differential equations using the evolution operator
17:05 – 17:30 Kazuaki Tanaka (Waseda University), On verified nu- merical computation for positive solutions to elliptic boundary value problems
17:30 – 17:55 Yoshitaka Watanabe (Kyushu University), Validated constructive error estimatations for bi-harmonic problems
19:00 – late Conference Banquet
Thursday, September 29, 2016
09:00 – 10:00 Plenary talk (Gamla salen) Weldon A. Lodwick, University of Colorado Denver The Molecular Distance Geometry Problem Under Interval Uncertainty 10:00 – 10:30 Coffee break 10:30 – 12:10 Parallel Sessions Session G1: Multiprecision (Inre l¨as,Chair: Denis Gaidashev)
10:30 – 10:55 Valentina Popescu (LIP, ENS-Lyon), Rigourous error bounds for double-double operations
10:55 – 11:20 Nozomu Matsuda (The University of Electro-Communications), LILIB – Long Interval Library
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11:20 – 11:45 Hao Jiang (National University of Defense Technology), The Implementation of multi-precision package in multiple-component format in MATLAB
11:45 – 12:10 Clothilde Jeangoudoux (Sorbonne Universit´es,UPMC), A Decimal Multiple-Precision Interval Arithmetic Library
Session G2: General (Gamla salen, Chair: Vladik Kreinovich)
10:30 – 10:55 Evgenija D. Popova (Bulgarian Academy of Sciences), Enclosing the Solution Set to Interval Parametric Matrix Equation A(p)X = B(p)
10:55 – 11:20 Aymeric Grodet (Ehime University), Adaptive mesh refinement technique for the classical Plateau problem
11:20 – 11:45 Takuya Tsuchiya (Ehime University), Error Analysis of Lagrange Interpolation on Tetrahedrons
11:45 – 12:10 Ronald van Nooijen (Delft University of Technology), The properties of negation and zero in ringoids as defined by Kulisch
12:10 – 13:30 Lunch (Norrlands Nation) 13:30 – 14:30 Plenary talk (Gamla salen) Jean-Philippe Lessard, Universit´eLaval Rigorously verified computing for infinite dimensional nonlinear dy- namics: a functional analytic approach 14:30 – 15:20 Parallel Sessions Session H1: Control (Inre l¨as,Chair: Andreas Rauh)
14:30 – 14:55 Shinya Miyajima (Iwate University), Fast validated computation for solutions of algebraic Riccati equations arising in trans- port theory
14:55 – 15:20 Shinya Miyajima (Iwate University), Fast validated computation for solutions of discrete-time algebraic Riccati equations
Session H2: (Gamla salen, Chair: Luc Jaulin)
14:30 – 14:55 Radoslav Paulen (Technische Universit¨atDortmund), Model-based design of optimal experiments for guaranteed parameter estimation of nonlinear dynamic systems
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14:55 – 15:20 Harsh Purohit (Indian Institute of Technology Bom- bay), Automated tuning of robust fractional PID controller for interval plants using Kharitonov’s theorem
15:20 – 15:50 Coffee break 15:50 – 17:05 Parallel Sessions Session I1: General (Inre l¨as,Chair: Denis Gaidashev)
15:50 – 16:15 Milan Hlad´ık(Charles University), When dependencies do not matter?
16:15 – 16:40 Antoine Plet (LIP, ENS Lyon) Sharp error bounds for complex floating-point inversion
16:40 – 17:05 Vladik Kreinovich (University of Texas at El Paso), Decision Making Under Interval Uncertainty as a Natural Example of a Quandle
Session I2: Constraints (Gamla salen, Chair: Luc Jaulin)
15:50 – 16:15 BartlomiejKubica (Warsaw University of Life Sciences), A template-based C++ library for automatic differentiation and hull consistency enforcing
16:15 – 16:40 Benoit Desrochers (ENSTA Bretagne), Relaxed inter- section of thick sets
16:40 – 17:05 Stefan Ratschan (Czech Academy of Sciences), Safety Verification By Interval Based Quantified Constraint Solving
End of SCAN 2016
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Contents Plenary speakers Interval based checking algorithm fit to the parallel archi- tecture of GPUs with an application to circle covering problems, Bal´azsB´anhelyi,Zsolt Bag´oczkiand Tibor Csendes ...... 21 Geometric methods and computer assisted proofs for invari- ant manifolds in dynamical systems, Maciej Capi´nski. 23 Using Interval Methods to handle Large Numerical Simula- tions, Martine Ceberio ...... 24 Validated numerics for robust space mission design or Be- yond Gravity (2013), Mioara Joldes ...... 26 Computer-assisted methods for detecting global structure of dynamics, Hiroshi Kokubu ...... 28 Rigorously verified computing for infinite dimensional non- linear dynamics: a functional analytic approach, Jean- Philippe Lessard ...... 30 The Molecular Distance Geometry Problem Under Interval Uncertainty, Weldon Lodwick, Tiago M. Costa, Hen- ricus Bouwmeester and Carlile Lavor ...... 31 Orbital stability investigation for travelling waves in a non- linearly supported beam, Kaori Nagatou, P. Joseph McKenna (Storrs, USA) and Michael Plum (Karlsruhe, Germany) 33 Rigorous method for robust optimization and design of non- linear dynamic systems, Yao Zhao and Mark A. Stadtherr 35
Contributed talks Validated Numerics Methods for the Mixed Boundary Value Problem for the System of Elastostatics, Hussein Awala 37 Why superellipsoids: an explanation, Pedro Barragan and Vladik Kreinovich ...... 38
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A New Efficient Algorithm for Computing Validated Cheby- shev Approximations Solutions of Linear Differential Equations, Florent Br´ehard, Nicolas Brisebarre and Mioara Joldes ...... 40 Runge-Kutta Theory and Constraint Programming, Julien Alexandre dit Sandretto and Alexandre Chapoutot . . 42 Relaxed intersection of thick sets, Benoit Desrochers and Luc Jaulin ...... 44 VERROU: a CESTAC evaluation without recompilation, Fran¸cois F´evotte and Bruno Lathuili`ere ...... 46 Zero Verification in Systems of Equations: Interval-based Implementation of a Topological Test, Peter Franek, Marek Krˇc´aland Hubert Wagner ...... 48 Golden-mean universality for Siegel disks, Denis Gaidashev and Michael Yampolsky ...... 50 Fast determination of the tensorial and simplicial Bernstein enclosure, J¨urgenGarloff and Jihad Titi ...... 51 Adaptive mesh refinement technique for the classical Plateau problem, Aymeric Grodet and Takuya Tsuchiya . . . . 53 When dependencies do not matter?, Milan Hlad´ık...... 55 IeeeCC754++ – an advanced tool to check IEEE 754-2008 conformity, Matthias H¨usken ...... 57 Towards Fast, Accurate and Reproducible LU Factorization, Roman Iakymchuk, David Defour and Stef Graillat . . 59 Secure a zone with robots, Luc Jaulin and BenoˆıtZerr . . . 61 A Decimal Multiple-Precision Interval Arithmetic Library, Stef Graillat, Clothilde Jeangoudoux and Christoph Lauter ...... 63 The Implementation of multi-precision package in multiple- component format in MATLAB, Hao Jiang, Peibing Du, Kuan Li, Lin Peng ...... 65 A study on verified ODE solver from the standpoint of stiff- ness, Masahide Kashiwagi ...... 67
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Simplicial Branch and Bound in Interval Global Optimiza- tion, Ralph Baker Kearfott ...... 69 Optimal order constructive a priori error estimates for a full discrete approximation of the heat equation, Takuma Kimura, Teruya Minamoto and Mitsuhiro T. Nakao . 71 Decision Making Under Interval Uncertainty as a Natural Example of a Quandle, Mahdokht Afravi and Vladik Kreinovich ...... 73 A template-based C++ library for automatic differentiation and hull consistency enforcing, BartlomiejKubica . . 75 Efficient Dedekind reals in Haskell, Ivo List ...... 77 A framework for high-precision verified eigenvalue bounds by using finite element methods, Xuefeng LIU . . . . . 79 Interval Methods in the Calculation of Solutions to Fuzzy Interval Linear Systems, Weldon A. Lodwick, Lotfi Taher and Hana Veiseh ...... 81 LILIB – Long Interval Library, Nozomu Matsuda and No- bito Yamamoto ...... 82 Rigorous numerics of global trajectories for fast-slow sys- tems with an explicit range of multi-scale parameter, Kaname Matsue ...... 84 Fast validated computation for solutions of algebraic Riccati equations arising in transport theory, Shinya Miyajima 86 Fast validated computation for solutions of discrete-time al- gebraic Riccati equations, Shinya Miyajima ...... 88 Fast enclosure for matrix multiplication on a GPU, Yusuke Morikura, Yusuke Nozawa, Kouta Sekine, Masahide Kashiwagi and Shi’nichi Oishi ...... 90 Generalized intervals in global optimization, Arnold Neu- maier, Ferenc Domes, Tiago Montanher, Mihaly Markot and Hermann Schichl ...... 92 Linear Systems with the Exact Solution for Numerical Tests, Katsuhisa Ozaki and Takeshi Ogita ...... 93
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Model-based design of optimal experiments for guaranteed parameter estimation of nonlinear dynamic systems, Anwesh Reddy, Gottu Mukkula and Radoslav Paulen . 95 PROMISE: floating-point precision tuning with stochastic arithmetic, Stef Graillat, Fabi´enneJ´ez´ekiel,Romain Picot, Fran¸coisF´evotte and Bruno Lathuili`ere. . . . . 97 Sharp error bounds for complex floating-point inversion, Claude- Pierre Jeannerod, Nicolas Louvet, Jean-Michel Muller and Antoine Plet ...... 99 Rigourous error bounds for double-double operations, Mioara Joldes, Jean-Michel Muller and Valentina Popescu . . 100 Enclosing the Solution Set to Interval Parametric Matrix Equation A(p)X = B(p), Evgenija D. Popova . . . . . 102 Automated tuning of robust fractional PID controller for in- terval plants using Kharitonov’s theorem, Harsh Puro- hit, P.S.V. Nataraj ...... 104 Safety Verification By Interval Based Quantified Constraint Solving, Peter Franek, Jan Kuˇr´atko and Stefan Ratschan106 An Interval-Based Algorithm for Feature Extraction from Speech Signals, Andreas Rauh, Susann Tiede and Cor- nelia Klenke ...... 108 Interval-Based Identification of Friction and Hysteresis Mod- els, Andreas Rauh and Harald Aschemann ...... 110 Toward the Optimal Parameterization of Interval-Based Variable- Structure State Estimation Procedures, Andreas Rauh and Harald Aschemann ...... 112 HPC and interval computations, Nathalie Revol ...... 114 Tube Programming Applied to State Estimation, Simon Ro- hou, Luc Jaulin, Lyudmila Mihaylova, Fabrice Le Bars, Sandor M. Veres ...... 116 Sharp error bounds for the Gamma function over the whole floating-point range, Siegfried M. Rump ...... 118 The origin of interval arithmetic, Siegfried M. Rump . . . . 119
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A method of verified computation for convex programming, Ryo Kobayashi, Takuma Kimura and Shin’ichi Oishi . 120 Numerical computation of invariant objects with wavelets, Llu´ısAlsed`ai Soler and David Romero i S`anchez . . . 122 The Julia package ValidatedNumerics.jl and its applica- tion to the rigorous characterization of open billiard models, David P. Sanders, Luis Benet and Nikolay Kryukov ...... 123 A norm estimation for an inverse of linear operator using a minimal eigenvalue, Kouta Sekine, Kazuaki Tanaka and Shin’ichi Oishi ...... 125 Interval computations in the metrology, Semenov K.K., Solopchenko G.N. and Kreinovich V.Ya...... 127 Verified numerical computations for blow-up solutions of ODEs, Akitoshi Takayasu, Kaname Matsue, Takiko Sasaki, Kazuaki Tanaka, Makoto Mizuguchi and Shin’ichi Oishi ...... 131 On verification methods for parabolic partial differential equa- tions using the evolution operator, Akitoshi Takayasu, Makoto Mizuguchi, Takayuki Kubo and Shin’ichi Oishi 133 On verified numerical computation for positive solutions to elliptic boundary value problems, Kazuaki Tanaka, Kouta Sekine, and Shin’ichi Oishi ...... 135 Error Analysis of Lagrange Interpolation on Tetrahedrons, Kenta Kobayashi, Takuya Tsuchiya ...... 137 An implicit algorithm for validated enclosures of the solu- tions to variational equations for ODEs, Irmina Walawska and Daniel Wilczak ...... 139 The properties of negation and zero in ringoids as defined by Kulisch, Ronald van Nooijen and Alla Kolechkina . 141 Validated constructive error estimatations for bi-harmonic problems, Yoshitaka Watanabe, Takehiko Kinoshita and Mitsuhiro T. Nakao ...... 142
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Computing the Worst-Case Peak Gain of Digital Filter in Interval Arithmetic, Anastasia Volkova, Christoph Lauter and Thibault Hilaire ...... 143 Computer-assisted existence proofs for one-dimensional Schr¨odinger- Poisson systems, Jonathan Wunderlich ...... 145 Numerical verification of existence of homoclinic orbits in dynamical systems, Nobito Yamamoto, Kaname Mat- sue and Shun Yamano ...... 147 Verification method for system of linear equations by QR factorization, Yuka Yanagisawa, Shin’ichi Oishi and Fumi Noda ...... 149 An Accurate and Efficient Solution of Ill-conditioned Linear Systems by Preconditioning Methods, Yuka Kobayashi, Takeshi Ogita and Katsuhisa Ozaki ...... 151
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Interval based checking algorithm fit to the parallel architecture of GPUs with an application to circle covering problems Bal´azsB´anhelyi,Zsolt Bag´oczki,and Tibor Csendes University of Szeged P.O. Box 652, Szeged, Hungary [email protected]
Keywords: circle covering, CUDA, parallel implementation
Videocards are not only for graphic display. With their high speed video memories, lots of math units and parallelism, they provide a very powerful platform for general purpose computing tasks, in which we have to deal with large datasets, which are highly parallelizable, have high computational complexity, etc. Our selected platform for testing is the CUDA (Compute Unified Device Architecture) [4], that grants us direct reach to the virtual instruction set of the video card, and we are able to run our computations on dedicated computing kernels. In this parallel environment we implemented a reliable method with properly rounded interval arithmetic [2]. Our method is based on the branch-and-bound algorithm, with the purpose to decide whether or not any given property applies for an n-dimensional interval. This al- gorithm will give us the opportunity to use node level parallelization. This means, that our nodes are evaluated simultaneously on multiple threads we start on the GPU. This is called low-level, or type 1 paral- lelization, since we don’t modify the searching trajectories, neither do we modify the dimensions of the branch-and-bound tree. For testing, we choose the circle covering problem [1, 3]. It is the dual of circle packing, where our goal is to find the densest packing of a given number of congruent circles with disjoint interiors in a unit square. In circle covering, we aim for finding the full covering of the unit square with congruent circles of minimal radii. Overlapping in- teriors are allowed. For achieving easily scalable test cases, we will
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decide the covering with precalculated circles, where the diameter of the circles is the diagonal of the unit square, divided by a given num- ber, n, and the number of the circles is the square of n. We scaled the problem up to three dimensions, and ran tests with sphere covering problems, too, and we discuss the possibility of scaling the problem up to any dimensions easily. We report our parallelization results.
References: [1] Balazs´ Banhelyi,´ Endre Palatinus, and Balazs´ L. Levai´ , Optimal circle covering problems and their applications, Central European J. Operations Research, 22(2014) pp. 815–832. [2] Sylvain Collange, Marc Daumas, and David Defour, In- terval Arithmetic in CUDA, In: Wen-mei W. Hwu: GPU Com- puting Gems, Jade Edition, Morgan Kaufmann, 2011, pp. 99-107. [3] Kari J. Nurmela and Patric R.J. Ostergard¨ , Covering a square with up to 30 equal circles, Research report, HUT-TCS- A62, Helsinki University of Technology, 2000 NP-hard classes of linear algebraic systems with uncertainties, [4] Jason Sanders and Edward Kandrot, CUDA by Example An Introduction to General-Purpose GPU Programming, Edward Brothers, Ann Arbor, 2010
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Geometric methods and computer assisted proofs for invariant manifolds in dynamical systems Maciej Capi´nski AGH University of Science and Technology al. Mickiewicza 30, 30-059 Krak´ow,Poland [email protected]
Invariant manifolds can be used to determine global behaviour of dynamical systems. They can often be proved using geometric or topo- logical arguments. In the talk we demonstrate how to apply such tech- niques for proofs of stable/unstable manifolds of fixed points and for normally hyperbolic manifolds. We show how to combine these geo- metric methods with interval verified numerics, obtaining proofs for problems inaccessible using standard techniques.
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Using Interval Methods to handle Large Numerical Simulations Martine Ceberio Computer Science Department The University of Texas at El Paso 500 W. University, El Paso, Texas, USA [email protected]
Keywords: dynamic systems, interval computations, constraint solv- ing, proper-orthogonal decomposition (POD), reduced-order modeling (ROM).
Many natural phenomena can be modeled as ordinary or partial differential equations. A way to find solutions of such equations is to discretize them and solve the corresponding (possibly) nonlinear large systems of equations; see [1]. Major issues with solving such nonlinear systems include the fact that their dimension can be very large and that uncertainty, often present, is tricky to handle. Model- Order Reduction (MOR) has been proposed as a way to overcome the issues associated with large dimensions, the most used approach for doing so being Proper Orthogonal Decomposition (POD); see [2,3]. The key idea of POD is to reduce a large number of interdependent variables (snapshots) of the system to a much smaller number of uncor- related variables while retaining as much as possible of the variation in the original variables. On the other hand, interval constraint-solving techniques (ICST) [4] allow to handle uncertainty and ensure reliable results of systems of (possibly) nonlinear equations. In this presentation, we show how intervals and constraint solving techniques (ICST) can be used to compute all the snapshots at once (IPOD). We take advantage of using interval techniques to also show how IPOD can address not only dimension but also uncertainty. As a result, this new process gives us two advantages over the traditional POD method: 1. handling uncertainty in some parameters or inputs;
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2. reducing the snapshots computational cost. We go over numeri- cal examples of the use of IPOD and we then take a glimpse at the implications of this work: How does using interval constraint solving techniques and handling uncertainty affect the whole simulation pro- cess? How is the reduced-order model then solved?
Acknowledgment: Most of this work was supported by Stanford’s Army High-Performance Computing Research Center (AHPCRC) funded by the Army Research Lab (ARL), and by the National Science Foun- dation award 0953339.
References:
[1] J. Li and Y. Chen, Computational Partial Differential Equations Using MATLAB. CRC Press, Las Vegas (NV), 2008.
[2] W. H. Schilders and H. A. Vorst, Model Order Reduction: Theory, Research Aspects and Applications. Springer Science & Business Media, 2008.
[3] K. Carlberg, C. Bou-Mosleh, and C. Farhat, Efficient non-linear model reduction via a least-squares Petrov–Galerkin pro- jection and compressive tensor approximations, Int. J. Numer. Meth. Engng., vol. 86, pp. 155–181, 2011.
[4] L. Granvilliers, and F. Benhamou, RealPaver: An Interval Solver using Constraint Satisfaction Techniques. ACM Trans. on Mathematical Software 32(1), 138–156, 2006.
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Validated numerics for robust space mission design or Beyond Gravity (2013). Mioara Joldes LAAS-CNRS, 7 Avenue du Colonel Roche, 31077 Toulouse, France [email protected]
Keywords: validated numerics, symbolic-numeric method, space mis- sion, space rendez-vous, validated trajectories, collision probability In this talk we present an overview of several symbolic-numeric objects, algorithms and software tools developed for applications in optimal control and robust space mission design. In this domain, certi- fication of computations is at stake and we aim at providing computer- aided proofs of numerical values, with validated and reasonably tight error bounds, without sacrificing efficiency. One application is the computation of collision probabilities be- tween space objects in low Earth orbits. The large number of orbit- ing debris constitute a serious hazard for operational satellites. The trade-off between the collision risk and the inherent risks of performing a collision avoidance maneuver is a strong incentive to precisely esti- mate the collision probability between two orbiting objects, given the uncertainties on their locations. We present a new method for com- puting the probability of collision between two spherical space objects involved in a short-term encounter under Gaussian-distributed uncer- tainty. In this model of conjunction, classical assumptions reduce the probability of collision to the integral of a two-dimensional Gaussian probability density function over a disk. The computational method presented here is based on an analytic expression for the integral, de- rived by use of Laplace transform and D-finite functions properties. Analytic bounds on the truncation error are also derived and are used to obtain a very accurate algorithm.
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Another application is the computation of validated impulsive con- trol for the rendezvous problem of spacecrafts. The rendezvous (RdV) problem consists in designing a plan of maneuvers which takes one active spacecraft, originally moving on an initial orbit, to the final reference orbit of a passive target spacecraft e.g., International Space Station. The impulsive nature comes from the way the thrusters of most satellites work. Since the ’60s many ideas were developed, and today, we are interested in successful RdV which minimizes fuel con- sumption, with increased autonomy (no human operator). This im- plies that validation of computations and solutions is at stake. In this talk we discuss the fixed-time minimum-fuel rendezvous between close elliptic orbits, assuming a linear impulsive setting and a Kep- lerian relative motion. Firstly, the optimal velocity increments and impulses locations are numerically obtained with a new iterative algo- rithm with proven convergence. This is based on discretizing a semi- infinite convex optimization problem. Secondly, the obtained numeri- cal solutions are validated by propagating the trajectories solutions of the linear differential equations of the dynamics. These are computed as truncated Chebyshev series together with rigorously computed error bounds. Different realistic numerical examples illustrate these results. This talk is based on joint works with D. Arzelier, F. Br´ehard,N. Brisebarre, J.-B. Lasserre, C. Louembet, A. Rondepierre, B. Salvy, R. Serra.
27 SCAN 2016
Computer-assisted methods for detecting global structure of dynamics Hiroshi Kokubu Kyoto University Kyoto 606-8502, Japan [email protected]
Keywords: dynamical system, Conley-Morse graph, ODE integration
Morse decomposition is a decomposition of the chain recurrent in- variant set of a dynamical system into finite collection of invariant sets, called Morse sets, that are related in a gradient-like manner. Morse graph represents the gradient-like connection in terms of a finite di- rected graph whose nodes are Morse sets and edges exhibit (possibility of) flow-defined connections. If each node is associated with the topo- logical information of the dynamics, more precisely the Conley index, of the corresponding Morse set, the graph is called the Conley-Morse graph. In [1] and [2] we have developed a computer-assisted method for obtaining the Conley-Morse graphs of a multi-parameter family of dis- sipative dynamical systems, in particular, in the form of iterated maps. This method can also be applied to families of ODEs, by using time-T maps of the flows generated by the ODEs. However, there are several computational issues that need to be further investigated. In this talk, we shall first review the method of Conley-Morse graphs and show several examples of computations. We then discuss some issues of time integration of ODEs and show a benchmark com- parison of CAPD and COSY softwares, intended for the type of compu- tations for our purpose, namely multi-scale set-oriented computations ([3]). A modification of the method using time-series data from a dynam- ical system has been considered recently, which is mainly developed for application purposes, but is of theoretical interest as well.
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References:
[1] Z. Arai, W. Kalies, H. Kokubu, K. Mischaikow, H. Oka, P. Pilarczyk, A database schema for the analysis of global dy- namics of multiparameter systems, SIAM Journal on Applied Dy- namical Systems, 8 (2009), pp. 757–789.
[2] J. Bush, M. Gameiro, S. Harker, H. Kokubu, K. Mis- chaikow, I. Obayashi, P. Pilarczyk, Combinatorial-topological framework for the analysis of global dynamics, Chaos 22 (2012), 047508.
[3] T. Miyaji, P. Pilarczyk, M. Gameiro, H. Kokubu, K. Mis- chaikow, A study of rigorous ODE integrators for multi-scale set-oriented computations, Applied Numerical Mathematics, 107 (2016), pp. 34–47.
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Rigorously verified computing for infinite dimensional nonlinear dynamics: a functional analytic approach Jean-Philippe Lessard Universit´eLaval D´epartement de math´ematiqueset de statistique 1045 av. de la M´edecine,Qu´ebec, QC, G1V 0A6, Canada [email protected]
Studying and proving existence of solutions of nonlinear dynamical systems using standard analytic techniques is a challenging problem. In particular, this problem is even more challenging for partial dif- ferential equations, variational problems or functional delay equations which are naturally defined on infinite dimensional function spaces. The goal of this talk is to present rigorous numerical technique relying on basic functional analytic tools to prove existence of steady states, time periodic solutions, traveling waves and connecting orbits for the above mentioned dynamical systems. We will spend some time iden- tifying difficulties of the proposed approach as well as time to identify future directions of research.
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The Molecular Distance Geometry Problem Under Interval Uncertainty
Weldon Lodwick∗, Tiago M. Costa∗, Henricus Bouwmeester∗∗, and Carlile Lavor∗∗∗
∗ University of Colorado Denver, Department of Mathematical and Statistical Sciences, USA ∗∗ Metropolitan State University of Denver ∗∗∗ University of Campinas/IMECC [email protected]
More recent interval approaches are used to analyze the Distance Geometry Problem (DGP) under interval uncertainty where we restrict ourselves, in this presentation, to the molecular DGPs. The molecular DGP is the following: Given the n atoms of a molecule and distances dij between atoms i 3 and j in R , find coordinates, xk, of the atoms k = 1, .., n, such that the distances between xi and xj are equal to the given Euclidean distances dij, that is, x x = d . k i − jk2 ij The case which is of interest to this presentation focuses on an incomplete set of distances some of which are intervals due to mea- surement errors. This problem, for the case of incomplete real-valued distances, is, in general, NP-Hard. When the problem is restricted to the molecular distance geometry problem, for example, those arising from protein molecules where the real-valued distances are obtained from a nuclear magnetic resonance machine, the number of solutions are finite though there are 2n possible solutions. The inclusion of measurement errors from the nuclear resonance machine is a more realistic approach to the molecular DGP. The mea- surement errors we assume to be intervals. Any approach that con- siders a distance as a real-number within the given real-valued interval
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error will produce only one of the possible conformations. We attempt to find all possible conformations using interval analytic techniques. This presentation discusses the use of interval analytic methods in four facets of the molecular DGP:
1. The representation of the problem under interval uncertainty - we represent all intervals as a one variable parameterization;
2. The reduction of the interval uncertainty - properties of distances in R3 such as the triangle inequality and any molecule-specific extra information are applied to the given interval distance to obtain the tightest interval bounds on the distance data, though for this talk, we will assume that the interval distances, as given, are the tightest possible;
3. The propagation of interval errors - given the tightest interval dis- tance bounds, the single-valued parameterization representation is propagated through the 2n possible coordinate computations, that is, we propagate the symbolic representation of the interval as a single parameterization, not the interval itself;
4. The computation of a set of conformations - given the propagated symbolic values of the coordinates, the interval-values of the coor- dinates are computed as a one variable global optimization prob- lem.
This research was partially funded by CNPq grant 400754/2014-2.
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Orbital stability investigation for travelling waves in a nonlinearly supported beam
Kaori Nagatou∗, P. Joseph McKenna (Storrs, USA) and Michael Plum (Karlsruhe, Germany)
∗Institute for Analysis, Karlsruhe Institute of Technology ∗Englerstrasse 2, 76131 Karlsruhe [email protected]
We consider the fourth-order wave equation
+ ϕtt + ϕxxxx + f(ϕ) = 0, (x, t) R R , ∈ × with a nonlinearity f vanishing at 0. Solitary traveling waves ϕ = u(x ct) satisfy the ODE − 2 u0000 + c u00 + f(u) = 0 on R, and for the case f(u) = eu 1, the existence of at least 36 travelling waves was proved in [1] by computer− assisted means. We investigate the orbital stability of these solutions via computa- tion of their Morse indicies and using results from [2] and [3]. In order to achieve it we make use of both analytical and computer-assisted techniques. References: [1] B. Breuer, J. Horak,´ P. J. McKenna, M. Plum, A computer- assisted existence and multiplicity proof for travelling waves in a nonlinearly supported beam, Journal of Differential Equations, 224, pp. 60-97, 2006. [2] J. Shatah and W. Strauss, Stability Theory of Solitary Waves in the Presence of Symmetry, I, Journal of Functional Analysis, 74, pp. 160-197, 1987.
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[3] J. Shatah and W. Strauss, Stability Theory of Solitary Waves in the Presence of Symmetry, II, Journal of Functional Analysis, 94, pp. 308-348, 1990.
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Rigorous method for robust optimization and design of nonlinear dynamic systems Yao Zhao and Mark A. Stadtherr University of Notre Dame Department of Chemical and Biomolecular Engineering Notre Dame, Indiana, USA [email protected]
Keywords: robust design, robust optimization, nonlinear dynamic systems, interval methods, Taylor models
In engineering product or process design it is common to have to deal with uncertainties, which may arise due to variability in problem parameters (e.g., physical properties of materials) or to the possibil- ity of external disturbances (e.g., change in temperature or flow rate of a feed stream). A design is considered robust if it will continue to meet specified constraints (e.g., on safety and/or performance) for all potential values of the uncertain quantities. Thus, the goal is to identify a region in the design space for which this robustness property is achieved. If finding a minimum in some specified objective func- tion for performance is also desired, then robustness may be sought by formulating the problem as a min-max optimization problem. In this case, the problem may be thought of as finding the best possible performance in the worst-case scenario with regard to the uncertain- ties. For systems with nonlinear dynamic constraints, these robust design and optimization problems may become particularly difficult to solve rigorously, and in many cases existing methods rely on prob- lem simplifications or modifications (e.g., linearization, discrete time approximation). In this presentation we will review a framework for solving robust design and optimization problems using interval methods. This com- bines ideas from region transition modeling [1] and solution of static min-max optimization problems [2] with methods using Taylor models
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for rigorous solution of interval-valued ODE problems [3] and rigorous global optimization of nonlinear dynamic systems [4, 5]. A variety of example problems will be used.
References:
[1] H. Huang, C.S. Adjiman, N. Shah, Quantitative framework for reliable safety analysis, AIChE J., 48 (2002), pp. 78-96.
[2] S. Zuhe, A. Neumaier, C. Eiermann, Solving min-max prob- lems by interval methods, BIT, 30 (1990), pp. 742–751.
[3] Y. Lin, M. A. Stadtherr, Validated solutions of initial value problems for parametric ODEs, Appl. Numer. Math., 57 (2007), pp. 1145–1162.
[4] Y. Lin, M. A. Stadtherr, Deterministic global optimization of nonlinear dynamic systems, AIChE J., 53 (2007), pp. 866–875.
[5] Y. Zhao, M. A. Stadtherr, Rigorous global optimization for dynamic systems subject to inequality path constraints, Ind. Eng. Chem. Res., 50 (2011), pp. 12678–12693.
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Validated Numerics Methods for the Mixed Boundary Value Problem for the System of Elastostatics Hussein Awala Temple University Philadelphia, PA 19122, USA [email protected]
Elliptic boundary value problems with mixed Dirichlet and Neu- mann type boundary conditions arise naturally in connection with physical phenomena such as conductivity, heat transfer, elastic de- formations, and electrostatics. In my talk I will discuss recent well- posedness results for the mixed boundary problem for the system of elastostatics in infinite sectors in two dimensions. These results are ob- tained through a blend of Calderon-Zygmund theory methods, Mellin transform techniques, and validated numerics. This work is part of an ongoing collaboration project with Irina Mitrea and Warwick Tucker.
References:
[1] I. Mitrea, W. Tucker, Some Counterexamples for the Spectral Radius Conjecture, Differential and Integral Equations, 16 (2003), No. 12, pp. 1409 -1439.
[2] I. Mitrea, W. Tucker, Interval Analysis Techniques for Bound- ary Value Problems of Elasticity in Two Dimensions, Journal of Differential Equations, 233 (2007), pp. 181 -198.
[3] H. Awala, I. Mitrea, K. Ott, On the Solvability of the Zaremba Problem in Infinite Sectors and the Invertibility of Associated Sin- gular Integral Operators, preprint 2016.
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Why superellipsoids: an explanation Pedro Barragan and Vladik Kreinovich University of Texas at El Paso 500 W. University, El Paso, TX 79968, USA [email protected]
Keywords: superellipsoid, uncertainty, probabilities
Need to describe uncertainty domains. The intent of the mass production of a gadget is to produce gadgets with identical values (x1, . . . , xn) of the desired characteristics xi. In reality, of course, dif- ferent gadgets end up having slightly different values xi of these char- acteristics: ∆x def= x x = 0. For each of these characteristics x , i i − i 6 i we usually have a tolerance bound ∆xi for which ∆xei ∆i, so that possible values of ∆x form an interval [ ∆ , ∆ ].| Thus,| ≤ possible val- ei − i i ues of the deviation vector ∆x = (∆x1,..., ∆xn) are located in the box [ ∆1, ∆1] ... [ ∆n, ∆n]. In practice, not all vectors ∆x from this box− are possible.× × It− is therefore desirable to describe the set of all possible deviation vectors ∆x. This set is known as the uncertainty domain. Shall not we also determine probabilities? At first glance, it seems that we should be interested not only in finding out which de- viation vectors ∆x are possible and which are not, but also in how frequent different possible vectors are. In other words, we should be interested not only in the uncertainty domain, but also on the probabil- ity distribution on this domain. In reality, however, it is not possible to find these probabilities. Indeed, the manufacturing process may slightly change (and often does change). After each such change, the tolerance intervals and the resulting uncertainty domain remain largely unchanged, but the probabilities change (often drastically). Empirical shapes of uncertainty domains. Empirical analysis has shows that in many practical cases, the uncertainty domain can be well
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n ∆x p approximated by a super-ellipsoid | i| C for some values σ ≤ i=1 i σi, p, and C, and the accuracy of thisP approximation is higher than for other approximation families with the same number of parameters. What we do in this paper. In this paper, we provide a theoretical explanation for this empirical phenomenon.
Our idea. In reality, there is some probability distribution ρi(∆xi) for each of the random variables ∆xi. Since we have no reason to as- sume that positive values are more probable than negative values or vice versa, it makes sense to assume that they are equally probable, i.e., that each distribution ρi(∆xi) is symmetric: ρi(∆xi) = ρi( ∆xi ). Similarly, since we have no reasons to believe that different deviations| | are statistically dependent, it makes sense to assume that the corre- sponding random variables are independent. In this case, the overall n probability density function (pdf) has the form ρ(∆x) = ρi( ∆xi ). i=1 | | Usually, we consider a deviation vector possible if itsQ probability def exceed a certain threshold t. Thus, the desired set has the form St = ∆x : ρ(∆x) t . Numerical values of the deviations ∆xi depend on the{ choice of a≥ measuring} unit; if we replace the original unit by a unit which is λ times smaller, then for the exact same physical situation, we get the new numerical values ∆xi0 = λ ∆xi. Since the physics remains the same, it makes sense to require that· the uncertainty domains do not change under such a re-scaling. To be more precise, the pdf threshold t may change, but the family of such sets should remain unchanged: St0 t = St t, where St0 corresponds to the re-scaled pdf ρ0(∆x) = const{ } ρ(λ{ ∆).} We· prove· that under this scale-invariance, the corresponding sets St are exactly super-ellipsoids. Thus, we get the desired explanation. References: [1] I. Elishakoff, F. Elettro, Interval, ellipsoid, and super-ellipsoid calculi for experimental and theoretical treatment of uncertainty: which one ought to be preferred?, International Journal of Solid Structures, 51 (2014), No. 7, pp. 1576–1586.
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A New Efficient Algorithm for Computing Validated Chebyshev Approximations Solutions of Linear Differential Equations
Florent Br´ehard∗, Nicolas Brisebarre∗∗ and Mioara Joldes∗
∗LAAS-CNRS, 7 Avenue du Colonel Roche, 31077 Toulouse, France ∗∗CNRS, LIP, ENS Lyon, 46 All´eed’Italie, 69364 Lyon, France [email protected] [email protected] [email protected]
Keywords: validated numerics, LODE, Chebyshev series, Newton method, fixed-point operator, error bound In this work we develop a validated numerics method for the solu- tion of linear ordinary differential equations (LODEs). It is well known that the Picard-Lindel¨oftheorem ensures under mild assumptions the existence and uniqueness of the solution to such an equation. How- ever, in general this solution lacks an easily computable closed form. A wide range of algorithms (i.e., Runge-Kutta, collocation, spectral methods) exist for numerically computing approximations of the so- lutions. Most of these come with proofs of asymptotic convergence, but usually, provided error bounds are non-constructive. However, in some domains like critical systems and computer-aided mathematical proofs, one needs validated effective error bounds. For that, one solu- tion is to use so-called a priori validated methods, which provide at each iteration both an approximation and bounds on the error e.g., validated Taylor approximations [3]. In contrast, a posteriori valida- tion methods take as argument an approximate solution (computed by some numerical algorithm) and compute afterwards an error bound. Our contribution belongs to this second class and it relies on a fixed point argument of a contracting map [5]. The two main ingredients of our method are: (1) numerical approx- imation by Chebyshev truncated series of (regular enough) functions on compact intervals. While appearing more natural to use, Taylor ap- proximation techniques may not be valid when the function is not suffi- ciently regular or has singularities in the complex disk surrounding the
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real interval considered. In contrast, Chebyshev approximations only require mild assumptions, typically a Lipschitz condition. Moreover, for a fixed degree, Chebyshev truncated series is guaranteed to be a near-best approximation. Also, standard operations on polynomials are not much harder in Chebyshev basis than in the monomial one [2]. (2) Expressing the solution as the unique fixed point of a compact linear integral operator, defined with multiplication by polynomials and integration. In Chebyshev basis, it has the form of an ”almost-banded” operator: the corresponding (infinite) matrix has non-zero coefficients only on few upper/lower diagonals and initial rows. Based on this, in [4], a linear time (with respect to the truncation degree) numerical algorithm has been proposed. Another linear complexity algorithm was given in [1], observing that when applying the operator, the coefficients of the expansions obey linear recurrence relations with polynomial co- efficients. For validating the solution, one possibility is to notice that after a finite number of iterations the operator becomes contracting [1]. We use a different approach, namely a pseudo-Newton method [5]. The key idea is to multiply the operator by an approximate inverse in fi- nite dimension, so that the result is close to the identity map, and the difference between them is hence a contracting operator. A technical contribution consists in correctly bounding the norm of this operator. Our new approach is illustrated by validating solutions of LODEs appearing in robust guidance algorithms for space trajectories. References [1] A. Benoit, M. Joldes, and M. Mezzarobba. Rigorous uniform approximation of d-finite functions using chebyshev expansions. Mathematics of Computation, (To appear), 2016. [2] N. Brisebarre and M. Joldes. Chebyshev interpolation polynomial-based tools for rigorous computing. In ISSAC ’10: Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation, page 147–154, 2010. [3] M. Neher, K. R. Jackson, and N. S. Nedialkov. On Taylor model based integration of ODEs. SIAM J. Numer. Anal., 45:236–262, 2007. [4] S. Olver and A. Townsend. A fast and well-conditioned spectral method. SIAM Review, 55(3):462–489, 2013. [5] N. Yamamoto. A numerical verification method for solutions of boundary value problems with local uniqueness by banach’s fixed-point theorem. SIAM Journal on Numerical Analy- sis, 35(5):2004–2013, 1998.
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Runge-Kutta Theory and Constraint Programming Julien Alexandre dit Sandretto and Alexandre Chapoutot U2IS, ENSTA ParisTech, Universit´eParis-Saclay, 828 bd des Mar´echaux, 91762 Palaiseau cedex France [email protected]
Keywords: Runge-Kutta, Butcher theory, Validated simulation
Introduction There exist many Runge-Kutta methods (explicit or implicit), more or less adapted to a given class of problems. Some of them have interest- ing properties such as A-stability for stiff problems or symplecticity for problems with energy conservation. Defining a new method, adapted to a given class of problems, has become a challenge. Indeed, the num- ber of stages and the order don’t stop to increase. This race to the “best” method is interesting but forgot an important problem. More precisely, the coefficients of a Runge-Kutta method are more and more difficult to compute and the result is often given in floating-point num- bers, which may lead to violate their definition rules. We propose a method using interval analysis tools to compute Runge-Kutta coeffi- cients by using a solver based on guaranteed constraint programming. Moreover, with a global optimization process and a well chosen cost function, we propose a way to define some novel optimal Runge-Kutta methods. One step of a Runge-Kutta integration scheme, applied on an or- dinary differential equationy ˙ = f(t, y), is obtained with
s s
yn+1 = yn + h biki, where ki = f t0 + cih, y0 + h aijkj . (1) i=1 j=1 ! X X
The coefficients ci, aij and bi, for i, j = 1, , s, fully characterize the Runge-Kutta methods and they are usually··· synthesized in a Butcher
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tableau [1] of the form: c1 a11 . . . a1s ...... cs as1 . . . ass b1 . . . bs Main idea Our approach consists on the generation of constraints defined by the Butcher theory in order to build a Runge-Kutta method. Then we solve these constraints with a Branch&Prune algorithm. We also propose a Branch&Bound approach to define new optimal methods w.r.t. an easy to obtain cost function. Interval coefficients preserve properties In a preliminary stage, we can verify that a Runge-Kutta method with interval coefficients preserves the Butcher rule and then that a method given for an order p has really a local truncature error in O(hp+1). We also propose three methods using interval tools to check the linear stability, algebraically stability and symplecticity properties. Main results First, our Branch&Prune based approach is used to find existing meth- ods and by the way re-discover the Runge-Kutta theory such as i) Gauss-Legendre is the only 2-stages 4-order method; ii) there is no 2-stages 5-order method; etc. Our Branch&Bound method finds the same results as Ralston [2]. Second, both of our methods is used to define new validated Runge-Kutta methods. References: [1] Butcher, John C., Coefficients for the study of Runge-Kutta integration processes, Journal of the Australian Mathematical So- ciety, 5 (1963), No. 3, pp. 185–201. [2] Ralston, Anthony, Runge-Kutta methods with minimum error bounds, Mathematics of computation, (1962), pp. 431-437.
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Relaxed intersection of thick sets
Benoit Desrochers and Luc Jaulin
Lab-STICC, ENSTA Bretagne, Brest, France
Relaxed Intersection : The q-relaxed intersection m sets X1,..., Xm n n of R , is the set of all x R which belong to all Xi’s except q at most. This notion is important∈ in the context of robust bounded error esti- mation [1]. Thick set. Denote by ( (Rn), ), the powerset of Rn equipped P ⊂ with the inclusion as an order relation. A thick set X of Rn is an ⊂ interval of ( (Rn), ). If X is a thick set of Rn , there exist [2] two Pn ⊂ subsets of R , called the subset bound and the supset boundJ K such that J K n X = X⊂, X⊃ = X (R ) X⊂ X X⊃ . { ∈ P | ⊂ ⊂ } Thick test. Given a thick set X , the corresponding thick test n J K J K t : R 0, ?, 1 associates to any x: 0 if x /X⊃, 1 if x X⊂ and ? → { } ∈ ∈ otherwise. J K J K Algorithm. Using a paver and an arithmetic on thick tests we show that we can easily obtain a guaranteed approximation of the relaxed intersection of m thick sets. TestCase: Let us take the following system of interval linear equa- tions [3]:
[2, 4] [ 2, 1] [ 1, 1] [0, 2] [−3, −1] [−5, 0] [ 6, 4]− [ 1,−0] x = [−6, 0] − − − · − [ 3, 2] [ 3, 2] [2, 7] − − − − [ 1, 1] [ 5, 4] [ 9, 3] − − − − −
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Each line corresponds to a thick set in R2. Using the relaxed in- tersection of thick tests, we obtain the characterization sets for q 1, 2, 3 as given by Figure 1. For q = 0, the solution set is empty. ∈ { }
Figure 1: Solution sets for q = 1 (Right), q= 2 (middle) and q = 3 (Right). Red boxes are inside X⊂, while blue ones are outside X⊃ and orange ones belong to ∆X = X⊃ X⊂. Other boxes are too small to be bisected and are undeterninated.\
References:
[1] Q. Brefort, L Jaulin, M. Ceberio, V. Kreinovich, To- wards Fast and Reliable Localization of an Underwater Object: An Interval Approach, Journal of Uncertain Systems, (2015).
[2] B. Desrochers, L. Jaulin, Computing a guaranteed approxima- tion the zone explored by a robot, IEEE Transaction on Automatic Control, (2016).
[3] V. Kreinovich and S. Shary, Interval Methods for Data Fitting under Uncertainty: A Probabilistic Treatment, Reliable Comput- ing, (2016).
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VERROU: a CESTAC evaluation without recompilation Fran¸coisF´evotte and Bruno Lathuili`ere EDF R&D 7 boulevard Gaspard Monge, F-91120, Palaiseau, FRANCE francois.fevotte, bruno.lathuiliere @edf.fr { }
Keywords: V&V, Numerical Verification, CESTAC, MCA, Valgrind As an industrial facility relying on numerical simulation to improve the safety and efficiency of its electricity production units, EDF is com- mitted to ensure that all the numerical simulation codes it develops and uses are correctly validated and verified. Within this context, the accuracy of floating-point operations has progressively become one of the important topics to study, especially since computing codes are ex- ploited on ever more powerful hardware to solve ever larger problems. The Verification and Validation (V&V) process should therefore in- clude the monitoring of inaccuracies introduced by floating-point arith- metic, as well as the verification that they are kept within acceptable limits. Numerous tools exist to diagnose floating-point problems, among which cadna [2] is one of the most advanced. It allows to assess floating-point inaccuracies, detect their origin in the source code, and follow their propagation throughout the computation. To this end, cadna requires instrumentating the source code to replace standard floating-point arithmetic by Discrete Stochastic Arithmetic (DSA), which is based on a synchronous cestac [4] method. cadna has already been successfully used on large industrial simulation codes [3], but instrumenting the source code of such tools can be hard, as there generally are numerous dependencies to third-party software libraries of which the development team only has limited understanding. We present the verrou tool, a valgrind-based system which im- plements an asynchronous cestac method to monitor the accuracy of floating-point operations without needing to instrument the source
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code or even recompile it. This tool is therefore well-suited to be part of an industrial V&V process. It has been successfully tested both on small-scale, well understood numerical applications, and on large-scale, more complex industrial computing codes such as Athena [1] which is used to simulate the propagation of ultrasonic waves in steel welds.
References:
[1] B. Chassignole, R. E. Guerjouma, M.-A. Ploix, and T. Fouquet, Ultrasonic and structural characterization of anisotropic austenitic stainless steel welds: Towards a higher re- liability in ultrasonic non-destructive testing. NDT & E Interna- tional 43, 4 (2010), 273 – 282.
[2] J.-L. Lamotte, J.-M. Chesneaux, and F. Jez´ equel,´ CADNA C: A version of CADNA for use with C or C++ programs. Computer Physics Communications 181, 11 (2010), 1925–1926.
[3] S. Montan, Sur la validation num´eriquedes codes de calcul in- dustriels. PhD thesis, Universit´ePierre et Marie Curie (Paris 6), France, 2013. In French.
[4] J. Vignes, A stochastic arithmetic for reliable scientific compu- tation. Mathematics and Computers in Simulation 35, 3 (1993), 233–261.
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Zero Verification in Systems of Equations: Interval-based Implementation of a Topological Test. Peter Franek, Marek Krˇc´aland Hubert Wagner IST Austria Am Campus 1, 3400 Klosterneuburg, Austria [email protected]
Keywords: Verification, Nonlinear equations, Computational Topol- ogy
Assume that B is a box in Rm and f : B Rn is continuous and our knowledge of f is limited. The function may→ come from imprecise measurements or previous numerical computations. One way to deal with the uncertainty is to represent f via an interval function. Our goal is to verify the existence of zero of f, if we only have access to an oracle I(f) that for a box B0 B outputs a box containing f(B0). If the dimensions of the domain⊆ and codomain are equal, then the problem is reduced to the computation of the topological degree and has been described in [3]. The degree test is provably stronger then older tests based on Brouwer, Miranda’s or Borsuk’s theorem. For under- determined systems, however, the situation is much more complex. A closely related problem asks whether f has a zero, if f is approx- imated via a given piecewise linear function g on a simplicial complex and we know that f g < 1. In [1] we showed that while this is undecidable in fullk generality,− k there exists a sequence of tests of in- creasing complexity, whereas a positive results of any of these tests certifies the zero of f. These tests can easily be adjusted to even ap- proximate the robustness of zero, that is, the maximal r so that each h with h f < r has a zero. The first and most accessible of these tests isk called− k the primary obstruction. In our current work, we implement the primary obstruction test in the framework of interval arithmetic. First experiments with random
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vector fields indicate that the primary obstruction is typically suffi- cient to detect zeros. In other words, higher obstructions (that we can detect in a slightly different simplicial setting) never occurred in the experiments [2].
References:
[1] Peter Franek, Marek Krcˇal´ , Robust Satisfiability of Systems of Equations, J. of the ACM, 2015
[2] Peter Franek, Marek Krcˇal,´ Hubert Wagner, Robust- ness of Zero Sets: Implementation, Preprint http://www.cs.cas.cz/~franek/rob-sat/experimental.pdf [3] Peter Franek, Stefan Ratschan, Piotr Zglizcynski, Quasi- Decidability of a Fragment of the First-order Theory of Real Num- bers, J. of Automated Reasoning, 2015
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Golden-mean universality for Siegel disks Denis Gaidashev and Michael Yampolsky University of Uppsala Box 480, Uppsala, Sweden [email protected]
Keywords: Siegel disks, renormalization, rigidity
We will describe a computer-assisted proof of one of the central open questions in one-dimensional renormalization theory – universal- ity of the golden-mean Siegel disks. We further show that for every function in the stable manifold of the golden-mean renormalization fixed point the boundary of the Siegel disk is a quasicircle which coin- cides with the closure of the critical orbit, and that the dynamics on the boundary of the Siegel disk is rigid.
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Fast determination of the tensorial and simplicial Bernstein enclosure J¨urgenGarloff 1,2, Jihad Titi 1 1Department of Mathematics and Statistics, University of Konstanz 2Institute for Applied Research, University of Applied Sciences/ HTWG Konstanz Konstanz, Germany [email protected] [email protected]
Keywords: Bernstein coefficients, tensorial Bernstein form, simplicial Bernstein form, range enclosure, subdivision
The underlying problems of our talk are unconstrained and con- strained global polynomial optimization problems over boxes and sim- plices. One approach for their solution is based on the expansion of a (multivariate) polynomial into Bernstein polynomials of the objective function and the constraints polynomials. This approach has the ad- vantage that it does not require function evaluations which might be costly if the degree of the polynomials involved is high.
The coefficients of this expansion are called the Bernstein coeffi- cients. These coefficients can be rearranged in a multi-dimensional array, the so-called Bernstein patch. From these coefficients we get bounds for the range of the objective function and the constraints over a box or a simplex, see [1, 2]. We can improve the enclosure for the range of the polynomial under consideration by elevating the degree of its Bernstein expansion or by subdivision of the region. Subdivision is more efficient than degree elevation. The complexity of the traditional approach for the computation of these coefficients is exponential in the number of the variables.
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In [1] Garloff proposed a method for computing the Bernstein coef- ficients of a bivariate polynomial over the unit box and the unit trian- gle by using a forward difference operator. We propose some efficient matrical methods that involve only matrix operations such as multipli- cation, transposition, and reshaping as Ray and Nataraj’s method, see [3]. Our methods are superior over Garloff’s and Ray and Nataraj’s method for the computation of the Bernstein coefficients over the unit and a general box and the standard simplex. We also propose a ma- tricial method for the computation of the Bernstein coefficients over subboxes and subsimplices when the original box and simplex are sub- divided, respectively.
References:
[1] J. Garloff, Convergent bounds for the range of multivariate poly- nomials, Interval Mathematics 1985, K. Nickel, Ed., Lecture Notes in Computer Science, 212(1986), Springer, Berlin, Heidelberg, New York, pp. 37–56.
[2] J. Titi and J. Garloff, Matrix methods for the tensorial Bern- stein form and for the evaluation of multivariate polynomials, sub- mitted.
[3] R. Leroy, Convergence under subdivision and complexity of poly- nomial minimization in the simplicial Bernstein basis, Reliable Computing, 17(2012), pp. 11–21.
[2] S. Ray and P.S.V. Nataraj, A matrix method for efficient com- putation of Bernstein coefficients, Reliable Computing, 17(2012), No. 1, pp. 40–71.
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Adaptive mesh refinement technique for the classical Plateau problem Aymeric Grodet and Takuya Tsuchiya Graduate School of Science and Engineering, Ehime University Matsuyama, Ehime, Japan [email protected]
Keywords: mesh refinement, error estimation, minimal surface
Let D = (u, v) R2 u2 + v2 < 1 be the unit disk with boundary ∈ ∂D = S1 the unit circle. Let Γ Rd be an arbitrary Jordan curve. We ⊂ would like to compute a minimal surface ϕ : D Rd with ϕ(∂D) = Γ. For such a minimal surface, it exists the following→ variational principle: H1(D; Rd) being the ordinary Sobolev space, let
d 1 d XΓ = ψ C(D; R ) H (D; R ) ψ(∂D) = Γ, ψ ∂D : monotone ∈ ∩ | where monotone means that if ∂D is traversed in a given direction, Γ is also traversed in the corresponding direction, although we allow arcs of ∂D to map onto single points of Γ. To such a map ϕ, we denote its Dirichlet integral by
d ∂ϕ 2 ∂ϕ 2 D(ϕ) = ϕ 2 dx = k + k dudv. D|∇ | D " ∂u ∂v # Z ZZ Xk=1 Then, ϕ XΓ is a minimal surface if and only if it is a stationary point of D∈(ϕ). The problem of finding such points is called the clas- sical Plateau problem [1]. The mapping ϕ is actually obtained by minimizing its Dirichlet integral. We need a normalization condition to determine a minimal surface (locally) uniquely. Several exist. For example, one can specify the images of three points on ∂D to define the same orientation on ∂D and Γ. In order to find a minimum, we have to move the interior
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points (inside the domain defined by the curve Γ) but also boundary points (on Γ), except for the ones we set as fixed points. If one apply a finite element approximation of minimal surfaces on the unit disk, with piecewise linear functions on each triangle of one triangulation, some inaccuracies can occur, in particular at singular- ities of Γ. We propose to overcome this difficulty by introducing a posteriori error estimation and an adaptive mesh refinement. This al- lows us to refine the mesh only in the local area corresponding to the problematic one on the domain defined by Γ. In this talk, we explain how to perform the finite element approximation and present several numerical examples to highlight what problems can occur and how to solve them.
References
[1] U. Dierkes, S. Hildebrandt, F. Sauvigny, Minimal Sur- faces, Springer, Berlin, 2010.
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When dependencies do not matter? Milan Hlad´ık Charles University, Faculty of Mathematics and Physics, Department of Applied Mathematics, Malostransk´en´am.25, 118 00, Prague, Czech Republic, [email protected]
Keywords: dependencies, interval linear equations, interval linear in- equalities
Dependencies are one of the most intractable obstacles in inter- val computation because they cause overestimation when evaluating expressions by interval arithmetic. Nevertheless, not always depen- dencies must cause overestimation. Hansen [1] presented an endpoint analysis to check if the enclosure computed by a natural interval ex- tension is optimal. There are also classes of interval problems, where dependencies do not matter. As shown by [3, 4], the (united) solution set of the interval system Ax = b is the same as the solution set of interval inequalities (ignoring depen- dencies) Ax b, Ax b. ≤ ≥ This property was then generalized [2] to the so called AE solutions and under some assumptions to AE solvability. As another example, a system Ax = b is strongly solvable if and only if the system
Ax1 Ax2 = b, x1, x2 0 − ≥ is strongly solvable. We will show more examples of problems, where dependencies can be ignored and the problem will not change. Moreover, we will show
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possible transformations of interval linear systems from equations to inequalities and vice versa, related to weak and strong solvability. Such transformations can be very useful in deriving new properties or simpli- fying the complicated proofs characterizing various kinds of solvability. We also discuss possible extensions to parametric systems.
References
[1] E.R. Hansen, Sharpness in interval computations, Reliable Com- puting, 3 (1997), No. 1, pp. 17–29.
[2] M. Hlad´ık, Transformations of interval linear systems of equations and inequalities, submitted.
[3] W. Li, A note on dependency between interval linear systems, Optimization Letters, 9 (2015), No. 4, pp. 795–797.
[4] J. Rohn, Miscellaneous results on linear interval sys- tems, Freiburger Intervall-Berichte 85/9 (1985), Albert-Ludwigs- Universit¨at,Freiburg.
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IeeeCC754++ – an advanced tool to check IEEE 754-2008 conformity Matthias H¨usken University of Wuppertal Gaußstr. 20, 42119 Wuppertal, Germany [email protected]
Keywords: IeeeCC754++, IEEE 754-2008, compliance checker Running numerical algorithms on a given computing platform in- volves a lot more components than visible at first glance, amongst which are the programming language of choice, the compiler, and the hardware on which the program is executed. Ideally, it should not be necessary for a user to fully understand all parts involved and the al- gorithms should give reproducible, platform-independent results. The user should thus rely on the means provided by the chosen program- ming language and trust that all other components work together fol- lowing the philosophy of the IEEE 754-2008 standard, or more pre- cisely, that all components adhere to this standard as far as needed to supply a conforming platform as a whole. With this in mind, IEEE 754-2008 explicitly specifies that it “pro- vides a discipline for performing floating-point computation that yields results independent of whether the processing is done in hardware, soft- ware, or a combination of the two.”[IEEE2008, p. iv], and it states that “conformance to this standard is a property of a specific implementa- tion of a specific programming environment, rather than of a language specification.”[IEEE2008, p. 2] In this talk, we introduce IeeeCC754++, a flexible testing tool that together with an extensive evaluation framework enables the user to evaluate the IEEE 754-2008 conformity of the computing platform that is used to run the target numerical application. Furthermore, IeeeCC754++ comes with a large selection of ports that enable testing a wide variety of architectures, specific hardware floating-point units, and software libraries such as MPFR.
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IeeeCC754++ is based on IeeeCC754 [VCV01a, VCV01b] and closely follows its testing philosophy: checking for standard conformity by ex- ecuting a large number of test vectors whose correct results are hard to achieve, e. g. because they are hard to round. Furthermore, test vectors that check for special representations like NaNs or infinities as well as test vectors to verify standard-conforming use of exception flags are included. With this systematic approach of checking partic- ularly difficult cases, it is possible to identify flaws in a floating-point implementation which might go undetected otherwise. We present the following extensions that were implemented in our new tool IeeeCC754++: Support for the IEEE 754-2008 standard.. • A default testing mode to check the current user environment, • taking the chosen compiler and compiler options into account. Ports for a number of hardware architectures and their floating- • point units. An evaluation framework including advanced analysis capabilities. • We will conclude with test results from selected architectures that em- phasize the need for testing tools like IeeeCC754++. References: [IEEE08] IEEE Std 754-2008, Standard for Floating-point Arithmetic, IEEE, 2008. [VCV01a] B. Verdonk, A. Cuyt, and D. Verschaeren. A precision- and range-independent tool for testing floating-point arithmetic I: basic operations, square root and remainder. ACM Transactions on Mathematical Software, 27:92–118, 2001. [VCV01b] B. Verdonk, A. Cuyt, and D. Verschaeren. A precision- and range-independent tool for testing floating-point arithmetic II: conversions. ACM Transactions on Mathematical Software, 27:119–140, 2001.
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Towards Fast, Accurate and Reproducible LU Factorization Roman Iakymchuk, David Defour and Stef Graillat CST/PDC, CSC, KTH Royal Institute of Technology Lindstedtsv¨agen5, 11428 Stockholm, Sweden [email protected] DALI–LIRMM, Universit´ede Perpignan 52 avenue Paul Alduy, F-66860 Perpignan, France [email protected] Sorbonne Universit´es, UPMC Univ Paris 06, UMR 7606, LIP6 4 place Jussieu, F-75005 Paris, France [email protected]
Keywords: Triangular solver, matrix-vector product, LU factoriza- tion, reproducibility, accuracy, long accumulator, error-free transfor- mation The process of finding the solution of a linear system of equations is often the core of many scientific applications. Usually, this pro- cess relies upon the LU factorization, which is also the most compute- intensive part of it. Although current implementations of the LU fac- torization may reach 70% of the peak performance, their accuracy and, even more, reproducibility cannot be guaranteed, mainly, due to the non-associativity of floating-point operations and dynamic thread scheduling. In this work, we address the problem of reproducibility of the LU factorization due to cancelations and rounding errors, resulting from floating-point arithmetic. Instead of developing a completely inde- pendent version of the LU factorization, we benefit from the hier- archical structure of linear algebra libraries and start from develop- ing/enhancing reproducible algorithmic variants for the kernel oper- ations – like the ones included in the BLAS library – that serve as building blocks in the LU factorization. In addition, we aim at ensur- ing the accuracy of these underlying BLAS routines.
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We consider an unblocked algorithmic variant of the LU factoriza- tion that can be expressed in terms of BLAS level 1 and 2 routines: triangular solver (TRSV), dot product (DOT) and matrix-vector prod- uct (GEMV). We prevent cancellations and rounding errors during the accumulation stage thanks to a long accumulator and error-free transformations [1]. Consequently, we tackle the problem of accuracy in the reproducible triangular solver (ExTRSV) [2] through iterative refinement. Moreover, we provide an accurate and reproducible al- gorithmic variant as well as an implementation of the matrix-vector product. Thus, following the bottom-up approach, we construct the reproducible algorithmic variant of the LU factorization. Up to our knowledge, this is the first implementation of a reproducible LU fac- torization. Finally, we present initial evidence that both accurate and reproducible higher-level linear algebra routines can be constructed following our hierarchical bottom-up approach.
References:
[1] S. Collange, D. Defour, S. Graillat, R. Iakymchuk, Nu- merical Reproducibility for the Parallel Reduction on Multi- and Many-Core Architectures, Parallel Computing, 49 (2015), pp. 83– 97.
[2] R. Iakymchuk, S. Collange, D. Defour, S. Graillat, Re- producible Triangular Solvers for High-Performance Computing, In the Proceedings of the 12th International Conference on Informa- tion Technology: New Generations (ITNG 2015), Special track on: Wavelets and Validated Numerics, April 13-15, 2015, Las Vegas, Nevada, USA, pp. 353-358.
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Secure a zone with robots
Luc Jaulin and BenoˆıtZerr
Lab-STICC, ENSTA Bretagne, Brest, France
Problem. We consider n underwater robots ,..., at po- R1 Rn sitions a1,..., an and moving in a 2D world [1]. Each robot has a visibility zone. If an intruder is inside the visibility zone of one robot, it is detected. The robots have to collaborate to guarantee that they is no moving intruder inside a subzone of a compact subset O of R2, representing the 2D ocean. Complementary approach. We assume that there exists a vir- tual intruder moving inside O satisfying the differential inclusion
x˙ (t) F(x(t)), ∈ where x(t) is the state vector. Moreover, we assume that each robot 1 i has a visibility zone of the form ga−i ([0, d]) where d is the scope. Our contributionR is to show that characterizing the secure zone translates into a set-membership set estimation problem [2] where x(t) is shown to be inside the set X(t) returned by our set-membership observer. Then we conclude that x(t) cannot be inside the complementary of X(t). This result can be formalized by the following theorem. Theorem. The virtual intruder has a state vector x(t) inside the set
1 X(t) = O dt F(X(t dt)) g− ([d(t), ]), ∩ · − ∩ ai(t) ∞ i \ where X(0) = O. As a consequence, the secure zone is
S(t) = projworld(X(t)).
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Figure 1: (left) Paving of the Biscay Bay, (right) Green: Secured zone
Proof (sketch). Two cases should be considered. If no actual intruder exists then S(t) cannot is secured. If the virtual intruder is a real one, its state x(t) is inside X(t) and its position (which is a part of the state) is inside projworld(X(t)). In both situations, the intruder can not be inside S(t). Method. Each robot follows a reference point. All reference points form a flat ellipsoid which plays the role of a barrier. The strategy is illustrated by Figure 1 for 10 robots. The set O corresponds to the blue area (left). On the right, S(t) is painted green. The observer has been implemented using interval analysis. References:
[1] L. Jaulin, Mobile robotics, ISTE editions, London, 2016.
[2] M. Kieffer, L. Jaulin, E. Walter, D. Meizel, Nonlinear Identification Based on Unreliable Priors and Data, with Applica- tion to Robot Localization, Robustness in Identification and Con- trol, 3 (1999), LNCIS 245, pp. 190–203.
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A Decimal Multiple-Precision Interval Arithmetic Library Stef Graillat, Clothilde Jeangoudoux, and Christoph Lauter Sorbonne Universit´es,UPMC Univ Paris 06, CNRS, LIP6 UMR 7606 4 place Jussieu, 75005 Paris, France [email protected], [email protected], [email protected]
Keywords: interval arithmetic, multi-precision, IEEE 754-2008 deci- mal format.
We describe the mechanisms and implementation of a library that define a decimal multiple-precision interval arithmetic. The aim of such a library is to provide guaranteed and accurate results in dec- imal. This library contains correctly rounded elementary operations and some transcendental functions such as sin, cos or exp. The application fields of such a library are wide from financial cal- culation to aeronautic engineering. In those fields, complex algorithms are generally designed in high level with decimal numbers. Thus there is a need for correctly rounded decimal arithmetic multiple-precision library. The IEEE 754-2008 [2] revision introduced the definition of decimal floating-point format, with the purpose of providing a basis for a robust and correctly rounded decimal arithmetic. Apart from being IEEE 754-2008 compliant, our library holds sev- eral useful properties, such as correct rounding for decimal arbitrary precision. The correctness of some functions will be proven. Further- more the interval arithmetic will be compliant with the new IEEE 1788-2015 Standard for Interval Arithmetic [3]. Studies of implementation of decimal arithmetic libraries have shown that one of the simplest and efficient way to implement decimal func- tions is to use their binary counterparts [1]. This method makes it possible to rely on the efficiency of the binary arithmetic and focus
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more effort only on some critical points. The implementation of this library is based on GMP and MPFR libraries.
References
[1] J. Harrison, Decimal Transcendentals via Binary, Proceedings of the 19th IEEE Sympoisum on Computer Arithmetic, 2009, pp. 187-194.
[2] IEEE Std 754-2008, IEEE Standard for Floating-Point Arithmetic, 2008.
[3] IEEE Std 1788-2015, IEEE Standard for Interval Arithmetic, 2015.
[4] GNU MPFR Library, multiple-precision floating-point computa- tions with correct rounding based on GMP.
[5] MPFI Library, A multiple precision interval arithmetic library based on MPFR.
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The Implementation of multi-precision package in multiple-component format in MATLAB Hao Jiang, Peibing Du, Kuan Li, Lin Peng, College of Computer, National University of Defense Technology 410072 Changsha, China haojiang,kuanli @nudt.edu.cn { }
Keywords: multi-precision, multiple-component, error-free transfor- mation, overloading.
High precision computations are required in many application ar- eas, such as algorithm’s verification. There are serval methods and responding libraries to obtain high precision, including arbitrary pre- cision library MPFUN, ARPREC, GMP, and MPFR. These libraries store numbers in a multiple-digit format similar to the symbolic com- putation package Mathematica and the symbolic toolbox and VPA in Matlab. Another well-known and interesting package is QD library [1], which uses the multiple-component format. The QD library de- fines a double-double number and a quad-double number, which are represented as an unevaluated sum of two and four double numbers, respectively. Trip-double number and its responding arithmetic are proposed by Lauter [2]. The packages described above are written in C or Fortran codes. Meanwhile, Matlab provides a familiar and efficient problem solving environment for many engineers. Some multi-precision matlab toolboxes based on the above arbitrary precision libraries have been proposed such as MPL[3] and mp[4]. Advanpix[5] is a commercial package having good performance, but the algorithms involved are in- visible. The literature [6] implements real double-double number and the corresponding operations in Matlab. In this paper, we implemented a multi-precision package in Mat- lab based on multiple component format. In the package, double- double(DD), triple-double(TD), and quad-double(QD) numbers are
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Table 1: Basic operations and functions implemented z = x + y z = exp(x) z = DD(x) ==, = z = x y z = sqrt(x) z = TD(x) <, >,∼ <=, >= z = x −y z = sqr(x) z = QD(x) z = real(x) z = x/y∗ z = ln(x) z = DDrand(x) z = imag(x) z = xn z = log10(x) z = T Drand(x) z = floor(x) z = xy z = inv(x) z = QDrand(x) z = fix(x) z = abs(x) z = conj(x) z = round(x) z = ceil(x) defined as a Matlab class including real and complex format. The ba- sic arithmetic operations, relational operation, some Matlab functions and so on are overloaded based on this class shown in Table 1. This package allows Matlab obtain the more accurate numerical results and run much faster than VPA, MPL and mp with some required accuracy. References: [1] Y. Hida, X.S. Li, D. H. Bailey, Algorithms for quad-double precision floating point arithmetic, ARITH, San Diego, CA, 2001. IEEE Computer Society, pp. 155–162. [2] C.Q. Lauter, Basic building blocks for a triple-double interme- diate format, Technical report LIP6,UCMP, Paris,France, 2005.. [3] W. Schreppers, F. Backeljauw, and A. Cuyt, MPL: A Multiprecision Matlab-Like Environment, LNCS, vol:3514, 295- 303, 2005. [4] www.sorceforge.net/projects/mptoolbox/. Multiple precision tool- box for MATLAB. [5] www.advanpix.com. Advanpix multiprecision computing toolbox [6] D. Tsarapkina and D. J. Jeffrey, Exploring Rounding Errors in Matlab using Extended Precision, Procedia Computer Science, vol:29, 1423-1432, 2014.
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A study on verified ODE solver from the standpoint of stiffness Masahide Kashiwagi Waseda University Tokyo, Japan [email protected]
Solving initial value problems of ordinary differential equations is very important and several studies for them have been proposed. We have proposed an algorithm for verified ODE solver for IVP using power series arithmeric and affine arithmetic [1]-[4]. Power series arithmetic defines operations between order-p polyno- mials like
2 p 1 p x(t) = x0 + x1t + x2t + xp 1t − + Xpt ··· − where the last coefficient Xp is interval. For IVP:
x0(t) = f(x(t), t), x(t0) = x0 , we use (t0-shifted) Picard type fixed point form:
t x(t) = x + f(x(t), t + t ) P (x(t)) 0 0 ≡ Z0 and if P (x∗(t)) x∗(t) for some interval polynomial, then x∗(t) en- closes true solution.⊂ For avoiding so-called wrapping effect, we always use Jacobian of the operator φti,ti+1 , where φti,ti+1 : x(ti) x(ti+1) . We use mean value form and affine arithmetic to connect solutions7→ over multi steps. For calculating φ0, we use variational equation w.r.t. initial value:
y0(t) = fx(x∗(t), t)y(t), y(t0) = I. We have tested two algorithms (Algo1, Algo2) for calculating y(t). Details will be shown at presentation.
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As a numerical example, we take up van der Pol equation: 2 x00 µ(1 x )x0 + x = 0 . − − It is known that the equation is non-stiff for small µ and become stiff for large µ. We show calculation results for µ = 1 and µ = 100 under the following environment: Intel Xeon CPU E5-2687W (3.10GHz), Ubuntu 14.04, gcc 4.8, kv-0.4.36, initial value (x, x0) = (1, 1), domain = [0, 200], order of polynomial is 24 and local error is machine epsilon. µ = 1 µ = 100
10 2.5 x 2 x 1.5 1 1 0.5
x x 0 -0.5 0.1 -1 -1.5 -2 0.01 -2.5 0 50 100 150 200 0 50 100 150 200 t t
algorithm1 1 algorithm1 1 algorithm2 algorithm2 awa 0.1 awa
0.1 0.01 stepsize stepsize 0.001 0.01 0.0001 0 50 100 150 200 0 50 100 150 200 t t algo1 2.276sec 53.820sec algo2 1.798sec 13.478sec AWA 3.815sec 176.78sec We can see that algo2 can take large step size and calculation become faster consequently in stiff case. References: [1] M, Kashiwagi, S. Oishi, Numerical Validation for Ordinary Dif- ferential Equations — Iterative Method by Power Series Arith- metic —, NOLTA’94, pp.243–246 (1994.10.7). [2] M, Kashiwagi, kv – C++ Numerical Verification Libraries, http: //verifiedby.me/kv/ . [3] M, Kashiwagi, Verified ODE(IVP) Solver, http://verifiedby. me/kv/demo/ode.cgi . [4] M, Kashiwagi, An algorithm to reduce the number of dummy variables in affine arithmetic, SCAN2012 (2012.9.28).
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Simplicial Branch and Bound in Interval Global Optimization Ralph Baker Kearfott University of Louisiana at Lafayette Department of Mathematics, U.L. Box 4-1010, Lafayette, LA 70504-1010 [email protected]
Keywords: rigorous global optimization, branch and bound, range enclosure
The predominant shape chosen for search regions in branch and bound (B&B) algorithms for global optimization of continuous func- n tions is a box x = x R xi xi xi, 1 i n , while the pre- dominant method of{ branching∈ | has≤ been≤ bisection≤ ≤ of one} of the coordi- nate directions. This scheme has been popular because of its simplicity, because a sub-region provides clear error bounds on individual param- eters, and because bounds on the coordinates occur naturally in many problems. However, other region shapes also have advantages and have been considered. One alternative is an n-simplex = P0,P1,...,Pn , the n S h i convex hull of n + 1 points Pi R . A relatively early B&B method dealing with continuous nonlinear∈ systems was proposed by Stenger [4], a method subsequently enhanced with more elegant formulas by Stynes [5, 6] and the present author [1, 2], although those techniques used a heuristic to decide when to branch. Due to certain advantages, simplexes as domains in B&B algo- rithms have received renewed scrutiny in more recent work. As sur- veyed in [3], Zilinskasˇ et al develop simplicial B&B to take advantage of certain naturally occurring symmetries is the objective and constraints. Here, in the spirit of Paulavˇciusand Zilinskas,ˇ we investigate sim- plex-based B&B algorithms in contexts in which the feasible region is best described by a simplex. However, we develop representations that
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allow interval evaluations (and hence mathematically rigorous bounds on ranges) with relatively small overestimation. We present appro- priate ways to represent the simplexes, we derive associated relatively sharp mean-value type interval extensions, and we propose and discuss branching processes related to these representations.
References
[1] Ralph Baker Kearfott. Computing the degree of maps and a general- ized method of bisection. Ph.D. thesis, Department of Mathematics, University of Utah, Salt Lake City, UT, USA 84112, 1977.
[2] Ralph Baker Kearfott. An efficient degree-computation method for a generalized method of bisection. Numerische Mathematik, 32(2):109–127, June 1979.
[3] Remigijus Paulavˇciusand Julius Zilinskas.ˇ Simplicial Global Opti- mization. Springer Verlag, 2014.
[4] Frank Stenger. Computing the topological degree of a mapping in Rn. Numerische Mathematik, 25(1):23–38, March 1975.
[5] Martin Stynes. A simplification of Stenger’s topological degree for- mula. Numerische Mathematik, 33(2):147–155, June 1979.
[6] Martin Stynes. On the construction of sufficient refinements for computation of topological degree. Numerische Mathematik, 37(3):453–462, July 1981.
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Optimal order constructive a priori error estimates for a full discrete approximation of the heat equation
Takuma Kimura∗, Teruya Minamoto† and Mitsuhiro T. Nakao‡
∗† Saga University, Saga 840-8502, Japan. ‡ Kyushu University, Japan. ∗ [email protected]
Keywords: parabolic problem, Galerkin methods, constructive a pri- ori error estimates
We consider the constructive a priori error estimates for an approx- imate solution of the following equations with homogeneous initial and boundary conditions:
∂u ν u = f(x, t) in Ω J, ∂t − 4 × u(x, t) = 0 on ∂Ω J, (1) × u(x, 0) = 0 in Ω.
d where Ω R (d 1, 2, 3 ) is bounded polygonal or polyhedral do- ⊂ ∈ { } mains, J := (0,T ) R (for a fixed T < ) is an open interval, ν is ⊂ 2 2 ∞ k a positive constant and f L (J; L (Ω)). Let Ph u be a full discrete approximation for (1) studied∈ in [1], [2], where h and k are mesh size for Ω and J, respectively. Our method is based on the finite element Galerkin method with an interpolation in time that uses the fundamen- tal solution for semidiscretization in space. In [1], the authors studied 2 1 2 2 the optimal order L (J; H0 (Ω)) and L (J; L (Ω)) error estimates for k 2 Ph u with the assumption that h = k . In this talk, assuming that f is sufficiently smooth, we show the optimal order error estimates can be obtained even if h = k2. In 6 the below, let CΩ(h) and CJ (k) denote constants determined by the approximation property with order of O(h) and O(k), respectively.
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Then we prove the following error estimates. 1 H0 -error estimates: k u P u 2 1 || − h ||L H0 ˆ 2 1 2 2 ∂ 2 C (h, k) f 2 2 + f( , 0) + f 2 2 + f 2 2 ≤ 1 ν || ||L L √2ν || · ||L2(Ω) || ||L L ||∂t ||L L q where Cˆ1(h, k) := max CΩ(h),CJ (k) . L2-error estimates: { } k ∂ u P u 2 2 Cˆ (h, k) f 2 2 + f 2 2 + P f( , 0) 2 k − h kL L ≤ 0 k kL L k∂t kL L k∇ 0 · kL (Ω) ˆ 8 2 2 2 2 where C0(h, k) := max ν C Ω(h) , √νCJ (k) ,CJ (k) and P0 is the L - projection. V 1-error estimates: ∂ k ∂ (u P u) 2 2 Cˆ (h, k)C( f 2 2 , f 2 2 , f( , 0) 2 ) k∂t − h kL L ≤ 3 || ||L L ||∂t ||L L ||∇ · ||L (Ω) where Cˆ (h, k) := max C (h),C (k) ( Cˆ (h, k)), 3 { Ω J } ≡ 1 ∂ C( f L2L2 , ∂tf L2L2 , f( , 0) L2(Ω)) || || || || ||∇ · || 1 2 2 ∂ 2 2 2 2 := (C (h) + ν)(2 f 2 2 + ν f( , 0) 2 ) + 2 f 2 2 ν Ω ||∂t ||L L ||∇ · ||L (Ω) || ||L L q n 1 o 2 ∂ 2 2 + ν f( , 0) 2 + f 2 2 ||∇ · ||L (Ω) ||∂t ||L L In the talk,n some numerical examples whicho confirm the expected rate of convergence will be presented. References: [1] M. T. Nakao, T. Kimura, T. Kinoshita, Constructive a priori error estimates for a full discrete approximation of the heat equa- tion, SIAM Journal on Numerial Analysis, 51 (2013), pp. 1525– 1541. [2] T. Kinoshita, T. Kimura, M.T. Nakao, On the a posteriori estimates for inverse operators of linear parabolic equations with applications to the numerical enclosure of solutions for nonlinear problem, Numerische Mathematik, 126 (2014), pp. 679–701.
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Decision Making Under Interval Uncertainty as a Natural Example of a Quandle Mahdokht Afravi and Vladik Kreinovich University of Texas at El Paso El Paso, Texas 79968, USA [email protected]
Keywords: decision making, interval uncertainty, quandle Need for decision making. In many real-life problems, we need to select an alternative a from the list of possible alternatives – e.g., we want to select a design and/or location of a plant, a financial invest- ment, etc. In many such situations, we have a well-defined objective function u(a) that describes our preferences. If we know the exact value of u(a) for each alternative a, then we select the alternative with the largest value of u(a). Decision making under interval uncertainty. In practice, we usu- ally only know the consequences of each decision with some uncertainty. Often, the only information that we have about the corresponding val- ues of u(a) is that it is somewhere between the known bounds u(a) and u(a), i.e., that u(a) [u(a), u(a)]. In such situations, to make a definite decision, we need to∈ assign, to each such interval, a single numerical value u0 describing the quality of the corresponding alternative. We will denote this value u0 by u(a) B u(a). Natural properties of the corresponding operation B. What are the natural properties of the operation a B b? First, if we know the exact value of u(a), i.e., if the corresponding interval has the form [x, x] for some x, then the corresponding equiva- lent value is simply equal to x: x B x = x. Another reasonable property is monotonicity: if x < x0, then x B y > x0 B y. (For continuous functions, monotonicity is equivalent to invertibility of x x y.) → B 73 SCAN 2016
To get the third property, let us consider a slightly more complex situation, when we know the lower bound z of the corresponding in- terval, but we do not know its upper bound: we only know that this upper bound is between y and x. We can analyze this situations in two different ways. First, we can say that since all we know about the upper bound is that it is between y and x, this upper bound is therefore equivalent to the value y B x. Now, after we have thus reduced the uncertain upper bound to a single number, the original information becomes simply an interval with an exact lower bound z and an exact upper bound x B y. We can now apply the operation B to estimate the equivalent value of this interval as (x B y) B z. There is also an alternative approach. For each possible value v between y and x, we have an interval [z, v] with equivalent value v B z. Due to the natural monotonicity, this equivalent value is the smallest when v is the smallest, i.e., when v = y, and it is the largest when v is the largest, i.e., when v = x. Thus, possible equivalent values form an interval [y B z, x B z]. The equivalent value of this interval is therefore (x B z) B (y B z). It is reasonable to require that these two approaches lead to the same value, i.e., that (x B y) B z = (x B z) B (y B z). Hurwicz optimism-pessimism criterion satisfies these proper- ties. One can easily check that the widely used Hurwicz criterion x y = α x + (1 α) y, with α > 0, satisfies the above properties. B · − · This is a quandle. Interestingly, the above three natural properties are well known in knot theory: sets with operations satisfying these properties are knows as quandles [1]. Since decision making under interval uncertainty naturally leads to a quandle, maybe we will be able to apply results from quandle theory to make better decisions? References: [1] M. Elhamdadi, S. Nelson, Quandles: An Introduction to the Algebra of Knots, American Mathematical Society, Providence, Rhode Island, 2015.
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A template-based C++ library for automatic differentiation and hull consistency enforcing Bart lomiejKubica Department of Applied Informatics, Warsaw University of Life Sciences Nowowiejska 159, Warsaw, Poland bartlomiej [email protected]
Keywords: automatic differentiation, hull consistency, C++ template meta-programming Automatic differentiation (AD) is often used in interval algorithms for optimization, nonlinear systems of equations or solving other deci- sion problems (see, e.g., [4]). The existing software for this purpose, like the one from the Toolbox of C-XSC library [1], has several limita- tions and drawbacks. As for this specific package: there are distinct classes for computing first or second derivatives • and for univariate or multivariate functions; there are global variables to distinguish the order of computed • derivative – these variables have to be checked at runtime, sev- eral times during the computation; they also affect multithreaded implementations [3]; computing derivatives of higher order would require to implement • a separate (but analogous) class; although, the developers of C-XSC have provided several useful • classes for sparse matrices and vectors, their AD code makes no use of it. What is more, if we need to use some hull consistency based narrowing (HC3, HC4), the representation of functions using AD classes is of little use to us.
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Many of these drawbacks can be overcome by using C++ template meta-programming [2]. This allows automatic generation of similar code for distinct data types (single intervals or their vectors/matrices, sparse or dense types, etc.). The author is going to present a novel C++ template library, hav- ing, in particular, the following features:
a single template class used for generating all derivatives and gen- • erating the procedure to create syntactic tree of the function – all of them at compile time;
generating (at compile time) separate functions to compute various • derivatives;
possibility of using both dense and sparse formats for gradients • and Hesse matrices;
potential possibility to computing higher derivatives; • integration with procedures enforcing hull consistency. •
References:
[1] C-XSC library, http://www.xsc.de.
[2] A. Alexandrescu, Modern C++ Design: Generic Programming and Design Patterns Applied, Addison-Wesley, 2001.
[3] B.J. Kubica, A. Wo´zniak, A multi-threaded interval algorithm for the Pareto-front computation in a multi-core environment, LNCS, 6126, published online.
[4] B.J. Kubica, A class of problems that can be solved using interval algorithms, Computing, 94 (2012), No. 2, pp. 271–280.
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Efficient Dedekind reals in Haskell Ivo List Faculty of Mathematics and Physics University of Ljubljana Ljubljana, Slovenia [email protected]
Keywords: Dedekind reals, ASD, Haskell
Exact reals may be described using Dedekind construction as cuts. In such construction the cut is described by two predicates that de- scribe lower and upper bounds. Predicates may be composed of arith- metics, functions, inequalities and quantifiers. Haskell with its flexible type system allows us to descibe concrete predicates without implementating them. On top of those descriptions we can implement different evaluation strategies. Concrete Haskell implementation will be presented with example descriptions of exact reals and evaluating them using different strate- gies. The basic evaluation strategy employs bisection and splitting of intervals. Although it is inefficient it serves as a proof of concept. More efficient strategy uses Kaucher arithmetics with back-to-front intervals and employs extended interval Newton method. Improving those strategies is currently in progress. On the other hand Haskell has some pitfalls, most pronounced in lack of support for efficient low-level floating-point and interval li- braries. Quite some time has been invested to overcome these.
References:
[1] A. Bauer, P. Taylor, The Dedekind reals in abstract Stone duality, Mathematical Structures in Computer Science; MSCS, 19 (2009), No. 4, pp. 757–838.
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[2] P. Taylor, A lambda calculus for real analysis, Journal of Logic and Analysis, 2 (2010), No. 5, pp. 1–115.
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A framework for high-precision verified eigenvalue bounds by using finite element methods Xuefeng LIU Niigata University 8050 Ikarashi 2-no-cho, Nishi-ku, Niigata City, Niigata 950-2181 Japan [email protected]
Keywords: differential operators, eigenvalue bounds, finite element method, Lehmann–Goerisch’s theorem For the purpose of high-precision eigenvalue bounds for differential operators, a general framework based on finite element methods is proposed. The high-precision eigenvalue bounds are obtained through the following two steps. First, rough but exact eigenvalue bounds can be obtained by us- ◦ ing lower order finite element methods along with a fundamental theorem in [1]. For example, for the Laplace operator defined on a bounded domain, the eigenvalues can be easily estimated by applying the Crouzeix–Raviart finite element. For biharmonic op- erators, the Fujino–Morley FEM can be adopted to give explicit eigenvalue bounds. Second, to obtain high-precision eigenvalue bounds, Lehmann– ◦ Goerisch’s theorem along with high-order finite element methods is adopted, where the rough eigenvalue bounds obtained in the first step is important [3, 2]. See Table 1 for a sample computa- tion result for eigenvalues of Laplacian with homogeneous Dirichlet boundary condition over square-minus-square domain, where there exist singularities of eigenfunction around the reentrant corners. By further adopting the interval arithmetic, the explicit eigenvalue bounds from numerical computations can be mathematically correct.
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As a computer-assisted proof, the verified eigenvalue bounds have been used to investigate the solution existence of semi-linear elliptic differ- ential equations; see, e.g., [4].
Table 1: Bounds for the leading 5 eigenvalues of Laplacian over square- minus-square domain [2]
λi Eigenvalue bounds (8, 8) 64 (1, 7) 1 9.1602137 89 2 9.1700861 89 3 9.1700861 4 9.1805680 52 (7, 1) 37 5 10.08984314 (0, 0)
References
[1] Liu, X., A framework of verified eigenvalue bounds for self-adjoint differential operators, Applied Mathematics and Computation, 267, pp.341-355, 2015 [2] Liu, X., Okayama, T. and Oishi, S., High-precision eigenvalue bound for the Laplacian with singularities. Computer Mathemat- ics, pp.311-323, Springer, 2014 [3] Liu, X. and Oishi, S., Guaranteed high-precision estimation for P0 interpolation constants on triangular finite elements, Japan Journal of Industrial and Applied Mathematics, 30, pp.635-652, 2013 [4] Takayasu, A., Liu, X. and Oishi, Verified computations to semi- linear elliptic boundary value problems, Nonlinear Theory and Its Applications, IEICE, 4(1), pp.34-61. doi:10.1588/nolta.4.34, 2013
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Interval Methods in the Calculation of Solutions to Fuzzy Interval Linear Systems Weldon A. Lodwick1, Lotfi Taher2 and Hana Veiseh2 1. University of Colorado Denver, Department of Mathematical and Statistical Sciences 2. Islamic Azad University, Hamedan Branch, Hamedan, Iran, Department of Applied Mathematics [email protected]
We present methods to obtain proper fuzzy interval vector solution sets (fuzzy intervals with nested α levels) arising from systems of fuzzy interval linear equations. Since computing− solutions, at times, do not produce proper fuzzy interval solutions, special care must be taken when obtaining solutions that are indeed fuzzy interval solutions. The theory of how to obtain proper fuzzy interval solutions to fuzzy linear systems is discussed. In particular, we present the conditions necessary for the solutions to fuzzy interval linear systems (both equal- ity and inequality) to exist as proper fuzzy interval vectors. We use more recent results that unify fuzzy interval linear systems as well as some older, early 1960tees, results from Oettli as it pertains to the computation of solutions to interval linear systems and the α level interval systems in fuzzy interval linear systems. −
References:
[1] R.E. Moore, R.B. Kearfott, M.J. Cloud, Introduction to Interval Analysis, SIAM, Philadelphia, 2009.
[2] A.V. Lakeyev, V. Kreinovich, NP-hard classes of linear al- gebraic systems with uncertainties, Reliable Computing, 3 (1997), No. 1, pp. 51–81.
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LILIB – Long Interval Library Nozomu Matsuda and Nobito Yamamoto The University of Electro-Communications 1-5-1 Chofu-ga-oka, Chofu, 182-8585, Tokyo, Japan [email protected]
Keywords: verified numerics, multiple precision arithmetic, center- radius form We have been developing a new computer program library on multi- ple precision arithmetic with verified computation named LILIB, Long Interval Library. It adopts interval arithmetic for verified computation. Whereas most of existing library for interval arithmetic, e.g. MPFI [1], represent an interval by its infimum and supremum, LILIB uses center- radius form in which an interval is represented by its center and radius. Center-radius form has advantages against inf-sup form in memory ca- pacity and speed on computation if the digits of the numbers is long enough. MPFI represents an interval number by floating point numbers of MPFR [2] as infimum and supremum and exploits the advantage of cor- rect rounding of MPFR for verified computation. Since the processes of computation for infimum and supremum are almost same, the com- putational time of interval arithmetic in MPFI takes twice of floating point number arithmetic of MPFR as well as the memory capacity. An interval number of LILIB is represented by a center value of long digits number and a radius of short digits. This structure comes from the multiple precision interval numbers in INTLAB and obviously it reduces the memory capacity of an interval number. Moreover it is supposed to reduce the computational time since the amount of com- putation of long center and short radius seems to be less than twice of long digits computation. On the other hand the processes of interval arithmetic for center-radius form are more complicated than inf-sup form. This makes it unclear for us whether less computational time would be attained or not. In order to get an answer to the question, we
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implemented center-radius form and floating point system in a previous version of LILIB [3] [4] , in which we had developed our original meth- ods for multiple precision arithmetic without using MPFR. Despite the fact that the previous LILIB is actually so slow in computational time, we have confirmed that the center-radius form is faster than twice of single floating point arithmetic in our system as long as the digits are long enough, which implies center-radius form is faster than inf-sup form. We have improved LILIB to be faster in computation. Our new version of LILIB is based on MPFR which is used in calculation of multiple precision numbers for both of center and radius. When the digits are long enough, the new version will be faster than MPFI. In our talk, we will explain the details of specification and implementation of the new version and show some numerical examples. Comparison with MPFI will be also mentioned if possible. References: [1] N.Revol and F.Rouillier, MPFI, http://perso.ens-lyon.fr/nathalie.revol/software.html. [2] Laurent Fousse, Guillaume Hanrot, Vincent Lefevre,´ Patrick Pelissier´ and Paul Zimmermann, MPFR:A Multiple- Precision Binary Floating-Point Library With Correct Rounding, ACM Transactions on Mathematical Software, Vol. 33, No. 2, 2007, Article 13. [3] Nozomu Matsuda and Nobito Yamamoto, On the basic oper- ations of Interval Multiple-Precision Arithmetic with center-radius form, Nonlinear Theory and Its Applications, IEICE 2(1), 2011, pp. 54–67. [4] Nozomu Matsuda, Development of interval multiple precision arithmetic library with center-radius form, doctor’s thesis for grad- uate school of The University of Electro-Communications, in Japanese, 2016.
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Rigorous numerics of global trajectories for fast-slow systems with an explicit range of multi-scale parameter Kaname Matsue1 1The Institute of Statistical Mathematics, 10-3, Midori-Cho, Tachikawa, Tokyo, 190-8562, Japan [email protected]
Keywords: geometric singular perturbation theory, rigorous numer- ics, covering relations, invariant foliations.
My talk is concerned with validations of global trajectories for the fast-slow system x = f(x, y, ), 0 (1) (y0 = g(x, y, ) with computer assistance, where 0 = d/dt is the time derivative and f, g are Cr-functions with r 1. The factor is a nonnegative real number and is considered as the≥ multi-scale parameter. Unless other- wise noted, the slow variable y is assumed to be one-dimensional. I focus on validations of
(i) the singular limit trajectory H0 for (1) with = 0; (ii) global trajectories H near H for all (0, ] 0 ∈ 0 at the same time, where 0 > 0 is an explicitly given number. “Grobal trajectories” here mean periodic, homoclinic and heteroclinic orbits. The strategy is based on topological notions such as covering rela- tions ([5]), isolating blocks and cones ([1]) as well as geometric singular perturbation theory (e.g. [2]). Invariant foliations of stable and unsta- ble manifolds are also applied to validating homoclinic and heteroclinic trajectories. Using these notions, normally hyperbolic slow manifolds
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can be validated in the standard way with computer assistance. Addi- tionally, I propose a new technique called the covering-exchange, which is the singular perturbation version of covering relations and is viewed as a topological counterpart of the Exchange Lemma (e.g. [2]). The covering-exchange consists of three parts: slow shadowing, drop and jump. These concepts topologically describe true trajectories which shadow the reference connecting orbits (with = 0) and normally hy- perbolic slow manifolds of (1). The covering-exchange automatically solves the matching problem between fast scale and slow scale dy- namics. Moreover, the assistance of rigorous numerics in reasonable processes enables us to validate (i) and (ii) simultaneously.
Details of this talk is shown in [3]. If the time permits, I also talk several extensions of the current work (e.g. [4]).
References:
[1] C. Conley, Isolated invariant sets and the Morse index, volume 38 of CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, R.I., 1978.
[2] C.K.R.T. Jones, Geometric singular perturbation theory, In Dy- namical systems (Montecatini Terme, 1994), volume 1609 of Lec- ture Notes in Math., 44–118. Springer, Berlin, 1995.
[3] K. Matsue, Rigorous numerics for fast-slow systems with one- dimensional slow variable: topological shadowing approach, arXiv: 1507.01462, to appear in Topological Methods in Nonlinear Analy- sis.
[4] K. Matsue, Rigorous verification of smooth neighborhoods around slow manifolds via (un)stable normal bundles, in preparation.
[5] P. Zgliczynski´ , Covering relations, cone conditions and the sta- ble manifold theorem, J. Differential Equations, 246(5):1774–1819, 2009.
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Fast validated computation for solutions of algebraic Riccati equations arising in transport theory Shinya Miyajima Iwate University 4-3-5 Ueda, Morioka, Iwate, Japan [email protected]
Keywords: algebraic Riccati equation, verified computation, M-matrix
Consider the following nonsymmetric algebraic Riccati equation (NARE) arising in transport theory:
XCX XE AX + B = 0, (1) − − n n where A, B, C, E R × are given by ∈ A = ∆ eqT ,B = eeT ,C = qqT ,E = D qeT . − − T T Here e = (1,..., 1) , q = (q1, . . . , qn) with qi = ci/(2ωi), ∆ = diag(δ1, . . . , δn) with δi = 1/(cωi(1 + α)), D = diag(d1, . . . , dn) with di = 1/(cωi(1 α)), 0 < c 1, 0 α < 1, 0 < ωn < < ω1 < 1, n − ≤ ≤ ··· i=1 ci = 1 and ci > 0, i = 1, . . . , n. For descriptions on how this equation arises in transport theory, see [1] and references cited therein. P In this talk, we consider enclosing the solution of (1). To the au- thor’s best knowledge, an algorithm for enclosing a solution of the NARE has not been written down in literatures. For continuous-time algebraic Riccati equation, well-established algorithms for enclosing the solution are proposed in [2–7]. In [4,5], the algorithms for discrete- time algebraic Riccati equation are also proposed. These algorithms require (n3) or larger operations. TheO purpose of this talk is to propose an algorithm for enclosing the solution of (1). This algorithm enables us to enclose the solution with only (n2) operations under a reasonable assumption by exploiting the O 86 SCAN 2016
special structure of (1). We finally report numerical results to show the effectiveness and efficiency of this algorithm.
References:
[1] J. Juang, W-W. Lin, Nonsymmetric algebraic Riccati equations and Hamiltonian-like matrices, SIAM Journal on Matrix Analysis and Applications, 20 (1998), pp. 228–243.
[2] T. Haqiri, F. Poloni, Methods for verified solutions to continuous- time algebraic Riccati equations, arXiv:1509.02015 [math.NA].
[3] B. Hashemi, Verified computation of symmetric solutions to continuous- time algebraic Riccati matrix equations, Proceedings of SCAN con- ference, Novosibirsk, 2012, pp. 54–56. http://conf.nsc.ru/files/ conferences/scan2012/139586/Hashemi-scan2012.pdf
[4] W. Luther, W. Otten, Verified calculation of the solution of algebraic Riccati equation, in: T. Csendes (ed.), Developments in Reliable Computing, Kluwer Academic Publishers, Dordrecht, 1999, pp. 105–118.
[5] W. Luther, W. Otten, H. Traczinski, Verified calculation of the solution of continuous- and discrete time algebraic Riccati equation, Schriftenreihe des Fachbereichs Mathematik der Gerhard- Mercator-Universitat¨ Duisburg, (1998), SM-DU-422.
[6] S. Miyajima, Fast verified computation for solutions of continuous- time algebraic Riccati equations, Japan Journal of Industrial and Applied Mathematics, 32 (2015), pp. 529–544.
[7] K. Yano, M. Koga, Verified numerical computation in LQ con- trol problem, Transactions of SICE, 45 (2009), pp. 261–267 (in Japanese).
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Fast validated computation for solutions of discrete-time algebraic Riccati equations Shinya Miyajima Iwate University 4-3-5 Ueda, Morioka, Iwate, Japan [email protected]
Keywords: discrete-time algebraic Riccati equation, verified compu- tation, stabilizing solution
Consider the following discrete-time algebraic Riccati equation (DARE):
H H H 1 H A XA X A XB(R + B XB)− B XA + Q = 0, − − n n n m m m where A, Q C × , B C × and R C × are given, Q and R are ∈H ∈ ∈ n n Hermitian, A denotes the conjugate transpose of A, and X C × is to be solved. The DARE appears in many important problems∈ in sci- ence and technology, e.g., discrete-time LQ-optimal control problems and an equation in ladder networks [1]. The solution X of interest is an Hermitian matrix, and the Hermitian solution X is called stabilizing if H 1 H all the eigenvalues of A B(R + B XB)− B XA are inside the unit disk. The stabilizing solution− is required in the practical applications. In this talk, we consider enclosing the solution of the DARE, specif- ically, computing interval matrices containing the solution. The pio- neering work is the algorithms proposed in [2,3]. In these algorithms, the special structure of the DARE is skillfully exploited. These algo- rithm require (n6) operations. The purposeO of this talk is to propose an algorithm for enclosing the solution of the DARE. This algorithm utilizes numerical spectral H 1 H decomposition of A B(R + B XB˜ )− B XA˜ , where X˜ denotes a numerical solution, requires− only (n3) operations, and is applicable H 1 H O when A B(R + B XB˜ )− B XA˜ is diagonalizable. The assumption −
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of the applicability is made just to accelerate the enclosure. The al- gorithm moreover verifies that the solution contained in the interval matrix is unique and the stabilizing one. We finally report numerical results to show the effectiveness and efficiency of this algorithm.
References:
[1] D.A. Bini, B. Iannazzo, B. Meini, Numerical Solution of Al- gebraic Riccati Equations, SIAM, Philadelphia, 2012.
[2] W. Luther, W. Otten, Verified calculation of the solution of algebraic Riccati equation, in: T. Csendes (ed.), Developments in Reliable Computing, Kluwer Academic Publishers, Dordrecht, 1999, pp. 105–118.
[3] W. Luther, W. Otten, H. Traczinski, Verified calculation of the solution of continuous- and discrete time algebraic Riccati equation, Schriftenreihe des Fachbereichs Mathematik der Gerhard- Mercator-Universitat¨ Duisburg, (1998), SM-DU-422.
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Fast enclosure for matrix multiplication on a GPU Yusuke Morikura1), Yusuke Nozawa1), Kouta Sekine1), Masahide Kashiwagi1) and Shi’nichi Oishi1,2) 1) Department of Applied Mathematics, Waseda University, 2) JST/CREST 1) 419 room Building 63, 3–4–1 Ookubo, Shinjuku-ku, Tokyo, Japan [email protected]
Keywords: matrix multiplication, interval arithmetic, GPU
In this talk, we are concerned with a fast inclusion of matrix mul- tiplication on a GPU. Let F be a set of floating-point number defined m n n p by IEEE 754 standard [1]. Let A F × and B F × be given. We m p ∈ ∈ define C, C F × by ∈ C A B C. (1) ≤ · ≤ The inclusion of (1) is one of the significant methods in verified numer- ical computation of linear problems (e.g. linear systems and eigenvalue problems [2]). On CPU, since settled rounding modes hold until we change it again, we can compute C and C such as the following by BLAS rou- tines1 [2]:
#include
1We assume that dgemm routine does not use a fast matrix multiplication algorithm such as Strassen’s method
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Here, the function fesetround is defined by fenv.h in C99, and changes the selected rounding mode. To compute matrix multipli- cation two times, we can obtain the inclusion of matrix multiplication. In this talk, we consider the other simple approach based on MAGMA BLAS routine on a GPU [3]. MAGMA BLAS is open source routines for numerical linear problems on GPU. MAGMA BLAS is written by CUDA [4] created by NVIDIA. Since recent NVIDIA’s GPUs support IEEE 754 standard, we can perform computation with rounding mode control using intrinsic functions. However, rounding modes of GPU can not change such as the function fesetround on CPU. Therefore, we directly revise the dgemm function in MAGMA BLAS in order to add the inclusion part of a matrix multiplication. In this method, since it requires no extra memory transfer from global memory to cache memory, we achieved less than twice of computing time, though we calculate two times of matrix mulplication. In addition, we show the optimization of the function dgemm in MAGMA BLAS for the case of the inclusion on a GPU. Finally, we illustrate details and the results of computation in the talk.
References:
[1] ANSI/IEEE 754-1985, Standard for Binary Floating-Point Arith- metic, 1985.
[2] S. Oishi and S.M. Rump, Fast verification of solutions of matrix equations, Numer. Math., 90(4):755–773, 2002.
[3] MAGMA., http://icl.cs.utk.edu/magma/icl.cs.utk.edu/magma/, (2016, 03, 30)
[4] CUDA C Programming Guide, NVIDIA, http://docs.nvidia. com/cuda/cuda-c-programming-guide/, (2016, 03, 30)
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Generalized intervals in global optimization Arnold Neumaier, Ferenc Domes, Tiago Montanher, Mihaly Markot and Hermann Schichl University of Vienna Faculty of Mathematics, Oskar-Morgenstern-Platz 1, A-1090 Wien, Austria [email protected]
Keywords: global optimization, union arithmetic, projective trans- formation
Union arithmetic and projective arithmetic are two generalizations of interval arithmetic that were found useful for speeding up branch and bound algorithms for solving nonlinear systems, constraint satisfaction problems, and global optimization problems. Union arithmetic represents sets of reals as unions of one or more disjoint intervals, and is thus able to take advantage of division by intervals containing zero in interval Gauss-Seidel methods. Projective arithmetic is a way to handle unbounded regions in constraint satisfaction problems and global optimization problems by using projective transformations that map unbounded regions into bounded ones [1].
References:
[1] H. Schichl, A. Neumaier, M.C. Markot, F. Domes, On solving mixed-integer constraint satisfaction problems with un- bounded variables, pp. 216–233 in: Integration of AI and OR Techniques in constraint programming for combinatorial optimiza- tion problems, Springer, Berlin 2013.
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Linear Systems with the Exact Solution for Numerical Tests Katsuhisa Ozaki and Takeshi Ogita Shibaura Institute of Technology 307 Fukasaku, Minumaku, Saitama 337-8570, Japan [email protected]
Keywords: Numerical Linear Algebra, Linear System, Numerical Computations This talk is concerned with a linear system Ax = b for numerical computations, where A is a coefficient matrix and b is a right-hand side vector. Let F be a set of binary floating-point numbers as defined by the IEEE 754 standard [1]. Let fl( ) denote that each operation in the parenthesis is evaluated by floating-point· arithmetic. All elements in the matrix and the vector are represented by the floating-point num- n n n bers, i.e., A F × and b F . Accuracy of a numerical solutionx ˜ of Ax = b is sometimes∈ monitored∈ by the residual Ax˜ b. However, a basic numerical analysis says that the residual is not− enough to show the accuracy of numerical results. If we know the exact solution x such that Ax = b in advance, then it is very useful to examine robustness of algorithms for solving linear systems, tendency of convergence for iterative methods, and overestimation of an error bound obtained by verified numerical computations. One may think that a right-hand n n n side vector b is generated by Ax from given A F × and x F . However, if rounding errors occur in fl(Ax), then∈Ax = b may not∈ be satisfied. Miyajima, Ogita and Oishi proposed a method [2] which produces m m m n n n A0 F × , x0, b0 F from given A F × and x F such that ∈ ∈ ∈ ∈ A0x0 = b0. As advantage of their method, we can set arbitrary vector x and condition number of A. On the other hand, the size of the matrix increases, i.e., m n in many cases. Moreover, the structure of A0 is ≥ different from that of A. For example, A0 becomes unsymmetric, even if A is symmetric.
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Assume that A is non-singular, and a condition number of A is 1 53 less than u− , where u is a roundoff unit, e.g., u = 2− for IEEE 754 binary64. For a brief introduction, we set x = (1, 1,..., 1)T in this n n n abstract. Our algorithm produces A0 F × and b F such that ∈ ∈ A0x = b with A0 A. We adopt ideas of error-free transformation of ≈ matrix multiplication [3] to this problem. The vector σ Fn is defined as ∈ β vi σi = 2 2 , β = log2 n , vi = log2 max aij . · d e d j | |e Let qij be a set of indices. The matrix A0 is obtained as follows.
aij0 = fl((fij + aij) fij), δij = fl(aij aij0 ), fij = max σk, − − k qij ∈ where the set qij depends on the structure of the matrix, e.g., qij = i, j for a symmetric matrix. Then, { } A = A0 + ∆, b := fl(A0x) = A0x.
The structure of A0 is the same to that of A. In the presentation, we introduce a method which produces A0 and b, assuming that each element in given x is a power of two. In addition, we demonstrate how to make ∆ as small as possible and how to preserve positive definiteness. References: [1] ANSI/IEEE, IEEE Standard for Binary Floating Point Arith- metic, IEEE, New York, 2008. [2] S. Miyajima, T. Ogita, S. Oishi, A method of generating linear systems with an arbitrarily ill-conditioned matrix and an arbitrary solution, Proc. 2005 International Symposium on Nonlinear The- ory and its Applications (NOLTA 2005), Bruges, Belgium, (2005), pp. 741–744. [3] K. Ozaki, T. Ogita, S. Oishi, S. M. Rump, Error-Free Trans- formation of Matrix Multiplication by Using Fast Routines of Ma- trix Multiplication and its Applications, Numerical Algorithms, 59:1 (2012), pp. 95–118.
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Model-based design of optimal experiments for guaranteed parameter estimation of nonlinear dynamic systems Anwesh Reddy Gottu Mukkula and Radoslav Paulen Process Dynamics and Operations Group, Technische Universit¨at Dortmund Emil-Figge-Strasse 70, Dortmund 44227, Germany Anwesh-Reddy.Gottu-Mukkula, Radoslav.Paulen{ @bci.tu-dortmund.de } Keywords: Optimal experiment design, Parameter estimation, Bounded noise Mathematical modeling has become an integral part of modern process design methodologies as well as in control system design and operations optimization. A typical model development procedure is divided into two main phases, namely specification of the model struc- ture and estimation of the unknown/uncertain model parameters. The latter phase, often referred to as model fitting, normally proceeds by determining parameter values for which the model predictions closely match (for example in least-squares sense) the available process mea- surements. Tailored approaches exist nowadays to strike against certain prob- lems encountered in classical parameter estimation. A guaranteed pa- rameter estimation of non-linear dynamic systems has been formu- lated [1] in a context of parameter estimation techniques that account for bounded measurement error. The problem consists of finding—or approximating as closely as possible—the set of all possible param- eter values such that the predicted values of certain outputs match their corresponding measurements within prescribed error bounds. A set-inversion algorithm is applied, whereby the parameter set is succes- sively partitioned into smaller boxes and exclusion tests are performed to eliminate some of these boxes, until a given threshold on the ap- proximation level is met.
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Optimal experiment design has been extensively studied in liter- ature as an approach that provides best available conditions for col- lection of information-rich data of a dynamic system [2]. Naturally this problem leads to dynamic optimization problem. The classical optimal experiment design is, however, not consistent with guaranteed parameter estimation as the results of the (classical) least-squares and guaranteed parameter estimation are not the same while the expec- tations of the realization of measurement noise differ in least-squares (unbounded normal distribution) and guaranteed (bounded arbitrary distribution) estimation. The work presented in this contribution provides a methodology for performing optimal experiment design in the context of guaranteed parameter estimation. The set of guaranteed parameter estimates is first over–approximated by a box using non-linear optimization. The problem of design of experiments, which determines a sequence of con- trol inputs for the optimal experiment design in the context of guar- anteed parameter estimation, is then formulated as a dynamic opti- mization problem over the non-linear optimization problem that over- approximates the set of guaranteed parameter estimates. The arising bi-level program is regularized which makes the resulting non-linear optimization with complementarity constraints well-conditioned. Acknowledgments: The research leading to these results has re- ceived funding from the European Commission under grant agreement number 291458 (ERC Advanced Investigator Grant MOBOCON). References: [1] G. Franceschini, S. Macchietto, Model-based design of ex- periments for parameter precision: State of the art, Chemical En- gineering Science, 63 (2008), pp. 4846–4872. [2] M. Kieffer, E. Walter, Guaranteed estimation of the param- eters of nonlinear continuous-time models: contributions of inter- val analysis, International Journal of Adaptive Control and Signal Processing, 25 (2011), No. 3, pp. 191–207.
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PROMISE: floating-point precision tuning with stochastic arithmetic Stef Graillat1, Fabienne J´ez´equel1,2, Romain Picot1,3, Fran¸coisF´evotte3 and Bruno Lathuili`ere3 1 Sorbonne Universit´es, UPMC Univ Paris 06, CNRS, UMR 7606, Laboratoire d’Informatique de Paris 6 (LIP6), Paris, France 2 Universit´ePanth´eon-Assas,12 place du Panth´eon,75231 Paris CEDEX 05, France 3 EDF R&D 7 boulevard Gaspard Monge, F-91120, Palaiseau, France Stef.Graillat,Fabienne.Jezequel,Romain.Picot @lip6.fr { Francois.Fevotte,Bruno.Lathuiliere @edf.fr} { }
Keywords: auto-tuning, Discrete Stochastic Arithmetic, floating-point arithmetic, mixed precision, numerical validation, round-off errors
Nowadays, most floating-point computations in numerical simula- tions are performed in IEEE 754 binary64 precision (double preci- 16 sion). This means that a relative accuracy of about 10− is provided for every arithmetic operation. Indeed, in practice, programmers tend to use the highest precision available in hardware which is the double precision on current processors. This approach can be costly in terms of computing time, memory transfer and energy consumption [1]. A better strategy would be to use no more precision than needed to get the desired accuracy on the computed result. The challenge of using mixed precision is to find some parts of codes (and so variables) that may be executed with lower precision. Unfortunately the amount of possible configurations is exponential in the number of variables. To overcome this difficulty, we propose an algorithm and a tool called PROMISE (PRecision OptiMISEd) based on the delta debug- ging search algorithm [2] that provide a mixed precision configuration with a worst-case complexity quadratic in the number of variables. From an initial C or C++ program and a required accuracy on the computed result, PROMISE automatically modifies the precision of
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variables. To estimate the numerical quality of results, PROMISE uses Discrete Stochastic Arithmetic (DSA) [3] which controls round- off errors in simulation programs. Unlike Precimonious [4], PROMISE does not focus on performance constraints, it can rather be seen as a tool helping developers to reduce the cost of double precision variables and improve the memory usage of their code. The PROMISE tool has been successfully tested on programs implementing several numerical algorithms including linear system solving and also on an industrial code that solves the neutron transport equations [5]. References: [1] M. Baboulin, A. Buttari, J. Dongarra, J. Kurzak, J. Langou, J. Langou,P. Luszczek, and S. Tomov, Accelerat- ing scientific computations with mixed precision algorithms, Com- puter Physics Communications, vol. 180, No. 12, pp. 2526–2533, 2009. [2] A. Zeller and R. Hildebrandt, Simplifying and isolating fail- ure-inducing input, IEEE Trans. Softw. Eng., vol. 28, No. 2, pp. 183–100, Feb. 2002. [3] J. Vignes, Discrete Stochastic Arithmetic for validating results of numerical software, Numerical Algorithms, vol. 37, no. 1–4, pp. 377– 390, Dec. 2004. [4] C. Rubio-Gonzalez,´ C. Nguyen, H. D. Nguyen, J. Dem- mel, W. Kahan, K. Sen, D. H. Bailey, C. Iancu, and D. Hough, Precimonious: Tuning assistant for floating-point preci- sion, Proceedings of the International Conference on High Perfor- mance Computing, Networking, Storage and Analysis, ser. SC’13. New York, NY, USA: ACM, 2013, pp. 27:1– 27:12. [5] F. Fevotte´ and B. Lathuiliere` , MICADO: Parallel implemen- tation of a 2D–1D iterative algorithm for the 3D neutron trans- port problem in prismatic geometries, Proceedings of Mathematics, Computational Methods & Reactor Physics, May 2013.
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Sharp error bounds for complex floating-point inversion Claude-Pierre Jeannerod1, Nicolas Louvet2, Jean-Michel Muller3, and Antoine Plet4 1 INRIA, 2 UCBL, 3 CNRS, 4 ENS Lyon Laboratoire LIP, Universit´ede Lyon ENS Lyon, 46, all´eed’Italie, 69364 Lyon cedex 07, France [email protected]
Keywords: floating-point arithmetic, numerical error, complex inver- sion, rounding error analysis. We study the accuracy of the classic algorithm for inverting a com- plex number given by its real and imaginary parts as floating-point numbers. Our analyses are done in binary floating-point arithmetic with an unbounded exponent range in precision p, and we assume that the basic arithmetic operations (+, , , /) are rounded to nearest, so p − × that the roundoff unit is u = 2− . We prove the componentwise relative error bound 3u for the complex inversion algorithm (assuming p 4), and we show that this bound is asymptotically optimal (as p ≥ ) when p is even, and reasonably sharp when using one of the basic→ IEEE ∞ 754 binary formats with an odd precision (p = 53, 113). This compo- nentwise bound obviously leads to the same bound 3u for the normwise relative error. However we prove that the significantly smaller bound 2.707131u holds (assuming p 24) for the normwise relative error, and we illustrate the sharpness of≥ this bound using numerical examples for the basic IEEE 754 binary formats (p = 24, 53, 113). This work has been published in [1]. References: [1] C.-P. Jeannerod, N. Louvet, J.-M. Muller, A. Plet, Sharp error bounds for complex floating-point inversion, Numerical Al- gorithms, 2016.
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Rigourous error bounds for double-double operations
Mioara Joldes∗, Jean-Michel Muller∗∗ and Valentina Popescu∗∗
∗LAAS-CNRS, 7 Avenue du Colonel Roche, 31077 Toulouse, France ∗∗LIP, ENS Lyon, 46 All´eed’Italie, 69364 Lyon, France [email protected]
Keywords: floating-point arithmetic, high precision arithmetic, multiple- precision arithmetic, double-double Most of today’s numerical computations are done using floating- point (FP) formats for representing real numbers. The two most widely implemented formats defined by the IEEE 754-2008 standard [1] for FP arithmetic are single-precision (binary32 ) and double-precision (bi- nary64 ). For some critical problems, such precisions do not suffice. Since higher precisions, such as quad-precision (binary128 ) is not hard- ware implemented on widely distributed processors, the only solution is to use software emulation for extended precision. There are mainly two ways of representing numbers in higher pre- cision. The first one, the multiple-digit method, uses integers for rep- resenting numbers as a sequence of possibly high-radix digits coupled with an exponent. The second one, the multiple-term method, repre- sents real numbers as unevaluated sums of several FP numbers, allow- ing for computations to be carried out naturally using the underlying FP hardware level. The later one is called a floating-point expansion when made up with an arbitrary number of terms and it makes use of well known error-free-transforms algorithms such as 2Sum, Fast2Sum and Fast2Mult (see [2]). In this work we focus on the “double-double” (DD) precision, i.e., the number x is represented as the sum of two double-precision FP 1 numbers, xh+x`, the high and the low part, that satisfy x` 2ulp xh . As a matter of fact, we should better talk about “double-word”| | ≤ preci-| | sion, since the algorithms and proofs we discuss can be used with any
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precision-p underlying FP format (p = 53 for double-precision). More precisely we talk about two algorithms that were first presented by Ka- han et. al. in [3]. The first one (Alg. 1) computes the sum of two DD numbers and returns also a DD number. In the initial paper they claim 2p to have a relative error bound of 2 2− , but no rigorous proof is given. During our tests we observed that· this bound is too optimistic, so we 2p p 2p 3p present here a slightly larger bound, 2− (5+9 2− +7 2− +6 2− ), for which we can provide a rigorous proof.· · · · Alg. 2 computes the product of a DD number with a standard dou- ble-precision one and returns also a DD number. In this case we proved a tighter error bound that the one given in [3]. The initial error bound 2p 2p p 2p was 4 2− and we provide a proof for 2− xy 3 + 4 2− + 2 2− . · | | · · Algorithm 1 (zh, z`) = (xh, x`) + (y , y ). Algorithm 2 (z , z ) = (x , x ) y. h ` h ` h ` ∗ 1: (s , s ) 2Sum(x , y ) 1: (c , c ) Fast2Mult(x , y) h ` ← h h h `1 ← h 2: (t , t ) 2Sum(x , y ) 2: c RN(x y) h ` ← ` ` `2 ← ` · 3: c RN(s + t ) 3: (t , t ) Fast2Sum(c , c ) ← ` h h `1 ← h `2 4: (v , v ) Fast2Sum(s , c) 4: t RN(t + c ) h ` ← h `2 ← `1 `1 5: w RN(t + v ) 5: (z , z ) Fast2Sum(t , t ) ← ` ` h ` ← h `2 6: (z , z ) Fast2Sum(v , w) 6: return (z , z ) h ` ← h h ` 7: return (zh, z`)
References: [1] IEEE Computer Society, IEEE Standard for Floating-Point Arithmetic, IEEE Standard 754-2008, Aug. 2008. [2] J.-M. Muller, N. Brisebarre, F. de Dinechin, C.-P. Jeannerod, V. Lefevre, G. Melquiond, N. Revol, D. Stehle, and S. Torres, Handbook of Floating-Point Arith- metic, Birkhauser Boston, 2010. [3] X.S.Li, J.W.Demmel, D.H.Bailey, G.Henry, Y.Hida, J.Iskandar, W.Kahan, A.Kapur, M.C.Martin, T.Tung, D.J.Yoo, Design, Implementation and Testing of Extended and Mixed Precision BLAS, ACM TOMS, vol. 28, no. 2, 2002.
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Enclosing the Solution Set to Interval Parametric Matrix Equation A(p)X = B(p) Evgenija D. Popova Institute of Mathematics and Informatics Bulgarian Academy of Sciences Acad. G. Bonchev str., block 8, 1113 Sofia, Bulgaria [email protected]
Keywords: linear matrix equation, dependent data, enclosure meth- ods
Consider the parametric matrix equation
A(p)X = B(p), p [p], ∈ where A(p) and B(p) are known m m and m n matrices whose × × elements are linear functions of uncertain parameters p = (p1, . . . , pk) varying within given intervals, and X = X(p) is the unknown matrix. We prove that the united solution set to the matrix equation and this solution set to the same parametric system considered as a sys- tem with multiple right hand sides have the same exact interval hull although, as shown in [1], they are different as sets. This implies that all known methods for enclosing AE-solution sets to parametric linear systems can be generalized to enclose the corresponding AE-solution sets to the parametric matrix equation without penalties on the com- putational cost. We present a generalization of the method from [2,3], which does not require strong regularity of the parametric matrix, and some improvements of the parametric Krawczyk iteration proposed in [1]. Comparison between some of the enclosure methods with respect to their scope of applicability and the quality of the solution enclosure will demonstrate some properties of the methods that have not been mentioned before.
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References:
[1] M. Dehghani-Madiseh, M. Dehghan, Parametric AE-solution sets to the parametric linear systems with multiple right-hand sides and parametric matrix equation A(p)X = B(p), Numerical Algo- rithms, (2016), doi: 10.1007/s11075-015-0094-3.
[2] A. Neumaier, A. Pownuk, Linear systems with large uncertain- ties, with applications to truss structures, Reliable Computing, 13 (2007), pp. 149–172.
[3] E.D. Popova, Improved enclosure for some parametric solution sets with linear shape, Comput. Math. Appl., 68, (2014), pp. 994– 1005.
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Automated tuning of robust fractional PID controller for interval plants using Kharitonov’s theorem Harsh Purohit, P.S.V. Nataraj Systems and Control Engineering Group Indian Institute of Technology Bombay Powai-400076, Mumbai [email protected], [email protected]
Keywords: Fractional PID controller, Interval plants, Kharitonov’s theorem
In this article, we propose a new automated method for synthesizing robust fractional PID controller for interval plant using Kharitonov’s theorem [1]. Only four Kharitonov polynomials are required to be Hurwitz and we can determine the stability of the interval system un- der the given parameter variation. A gain and phase margin tester compensator is incorporated to guarantee the concerned system with certain robust safety margins. This kind of controller synthesis pro- cedures have not closed form solutions and usually they are iterative and/or based on graphical interpretations. Specifically, a new graphical method for synthesizing robust PID controllers to achieve pre-specified safety margin is proposed for uncertain time delay systems [2].How- ever, with easy availability of computation power, we can automate this design procedure. We have chosen fractional order controller because of its promis- ing recent developments and application to engineering problems[3]. Considering the fractional order PID controller, we can obtain char- acteristic equation of the close loop system. And after selecting the design frequencies and gain-phase margin, we can pose this controller synthesis problem as a constrained satisfaction problem (CSP) with all vertex Kharitonov polynomials where unknowns for this CSP are
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controller parameters (Kp,Ki,Kd, λ, µ) with pre-specified initial search interval. In this work, we have solved this problem using constraint solver Realpaver [4]. Different consistency techniques like box, hull and 3B has been implemented in Realpaver. Finally, illustrative example with computer simulations is demonstrated to confirm the validity of the proposed methodology.
References:
[1] V.L. Kharitonov, Asymptotic stability of an equilibrium postion of a family systems of linear differential equations, Differential’nye Uraveniya , (2008), No. 14, pp. 2086-2088.
[2] Yuan-Jay Wang, Graphical computation of gain and phase mar- gin specifications-oriented robust PID controllers for uncertain sys- tems with time-varying delay, Journal of Process Control, (2011), No. 21, pp. 475–488.
[3] C. Monje, B. Vinagre, V. Feliu and Y. Chen, Tuning and autotuning of fractional order controllers for industry applications, Control Engineering Practice, (2008), No. 16(7), pp. 798–812.
[4] L. Granvilliers and F. Benhamou, Algorithm 852: Real- paver: an Interval Solver using Constraint Satisfaction Techniques, ACM TOMS , (2006), No. 32(1), pp. 138-156.
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Safety Verification By Interval Based Quantified Constraint Solving Peter Franek, Jan Kuˇr´atko, and Stefan Ratschan Institute of Computer Science, Czech Academy of Sciences Pod Vodarenskou vezi 2, 182 07 Praha, Czech Republic [email protected]
Keywords: ordinary differential equations, safety verification, con- straints, constraint solving
We consider the following problem (“safety verification”):
Given:
an ordinary differential equation • a set of states I (the initial states) • a set of states U (the unsafe states) • Verify: It is not possible for a solution of the ODE to start in a state in I and reach a state in U. Or equivalently: Every solution of the ODE with an initial state in I never reaches a state in U.
This verification task can be reduced [3] to a constraint solving problem containing logical quantifiers ( , ). Such problems have been traditionally solved by computer algebra∀ ∃ techniques [2], but methods based on interval computation have been shown to be competitive [4]. However, the constraint solving problem arising in safety verification has a specific structure. In the talk we present an interval based algo- rithm that exploits this structure. This work was supported by GACRˇ grant 15-14484S and by the long-term strategic develop- ment financing of the Institute of Computer Science (RVO:67985807).
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References
[1] B. F. Caviness and J. R. Johnson, eds., Quantifier Elimination and Cylindrical Algebraic Decomposition, Springer, Wien, 1998.
[2] G. E. Collins, Quantifier elimination for the elementary theory of real closed fields by cylindrical algebraic decomposition, in Sec- ond GI Conf. Automata Theory and Formal Languages, vol. 33 of LNCS, Springer Verlag, 1975, pp. 134–183. Also in [1].
[3] S. Prajna and A. Jadbabaie, Safety verification of hybrid sys- tems using barrier certificates, in HSCC’04, R. Alur and G. J. Pap- pas, eds., no. 2993 in LNCS, Springer, 2004.
[4] S. Ratschan, Efficient solving of quantified inequality constraints over the real numbers, ACM Transactions on Computational Logic, 7 (2006), pp. 723–748.
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An Interval-Based Algorithm for Feature Extraction from Speech Signals Andreas Rauh1, Susann Tiede2 and Cornelia Klenke2 1University of Rostock, Chair of Mechatronics D-18059 Rostock, Germany [email protected] 2Speech Therapists Bahnhofstraße 35, D-17087 Altentreptow, Germany [email protected], logopaedie [email protected]
Keywords: Branch-and-bound algorithm, Speech analysis, Phoneme and pattern recognition To develop a computer-based assistance system for speech therapy, it is essential to distinguish between the linguistic levels of lexicon, grammar, and pronunciation [1] and to deal with (i) the automatic transcription and preprocessing of speech involving erroneous pronun- ciation, (ii) the classification of pronunciation disorders, and (iii) a grammatical analysis. For the tasks (i) and (ii), this contribution ex- ploits an online frequency analysis of speech signals based on stochastic filtering approaches and the identification of points of time at which transitions between subsequent phonemes can be expected (cf. [3,4]). In general, phonemes can be classified into voiced and unvoiced sounds [2,3]. Voiced sounds (e.g. normal vowels) are characterized by several relatively sharp formant frequencies produced by vibrations of the vocal folds representing a fluidic resistance against the outflow of air expelled from the lungs. In contrast, unvoiced sounds (e.g. whis- pered vowels and fricatives such as ch, ss, sh, f) are caused by a turbu- lent, partially irregular, air flow with negligible vibrations of the vocal folds. To some extent, they are produced by fizzing sounds originat- ing between teeth and lips as well as between tongue and hard or soft palate. Here, sharp formant frequencies are characteristic for voiced
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phonemes, whereas wide frequency bands are typical for unvoiced ones. For both cases, a stochastic filtering approach [3] can be employed to estimate expected values of the formant frequencies and their associ- ated covariances, where the broad-band nature of unvoiced speech is reflected by (co-)variance estimates that are significantly larger than for the voiced case. Transitions between subsequent phonemes are in- dicated by rapid changes in the above-mentioned estimation results [4]. In this contribution, an interval algorithm is presented for the ex- traction of speech features from the estimated formant frequencies and signal amplitudes on the level of individual phonemes. Besides a pat- tern recognition for correctly pronounced sounds, this procedure helps to identify and classify disorders (together with expert knowledge of the speech therapist if these features are not yet included a phoneme database). Moreover, stuttering disorders (those characterized by rep- etitions of individual/ multiple phonemes/ syllables) can be detected from the extracted features. Numerical results for speech sequences without and with pronunciation disorders conclude this contribution. References: [1] J.S. Damico, N. M¨uller,and M.J. Ball, The Handbook of Language and Speech Disorders, ser. Blackwell Handbooks in Linguistics. Chichester, West Sussex, UK: Wiley, 2010. [2] P. Ladefoged and I. Maddieson, The Sounds of the World’s Lan- guages, ser. Phonological Theory. Chichester, West Sussex, UK: Wiley, 1996. [3] A. Rauh, S. Tiede, and C. Klenke, Observer and Filter Approaches for the Frequency Analysis of Speech Signals, Proc. of 21st IEEE Intl. Conf. on Methods and Models in Automation and Robotics, Poland, 2016, under review. [4] A. Rauh, S. Tiede, and C. Klenke, Stochastic Filter Approaches for a Phoneme-Based Segmentation of Speech Signals, Proc. of 21st IEEE Intl. Conf. on Methods and Models in Automation and Robotics, Miedzyzdroje, Poland, 2016, under review.
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Interval-Based Identification of Friction and Hysteresis Models Andreas Rauh and Harald Aschemann University of Rostock, Chair of Mechatronics D-18059 Rostock, Germany Andreas.Rauh,Harald.Aschemann @uni-rostock.de { }
Keywords: Interval-based identification, Friction, Hysteresis, Verified solution of non-smooth ODEs
Dynamic models with state- and input-dependent transitions be- tween different continuous-time representations are common for the mathematical modeling of a large variety of technical systems. Among these, especially the description of friction and hysteresis is widely used in both mechanical and control engineering [1, 3, 4, 5]. In previous work, interval-based identification routines have been derived where state- and input-dependent transitions were included in piecewise-defined ordinary differential equations (ODEs) [2]. In this framework, the experimental identification of a mathemat- ical model for a drive train test rig was considered. The correspond- ing non-smooth set of ODEs was characterized by an automaton rep- resentation for all possible transitions between static friction and a piecewise linear, velocity-proportional model for sliding friction. In this work, the coefficients for the rotary mass moment of inertia, the velocity-proportional sliding friction, and the initial breakaway torque were identified in terms of interval parameters on the basis of measured data with an additive bounded uncertainty [3, 4]. To make the identification procedure applicable for further — es- pecially more complex — system models, algorithmic extensions are developed which allow for preventing an excessive growth of the com- putational effort if a larger number of uncertain parameters are consid- ered. During that process, the friction model used in [3, 4] is extended to a polynomial representation of the sliding friction behavior, to an
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inclusion of Stribeck characteristics near the breakaway point, to com- binations of friction and Bouc-Wen hysteresis models [1, 5], and to more detailed representations for the actuator dynamics. The Bouc- Wen model introduces further points into the set of non-smooth ODEs at which state-dependent transitions between individual system com- ponents occur. A numerical case study for the experimental parameter identification of a drive train test rig available at the Chair of Mecha- tronics at the University of Rostock concludes this paper with a focus on a verified exclusion of friction and hysteresis models that are defi- nitely not compatible with the given interval-bounded measured data. References: [1] F. Ikhouane and J. Rodellar, Systems with Hysteresis: Analysis, Identification and Control using the Bouc-Wen Model, Chichester, West Sussex, UK: John Wiley&Sons, Ltd., 2007. [2] N.S. Nedialkov and M. v. Mohrenschildt, Rigorous Simulation of Hybrid Dynamic Systems with Symbolic and Interval Methods, Proc. of American Control Conf. ACC, Anchorage, USA, 2002. [3] A. Rauh, L. Senkel, and H. Aschemann, Verified Parameter Identi- fication for Dynamic Systems with Non-Smooth Right-Hand Sides, Proc. of 16th GAMM-IMACS International Symposium on Scien- tific Computing, Computer Arithmetic, and Validated Numerics SCAN 2014, W¨urzburg, Germany, Vol. 9553 of Lecture Notes in Computer Science, Heidelberg: Springer, 2016. [4] A. Rauh, L. Senkel, and H. Aschemann, Experimental Compari- son of Interval-Based Parameter Identification Procedures for Un- certain ODEs with Non-Smooth Right-Hand Sides, Proc. of 20th IEEE Intl. Conf. on Methods and Models in Automation and Ro- botics, Miedzyzdroje, Poland, 2015. [5] A. Rauh, Ch. Siebert, and H. Aschemann, Verified Simulation and Optimization of Dynamic Systems with Friction and Hysteresis, Proc. of ENOC 2011, Rome, Italy, 2009.
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Toward the Optimal Parameterization of Interval-Based Variable-Structure State Estimation Procedures Andreas Rauh and Harald Aschemann University of Rostock, Chair of Mechatronics D-18059 Rostock, Germany Andreas.Rauh,Harald.Aschemann @uni-rostock.de { } Keywords: Sliding mode observers, Lyapunov functions, Itˆodifferen- tial operator, Time discretization In previous work, interval-based extensions of sliding mode state observes [4] have been presented by the authors which allow for a guaranteed stabilization of the associated error dynamics [2, 3]. The parametrization of these observers is so far based on the on- line evaluation of a suitable candidate for a Lyapunov function and its related time derivative. The variable-structure gain of this type of observer is determined in such a way that the time derivative of the Lyapunov function candidate can be guaranteed to be negative defi- nite despite bounded uncertainty in system parameters as well as in the measured output variables. Besides the treatment of interval uncer- tainty, extensions were developed which guarantee asymptotic stability in cases in which stochastic measurement noise influences the system dynamics. These stochastic disturbances are taken into account by a replacement of the “classical” time derivative of the Lyapunov function candidate by the so-called Itˆodifferential operator [1, 2]. The limitations of this estimation approach are the assumption of negligibly small time discretization errors in a • quasi-continuous implementation of the interval-based estimation procedure and the use of the stability requirement of the error dynamics (with an • a-priori fixed underlying observer gain for all linear system com- ponents) as the only design requirement.
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In this contribution, the following extensions of the interval-based variable-structure state and parameter estimator are developed and demonstrated for suitable application scenarios: Firstly, time discretiza- tion errors are taken into account in a discrete-time implementation of the stability proof for the state estimation procedure which needs to be performed in real time. Secondly, a first attempt toward the optimal parameterization of the underlying linear state observer as well as extensions of the variable-structure gain computation are pre- sented. These extensions aim at choosing the gain values in such a way that the rate of convergence of the estimated states toward their true values (which are, however, in practical applications unknown) can be optimized. There, it is necessary to find a trade-off between the maximum rate of convergence, on the one hand, and the robustness of the estimation as well as the achievable tightness of the computed interval bounds, on the other hand. Simulation results for prototypical applications in control engineering conclude this contribution. References: [1] H. Kushner, Stochastic Stability and Control, New York: Academic Press, 1967. [2] L. Senkel, A. Rauh, and H. Aschemann, Sliding Mode Techniques for Robust Trajectory Tracking as well as State and Parameter Estimation, Mathematics in Computer Science, Vol. 8 (2014), Is- sue 3–4, pp. 543–561. [3] L. Senkel, A. Rauh, and H. Aschemann, Sliding Mode Approaches Considering Uncertainty for Reliable Control and Computation of Confidence Regions in State and Parameter Estimation, Proc. of 16th GAMM-IMACS International Symposium on Scientific Com- puting, Computer Arithmetic, and Validated Numerics SCAN 2014, W¨urzburg, Germany, Vol. 9553 of Lecture Notes in Computer Sci- ence, Heidelberg: Springer, 2016. [4] V. Utkin, Sliding Modes in Control and Optimization, Berlin, Hei- delberg: Springer, 1992.
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HPC and interval computations Nathalie Revol Inria - University of Lyon, LIP (UMR 5668 CNRS - ENS de Lyon - INRIA - UCBL) ENS de Lyon, 46 all´eed’Italie, 69364 Lyon Cedex 07, France [email protected]
Keywords: interval arithmetic, high-performance computing Interval computations can benefit from high-performance environ- ments: larger problems can be solved, execution time can be reduced. Furthermore, as each numerical value is used several times for each operation in interval arithmetic, the ratio between computations and data moves is favourable. However, it is not totally clear whether HPC environments offer the required facilities, in particular whether directed rounding modes are properly set by these environments.
Reciprocally, high-performance computations may need interval com- putations. Typically, fault-tolerant or resilient algorithms, which per- form large amounts of computations, also perform quick computations to detect whether a fault has occured. With floating-point arithmetic, these quick computations frequently return results which differ from the results of the long computations, because of roundoff errors, cf. [1]. An issue is to determine whether this difference is due to floating- point computations being non reproducible or to a failure. Interval arithmetic could be useful to compute these certificates: it would re- turn a value with bounds and thus the comparison between the result of the long computation and the certificate would boil down to an in- clusion test. Although much slower than floating-point arithmetic, the overhead induced by interval arithmetic would apply only to a small computation. Current resilient algorithms such as the ones in [2] apply an equality test up to a threshold, and this threshold can be difficult to choose. We will detail if, how and when interval arithmetic can suppress the use of this threshold.
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References:
[1] Revol, Nathalie, and Philippe Theveny´ , Numerical repro- ducibility and parallel computations: Issues for interval algorithms, IEEE Transactions on Computers, 63 (2014), No. 3, pp. 1915– 1924.
[2] S. Cools, E.F. Yetkin, E. Agullo, L. Giraud, W. Van- roose, Analysis of rounding error accumulation in conjugate gra- dients to improve the maximal attainable accuracy of pipelined CG, arXiv:1601.07068, (2016), 27 pages.
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Tube Programming Applied to State Estimation Simon Rohou, Luc Jaulin, Lyudmila Mihaylova, Fabrice Le Bars, Sandor M. Veres Lab-STICC, ENSTA Bretagne, Brest, France University of Sheffield, Western Bank Sheffield S10 2TN, UK [email protected]
Problem. We consider the guaranteed non-linear state estimation of a robot described by its state x Rn and measuring distances to beacons, see Fig. 1. x| = x, y, θ, v ∈. System’s description is given by: { } ˙x = f(x, u) (1) R ri(ti) = gj(x(ti)) With f : Rn Rm Rn, the continu- ous evolution×function→ based on robot’s m state x and input u R , and gj : ∈ Rn R, a sporadic observation func- → tion giving a range value ri R between ∈ the robot and the j -th beacon at time ti. Main approach. In a set member- ship approach, x and ˙x respectively be- long to bounded signals evolving with time: t , x(t) [x](t) , ˙x(t) [ ˙x](t). These ∀trajectories∈ are represented∈ with tubes [1], see Fig. 2. The more uncer- Figure 1: Localization of tain a trajectory is, the thicker the tube robot among beacons containing it will be. When trajectory’s R estimation is improved, the tube needs to be contracted, thus becom- ing thinner. To this end, we propose to break down the problem into several elementary constraints involving contractors on tubes.
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[f]
t
Figure 2: a tube: guaranteed representation of a set of trajectories (in light gray). A signal s( ) (in orange) belongs + · to an interval functionFigure so that: 2: At, s( tubet) [f]( [ft)]( = [)f − implemented(t), f (t)]. This ensures with a a guaranteed set of slices representation of signals. + ∀ ∈ Functions f −(t) and f (t) are represented with· a set of boxes (in blue).
To this end,Break we implemented down. Leta new us C++ consider library for the tubes given representations. equations: Codex ˙ is= freelyv cos available(θ), on the 2 2 · GitHub softwarex(t1) development+ y(t1) platform:= r1, the following constraints can be established: p IBEX-Robotics library A complementary C++ library of IBEX for roboticsa = purposes.cos(θ) https://github.com/ibex-team/ibex-roboticscontinuous constraints b = v a · x = b Future related paper: 2 2 sporadic constraint x(t1) =R r1 y(t1) Tube contractions applied to guaranteed non-linear state estimation− with time-uncertainties • SimonTube Rohou, Luc contraction Jaulin, Lyudmila. Our Mihaylova, contribution Fabrice Le Bars, is to Sandorp perform M. Veres a reliable state estimation with a simple and general method involving continuous or 3.3 Guaranteedfleeting [2] integrationconstraints on tubes. In the continuous case, constraints This researchare managed on tubes and with their related tube tools arithmetic: – such as contractors, variables Sectiona and 3.2 – bringsb are new tubes ideas and too. enable us to consider[x]( the) problem= [x]( of) guaranteed[b]( ). integration In the of differentialsporadic equations case, we [3, 19, will 11, 8, show 17, 12]. that From to an initial box [x] (0), guaranteed· integration· ∩ [20]· provides a set of techniques based on interval arithmetic to compute a box- valued functionbe compliant[x](t) (a tube with) containing tube’s true representation values for the initial [x value]( ), problem. such a These contraction methods provide can interval counterpartsonly of be Euler done [16], Runge-Kutta consideringR [7] or the Taylor derivative [18] integration tube and· [ validatex ˙]( ). results using the Picard Theorem. · In robotics, guaranteed integration provides a reliable knowledge about robots trajectories. In this work, we pro- pose a newReferences: approach based on tube arithmetic. Current experiments show interesting results when computing robot trajectories. This has been compared with existing methods, namely the CAPD DynSys library, well recognized in the community.[1] A. Bethencourt, Our approach is in fact L. particularly Jaulin, suitable Solving for non-linear robots trajectories constraint and seems satis- to offer better results in thisfaction context. We problems plan to open involving discussions about time-dependant this work during next functions, conferencesMathematics SWIM’16 and SCAN’16. in Computer Science, 2014. Future related paper: Guaranteed integration of robot trajectories • [2] F. Le Bars, J. Sliwka, O. Reynet, L. Jaulin, State estima- Simon Rohou, Luc Jaulin, Lyudmila Mihaylova, Fabrice Le Bars, Sandor M. Veres tion with fleeting data, Automatica, 2012. 4 Issues/Problems preventing from achieving PhD Targets
None, except time. 117
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Sharp error bounds for the Gamma function over the whole floating-point range Siegfried M. Rump Institute for Reliable Computing Hamburg University of Technology Schwarzenberg-Campus 1, 21071 Hamburg, Germany and Waseda University Faculty of Science and Engineering 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan [email protected]
Keywords: Gamma function, rigorous error bounds, floating-point
The Matlab built-in Gamma function delivers approximations with relative error up to 100%. We present an algorithm for computing rigorous error bounds for the Gamma function over the whole floating-point range. More pre- cisely, for a given floating-point number x, an inclusion of Γ(x) is 16 computed with relative error less than 6 10− . · If the input argument is an interval X IF, then an inclusion of Γ(x): x X of similar quality is computed.∈ The presented function{ is part∈ of} INTLAB, the Matlab/Octave toolbox for Reliable Computing.
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The origin of interval arithmetic Siegfried M. Rump Institute for Reliable Computing Hamburg University of Technology Schwarzenberg-Campus 1, 21071 Hamburg, Germany and Waseda University Faculty of Science and Engineering 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan [email protected]
Keywords: error bounds, intervals, reliable computing
Much has been said and published on the origin of interval arith- metic. In this talk we give a brief overview and concentrate on the seminal Master Thesis by Sunaga, handwritten in Japanese, submitted at the University of Tokyo 60 years ago on February 29, 1956.
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A method of verified computation for convex programming Ryo Kobayashi1, Takuma Kimura2 and Shin’ichi Oishi1 1Waseda University, Tokyo 169-8555, Japan 2Saga University, Saga 840-8502, Japan [email protected]
Keywords: numerical verification, convex programming, Kantorovich’s theorem, continuous Newton method
In this paper, we are concerned with a convex programming
minimize f(x), subject to g(x) 0 and x 0, (1) ≤ ≥ T where g(x) = (g1(x), g2(x), . . . , gm(x)) , and f, gi (i = 1, . . . , m): Rn R are sufficiently smooth and convex. We assume that problem (1)→ satisfies Slater’s constraint qualification. In [1], Oishi and Tanabe have studied a numerical method of in- cluding an optimal solution to a linear programming provided that an approximation of an optimal solution is given. In this talk, we pro- pose a numerical method of including an optimal solution to the above convex programming. The KKT conditions of (1) are represented as a complementarity problem
x ( f(x) + m y g (x)) φ(x, y) := i=1 i i = 0, (2a) ◦ ∇ y g(x) ∇ ◦P subject to m f(x) + y g (x) 0, g(x) 0, x 0 and y 0, (2b) ∇ i∇ i ≥ ≤ ≥ ≥ i=1 X where y Rm is a Lagrange multiplier. Here, for u, v Rn, u v denotes an∈ element-wise product (u v) = u v (i = 1, . . .∈ , n). ◦ ◦ i i i
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Hereafter, we employ the notation z = (xT , yT )T . The purpose of this talk is to verify the accuracy of a numerically computed ap- proximation of an optimal solution to (2). We propose the following theorem: Theorem 1. Let z be an interior point such that
gi(x) < 0 and x > 0.
Suppose that φ0(z) is nonsingular and there exists positive numbers α, ω 1 1 such that φ0(z)− φ(z) α and φ0(z)− (φ0(z0) φ0(z00)) ∞ k nk+∞m ≤ − ∞ ≤ ω z0 z00 , z0, z00 R . If k − k ∞ ∀ ∈ 1 αω , (3) ≤ 4 n+m there exists an optimal point z∗ B(z, ρ) := z0 R z0 z ρ satisfying (2), which is in fact∈ unique in B(z,{ ρ∈), where| k − k∞ ≤ } 1 √1 3αω ρ = − − . (4) ω Theorem 1 enables us to verify the existence of an optimal solution to (2) around a given approximation and to obtain the error bound for the approximation. Moreover, Theorem 1 guarantees feasibility of the optimal solution. Theorem 1 is proved by using Kantorovich’s theorem and continuous Newton method [2]. Some numerical examples will be presented, which show that our method is useful for verifying solutions to convex programmings. References: [1] S. Oishi, K. Tanabe, Numerical Inclusion of optimal solution for Linear, JSIAM Letters, Vol. 1 (2009), pp. 5–8. [2] K. Tanabe, Continuous Newton-Raphson method for solving an underdeterminded system of nonlinear equations, Nonlinear Anal. T.M.A., Vol. 3 No. 4 (1979), pp. 495–503. [3] S. Boyd, L. Vandenberghe, Convex Optimization, Cambridge University Press, New York, 2004.
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Numerical computation of invariant objects with wavelets Llu´ısAlsed`ai Soler and David Romero i S`anchez Universitat Aut`onomade Barcelona Departament de Matem`atiques Edifici Cc, 08913 Cerdanyola del Vall`es,Barcelona, Catalonia [email protected]
Keywords: Wavelets, regularity, quasiperiodically forced system. In certain classes of dynamical systems invariant sets with a strange geometry appear. For example the iteration of two-dimensional quasi- periodically forced skew product, under certain conditions, gives us Strange Non-Chaotic Attractors. On the other side, when approximating dynamical invariant ob- jects, a usual standard approach is to use Fourier expansions (rather than Wavelet ones). However, to obtain an analytical approximation of the aforesaid objects it seems more natural to use wavelets instead of the Fourier approach. The aim of the talk is to describe an algorithm for the semi-analytical computation of the invariant object (numerical computations of the wavelet coefficients) using both Daubechies and Haar wavelets. One of the advantages of this approach is that wavelets also de- fine certain regularity spaces that provide a natural framework for the approximations that one gets. Indeed, the computation of the regu- larity (depending on parameters) can give some insight in the study of the fractalization or other routes of creation of Strange Non-Chaotic Attractors and help in detecting this bifurcation. Thus, the aim of the above exercise is to be able to detect, by means of the estimation of the regularity of the object, how a non-chaotic at- tractor “becomes” strange. The study of this regularity (depending on parameters) may give another point of view to the “fractalization routes” described in the literature and that are currently under dis- cussion.
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The Julia package ValidatedNumerics.jl and its application to the rigorous characterization of open billiard models David P. Sanders, Luis Benet and Nikolay Kryukov Department of Physics, Faculty of Sciences Universidad Nacional Aut´onomade M´exico(UNAM) & Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology, USA Ciudad Universitaria, Del. Coyoac´an,Ciudad de M´exico04510, Mexico [email protected]
Keywords: validated software, billiard models, decorations
We firstly present ValidatedNumerics.jl [1], a new software pack- age for validated numerics using the Julia programming language, writ- ten from scratch. Julia provides a unique combination of mathematical- style syntax, high-level interactive usability, and C-like performance, making it an excellent option for technical computing, in particular for interval arithmetic. The package is well on the way to being con- formant with the IEEE Standard 1788-2015, including decorations. It is free/libre open-source software (FLOSS), with a permissive MIT li- cense, and a simple code base that allows working with both double and arbitrary precision. Due to the excellent composability of Julia, it interacts seamlessly with other Julia packages, which provide, for example, generic linear algebra and automatic differentiation capabili- ties that immediately work with intervals, with (almost) no additional coding. We then apply this to study open billiard models, which are deter- ministic dynamical systems in which point particles collide with fixed obstacles. Despite the relevance of billiard models, there seems to have been little previous work on applying interval methods to them. We study open billiard models, in which particles may escape to infinity,
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so that the billiard map, which finds the next collision with an ob- stacle, has a non-trivial domain; thus the use of decorated intervals arises naturally in these systems. We find enclosures of the domain of definition for the n-collision billiard map, as well as periodic orbits.
References:
[1] David P. Sanders and Luis Benet, ValidatedNumerics.jl Julia package, https://github.com/dpsanders/ValidatedNumerics.jl
[2] Nikolay Kryukov, David P. Sanders and Luis Benet, Rig- orous calculation of periodic trajectories in open billiard models, In preparation
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A norm estimation for an inverse of linear operator using a minimal eigenvalue Kouta Sekine, Kazuaki Tanaka and Shin’ichi Oishi Waseda university 3-4-1 Okubo, Tokyo, Japan [email protected]
Keywords: Numerical verification, Differential equations
2 1 2 Let A : H (Ω) H0 (Ω) L (Ω) be an elliptic operator with Dirich- ∩ → 1 2 let boundary conditions and N : H0 (Ω) L (Ω) be a linear operator which is given. We define a linear operator→ L := A + N. The aim of this talk is to estimate a constant K satisfying
2 1 φ 1 K Lφ L2 , φ H (Ω) H (Ω) (1) k kH0 ≤ k k ∈ ∩ 0 using an eigenvalue problem. The evaluation of the constant K plays an important role for a verified numerical proof to solutions of semi- linear partial differential equations, and have been studied [1-4]. In this talk, we first proof the following theorem:
Theorem 1. Let λ1 be an minimal eigenvalue satisfying an eigenvalue problem
2 1 1 Find u H (Ω) H (Ω), λ R s.t. LA− L∗u = λu, (2) ∈ ∩ 0 ∈ where L∗ is a dual operator of L. If λ1 = 0 holds, then L is nonsingular, and the constant 6 1 K = √λ1 satisfies the inequality (1).
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Next, we show how to compute the minimal eigenvalue λ1. Let σ be a positive constant satisfying (Du, u)L2 0, where D := A + σI 1 ≥ − (N + N ∗) + N ∗A− N. We put µ = λ + σ, and the eigenvalue problem (2) is transformed into
Du = µu. (3)
Then, we can compute the minimal eigenvalue µ1 using Liu-Oishi’s theorem[5], and obtain the constant K.
References:
[1] S. Oishi, Numerical verification of existence and inclusion of solu- tions for nonlinear operator equations, Journal of Computational and Applied Mathematics, 60.1 (1995), pp. 171-185.
[2] M.T. Nakao, K. Hashimoto, and Y. Watanabe, A numer- ical method to verify the invertibility of linear elliptic operators with applications to nonlinear problems, Computing, 75.1 (2005), pp. 1-14.
[3] K. Tanaka, A. Takayasu, X. Liu, and S. Oishi, Verified norm estimation for the inverse of linear elliptic operators using eigen- value evaluation, Japan Journal of Industrial and Applied Mathe- matics, 31:3 (2014), pp. 665-679.
[4] Y. Watanabe, K. Nagatou, M. Plum, and M.T. Nakao, Norm bound computation for inverses of linear operators in Hilbert spaces, Journal of Differential Equations, 260.7 (2016), pp. 6363- 6374.
[5] X. Liu, and S. Oishi, Verified eigenvalue evaluation for the Lapla- cian over polygonal domains of arbitrary shape, SIAM Journal on Numerical Analysis, 51.3 (2013), pp. 1634-1654.
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Interval computations in the metrology Semenov K.K.1, Solopchenko G.N.1 and Kreinovich V.Ya.2 1 Peter the Great St. Petersburg Polytechnic University 29, Polytechnicheskaya str., St. Petersburg, 195251, Russia [email protected], [email protected] 2 University of Texas at El Paso 500 W. University, El Paso, TX 79968, USA [email protected]
Keywords: interval computations, metrology, heat measuring
The modern informational world is antagonistic to the absolute pre- cision [1]. The using of the concept ’the absolute accuracy’ is possible only for mental theoretical constructs, but it isn’t possible for the real- life applications because we need to take into account the used data uncertainty. The human’s reasoning in the engineering practice and the decision-making in the automatic control systems are based on the measurements results that are distorted by the errors and that are of- ten used as the initial data. Moreover, in practice, we operate with the inaccurate results of the uncertain data mathematical processing, so our decisions should take this circumstance into account, otherwise we may be wrong in our conclusions. We often obtain the empirical data about object under observation in the interval form from the direct measurements results, the metrological assurance and the accuracy of which are provided by the metrology. When this derived uncertain in- formation is mathematically processed, the accuracy estimation of its results is also assigned to the metrology. The solution of this problem for the big amount of heterogeneous initial data is difficult because of the big number of their possible values combinations. However, at present time, this problem turns out to be solvable in the area of metrology of computer programs because of using the theory of fuzzy sets and the interval calculus.
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The interval age in computational techniques began in the 60-ies of the XX century with the IFIP’s (that was headed by Wilkinson) propositions on the accompanying the data processing software with the interval-valued estimation of the results inaccuracy. In 1962, Moore systematically considered the interval arithmetic [2] and offered the first practical manuals for its applications. In the 70-ies of the XX century, the first software shells (Linpack, for example) and libraries appeared for the linear computations. They allowed users to get the final results as the intervals of their possible values. L. Zadeh expanded the scope of the term ’uncertainty’ from only instrumental causes (rounding errors) to the initial data inaccuracy and the expert a priori information by introducing the fuzzy variables. To represent the measurement results and a priori information about them and their uncertainty as the intervals, L. Reznik introduced the fuzzy intervals [3]. V. Kreinovich and H. T. Nguyen considered the nested intervals as the measure of the fuzzy variables uncertainty [4]. This development trend expressed the real-life applications’ necessity to operate with the inaccurate initial data. The expanding using of the computational techniques in the mea- surement systems and the data processing performed by them caused the problem of the metrological software tests organization. This cir- cumstance induced the metrological community to take into account the success of the interval computations and the fuzzy variables and to include them into the metrological practice. The first step in this direction was made in 1985 with the special issue of the magazine ”Measurement Techniques” that was dedicated to the problem of the metrological support of the data processing in the metrology and the measuring. To date, the different interval approaches and methods were pro- posed and used in the various metrological applications, particularly the method [5,6] that corresponds with the requirements of the metro- logical norms and standards. To illustrate how deeply the interval computations and their expan- sions can be spread in the real life, we consider the example from our everyday practice: the calculations in the systems of heat metering.
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We cannot measure the consumed heat quantity directly, so its value is calculated from the indirect measurement results by the computa- tional block of the meter (that is called calculator). The initial data for these calculations are received from two sensors signals of the heat conveying liquid amount in the forward and return directions of the flow, two sensors signals of the temperature in the forward and return flows of heat-transfer agent, the signal of the temperature difference between these two flows and the sensor signal of the pressure of the liq- uid. The instrumental inaccuracy of all used sensors is standardized by the special European normative document [7]. The uncertainty of the measured heat amount value results to the inaccuracy in determining how much the heat was really consumed and how much the consumer should pay. For the heat calculator, only the uncertainty of temperature per- ception is normalized, but not of any other signals mentioned above. This circumstance draws some criticism. That’s why the real commer- cial task is to estimate uncertainty of the heat calculator results with taking into consideration the uncertainties of all initial data, not only the temperatures difference. In full, this problem can be solved only using the interval computations tools in the sense of the works [5,6]. In the report, the particular examples are presented for the interval computations that provide the reliable intervals of the uncertainty of the consumed and paid heat. References: [1] V. Knorring, History and mthodoloy of science and technics: in- formarmational scope of human’s agency from from aincient times to the beginning of XVI cent., St.Petersburg State Polytechnic Uni- versity, St.Petersburg, 2013 (in Russian). [2] R. Moore, Interval arithmetic and automatic error analysis in digital computing, a thesis submitted for the degree of Doctor of Philosophy, Stanford University, 1962. [3] L. Reznik, Mathematical toolkit for processing fuzzy informa- tion of experimenter in data processing systems, Proceedings of
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All-Russian Scientific Research Institute on Electrical Instrument- Making, 1983, pp. 45–55 (in Russian).
[4] H. Nguyen, V. Kreinovich, Nested intervals and sets: concepts, relations to fuzzy sets, and applications, Applications of interval computations, 1996, pp. 245–290.
[5] K. Semenov, G. Solopchenko, Combined method of metrolog- ical self-tracking of measurement data processing programs, Mea- surement Techniques, 54 (2011), No. 4, pp. 378–386.
[6] K. Semenov, L. Reznik, G. Solopchenko, Fuzzy Intervals Application for Software Metrological Certification in Measure- ment and Information Systems, Int. Journal of Uncertainty, Fuzzi- ness and Knowledge-Based Systems, 23 (2015), Suppl. 1, pp. 95– 104.
[7] European Standard prEN 1434, Heat meters, In 6 parts. Euro- pean Committee for Standardization, Brussels, 2006.
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Verified numerical computations for blow-up solutions of ODEs Akitoshi Takayasu1, Kaname Matsue2, Takiko Sasaki3, Kazuaki Tanaka3, Makoto Mizuguchi3 and Shin’ichi Oishi3 1University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8573, Japan 2The Institute of Statistical Mathematics 3Waseda University [email protected]
Keywords: ordinary differential equations, blow-up solutions, com- pactifications, Lyapunov functions, verified numerical computations This talk is concerned with blow-up solutions of ordinary differential equations (ODEs) in Rm (m N): ∈ dy(t) = f (y(t)) , y(0) = y , (1) dt 0 where t [0,T ) with 0 < T , f : Rm Rm is a C1 function and m∈ ≤ ∞ → y0 R . Unless otherwise noted, f is assumed to be a polynomial, whose∈ coefficients are real numbers. The blow-up solutions are a class of solutions of (1).
Definition. Define tmax > 0 as
t := sup t¯: a solution y C1([0, t¯)) of (1) exists . max ∈ We say that the solution y of (1) blows up if t < . In such a case max ∞ tmax is called the blow-up time of (1).
We provide a method for verifying blow-up solutions with their blow-up times on the basis of compactifications and verification of the Lyapunov function.
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Blow-up solutions are not compatible with numerical computations without any ideas, since their norm goes to infinity in finite time. To overcome this difficulty, we analyze blow-up solutions with two instruments. The first one is compactification of base spaces and of dynamical systems on them, following Poincar´e([1] see also [2,3]). The second one is verification of Lyapunov functions. The monotonicity of the Lyapunov function enables us to derive a re-parameterization of trajectories, which is called Lyapunov tracing proposed in [4]. This re- parameterization yields explicit estimates of blow-up times. The above two instruments with standard methodologies of numerical verification methods [5,6] and dynamical systems lead to verification of blow-up solutions with their blow-up times. References: [1] H. Poincare´, M´emoire sur les Courbes D´efiniespar une Equation Diff´erentielle, Oeuvres., 1881. [2] U. Elias, H. Gingold, Critical points at infinity and blow up of solutions of autonomous polynomial differential systems via com- pactification, J. Math. Anal. Appl., 318 (2006), pp. 305–322. [3] J. Hell, Conley index at infinity, Ph.D. Thesis in Freie Universit¨at Berlin, 2010. [4] K. Matsue, T. Hiwaki, and N. Yamamoto, On the con- struction of Lyapunov functions with computer assistance, arXiv preprint arXiv:1604.05953, 2016. [5] K. Kashiwagi, M. Kashiwagi, Numerical Verification of ordi- nary differential equations using Affine Arithmetic and Mean Value Form, Transactions of the Japan Society for Industrial and Applied Mathematics, 21 (2011), No. 1, pp. 37–58 (in Japanese). [6] M. Kashiwagi, kv - C++ Numerical Verification Libraries. http://verifiedby.me/kv/
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On verification methods for parabolic partial differential equations using the evolution operator Akitoshi Takayasu1, Makoto Mizuguchi2, Takayuki Kubo1 and Shin’ichi Oishi2 1University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8573, Japan 2Waseda University [email protected]
Keywords: evolution operator, parabolic PDEs, wrapping effect
In this talk, we provide verification methods for an initial-boundary value problem of parabolic partial differential equations:
∂tu(t, x) ∆u(t, x) = f(u(t, x)) 0 < t T, x Ω, u(t, x) =− 0 0 < t ≤ T, x ∈ ∂Ω, (1) u(0, x) = u (x) x Ω≤, ∈ 0 ∈ d d 2 where T > 0, Ω = (0, 1) R (d = 1, 2, 3), u0 L (Ω), and the map f satisfies suitable assumptions.⊂ Such a method is∈ based on the evolution operator U(t, s), which is the solution operator of the homogeneous initial value problem:
∂ u + A(t)u = 0, 0 s < t T, t (2) u(s) = φ, ≤ ≤ where A(t) 0 t T is a certain family of unbounded operators and φ L2(Ω).{ The} ≤ ≤ evolution operator gives the formula u(t) = U(t, s)φ for∈ representing a solution of (2). From the classical theory of the evolution operator [1, 2, 3, 4], the criteria of generating the evolution operator are that A(t) for each t (s, T ] is sectorial and the family A(t) is H¨oldercontinous with respect∈ to t. { }
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In the talk, we introduce the precise assumptions for providing the verification method and give several estimates associated with the evolution operator. By using the evolution operator, an fixed-point formulation is obtained in order to prove the existence of the mid solution of (1). A sufficient condition for a solution to be enclosed within a neighborhood of a numerical solution is also derived from Banach’s fixed point theorem. Furthermore, if the sufficient condition is satisfied, another fixed- point formulation is used for estimating the error at a fixed time be- tween the enclosed mid solution and the numerical solution. This for- mulation is based on the generalized semigroup property of the evolu- tion operator: U(t, s) = U(t, r)U(r, s), 0 s r t T, U(s, s) = I, 0 ≤ s ≤ T.≤ ≤ ≤ ≤ Consequently, this formulation is similar to the techniques for reducing the wrapping effect when solving ordinary differential equations by interval methods.
References
[1] H. Tanabe, On the equations of evolution in a Banach space, Os- aka Mathematical Journal, 12:2 (1960), pp. 363–376. [2] P. E. Sobolevskii, On equations of parabolic type in Banach space with unbounded variable operator having a constant domain, Akad. Nauk Azerbaidzan. SSR Doki, 17:6 (1961) (in Russian). [3] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer, New York, 1983. [4] A. Yagi, Abstract Parabolic Evolution Equations and their Appli- cations, Springer-Verlag, Berlin Heidelberg, 2010.
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On verified numerical computation for positive solutions to elliptic boundary value problems Kazuaki Tanaka1, Kouta Sekine1, and Shin’ichi Oishi2,3 1Graduate School of Faculty of Science and Engineering, Waseda University. 2Faculty of Science and Engineering,Waseda University. 3CREST, JST. 1,2Building 63, Roeom 419, Okubo 3-4-1, Shinjuku, Tokyo 169-8555, Japan. [email protected]
Keywords: elliptic equation, positive solution, verified numerical com- putation
We are concerned with verified numerical computations for solu- tions to the following elliptic problem:
Lu = f (u) (1a) − and
u > 0 in Ω (1b) with an appropriate boundary condition. Here, Ω is a bounded domain (i.e., an open connected bounded set) in Rn (n = 1, 2, 3, ), f is a given nonlinear operator from V (an appropriate solution··· space) to L2 (Ω), and L is a uniformly elliptic self-adjoint operator from its domain D(L) to L2 (Ω) (the domain D(L) depends on the smoothness of the boundary ∂Ω). Verified numerical computation methods for elliptic equations have been developed by many researchers (see, e.g., [1,2,3]). These methods enable us to numerically obtain a concrete ball containing a solution to target equations, typically in the sense of the norms 2 and k∇·kL (Ω) 135 SCAN 2016
. No matter how small the radius of the ball is, however, some L∞(Ω) solutionsk·k have the possibility not to be positive. For example, in the homogeneous Dirichlet case, it is possible for a verified solution not to be positive near the boundary ∂Ω. To verify solutions to (1), we propose two numerical methods for proving the positiveness of a function u satisfying (1a). One is an extended result of [4, Theorem 2], which provides a sufficient condition for the positiveness on the basis of the strong maximum principle. The other method computes the upper bound of the number of the nodal domains of a verified solution u∗ to (1a), by regarding u∗ as an eigenfunction of a self-adjoint elliptic operator; if the number is at most one, we can ensure the positiveness of u∗. Numerical examples for some concrete elliptic boundary value prob- lems will be presented.
References:
[1] M.T. Nakao, Numerical verification methods for solutions of ordi- nary and partial differential equations, Numerical Functional Anal- ysis and Optimization, 1 (2001), pp. 321–356.
[2] M. Plum, Computer-assisted enclosure methods for elliptic differ- ential equations, Linear Algebra and its Applications, 324 (2001), pp. 147–187.
[3] A. Takayasu, X. Liu, and S. Oishi, Remarks on computable a priori error estimates for finite element solutions of elliptic prob- lems, Nonlinear Theory and Its Applications, IEICE 5 (2014), pp. 53–63.
[4] K. Tanaka, K. Sekine, M. Mizuguchi, and S. Oishi, Numer- ical verification of positiveness for solutions to semilinear elliptic problems, JSIAM Letters 7 (2015), pp. 73–76.
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Error Analysis of Lagrange Interpolation on Tetrahedrons Kenta Kobayashi Takuya Tsuchiya Hitotsubashi University Ehime University Kunitachi, 186-8601 Japan Matsuyama, 790-8577 Japan [email protected]
Keywords: Lagrange interpolation, tetrahedrons, generalized circum- radius, finite elements 3 Let K R be a tetrahedron with vertices xi, i = 1, , 4. Let ⊂ ··· λi be its barycentric coordinates with respect to xi. By definition, 4 we have 0 λi 1, i=1 λi = 1. Let N0 be the set of nonnegative ≤ ≤ 4 integers, and γ = (a1, , a4) N0 be a multi-index. Let k be a P··· 4 ∈ positive integer. If γ := i=1 ai = k, then γ/k := (a1/k, , a4/k) can be regarded as| a| barycentric coordinate in K. The set··· Σk(K) of points on K is defined by P
k γ 4 Σ (K) := K γ = k, γ N . k ∈ | | ∈ 0 n o For k, is the set of all polynomials of three variables whose degree k is at mostP k. For a continuous function v C0(K), the Lagrange interpolation k v of degree k is defined∈ as IK ∈ Pk v(x) = ( k v)(x), x Σk(K). IK ∀ ∈ In this talk We present an error estimation of v k v in terms | − IK |m,p,K of hK := diamK and the generalized circumradius GK of K, which is defined in the following. Let B be a facet of K and hB := diamB. Let RB be the circumra- 3 dius of B. Consider a projection δθ from R into a 2-dimensional affine linear space such that δθ(B) is a segment. Then, δθ(K) is always a tri- angle on the plane. Let RP be the maximum value of the circumradius of δθ(K). We define the generalized circumradius GK by
RBRP GK := min , B hB
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where the minimum is taken for all facets of K. The following is the main theorem.
Theorem 1 Let K be an arbitrary tetrahedron. Let hK := diamK and GK be the generalized circumradius of K. Let 1 p and k, m be integers with k 1 and 0 m k. Then,≤ there≤ ∞ exists a constant C = C(k, m, p≥) independent≤ of K≤such that, for arbitrary v W k+1,p(K), ∈ k m k+1 2m v Kv m,p,K CGKhK − v k+1,p,K | − I | ≤ m | | GK k+1 m = C h − v . h K | |k+1,p,K K Note that no geometric condition is imposed in the main theorem.
References:
[1] K. Kobayashi, T. Tsuchiya, A Babuˇska-Aziz type proof of the circumradius condition, Japan J. Indust. Appl. Math., 31 (2014), 193-210.
[2] K. Kobayashi, T. Tsuchiya, On the circumradius condition for piecewise linear triangular elements, Japan J. Indust. Appl. Math., 32 (2015) 65–76.
[3] K. Kobayashi, T. Tsuchiya, A priori error estimates for La- grange interpolation on triangles. Appl. Math., Praha, 60 (2015), 485–499.
[4] K. Kobayashi, T. Tsuchiya, Extending Babuˇska-Aziz theorem to higher order Lagrange interpolation. Appl. Math., Praha, 61 (2016), 121–133.
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An implicit algorithm for validated enclosures of the solutions to variational equations for ODEs. Irmina Walawska and Daniel Wilczak Institute Of Computer Science and Computational Mathematics, Jagiellonian University Lojasiewicza 6 30-348 Krak´ow POLAND [email protected]
Keywords: validated numerics, variational equations, initial value problem We propose a new algorithm for computing validated bounds for the solutions to the first order variational equations associated to ODEs: x˙(t) = f(x(t)), V˙ (t) = Df(x(t)) V (t), (1) x(0) [x ], · 0 V (0) ∈ [V ], ∈ 0 n n n n2 where f : R R is a smooth function and [x0] R ,[V0] R are sets of initial→ conditions. Solutions the variational⊂ equation⊂V (t) give us an information about sensitivities of trajectories with respect to ini- tial conditions. They proved to be useful in finding periodic solutions, proving their existence and analysis of their stability. They are used to estimate invariant manifolds of periodic orbits. Derivatives with respect to initial conditions are used to prove the existence of connect- ing orbits. The method that we propose for computation of validated solutions to (1), uses a high-order Taylor method as a predictor step and an implicit method based on Hermite-Obreshkov interpolation as a corrector step. The proposed algorithm is an improvement of the 1-Lohner algorithm proposed by Zgliczy´nskiand it provides sharper C 139 SCAN 2016
bounds. Our algorithm, cannot produce worse estimations than the 1-Lohner algorithm. Complexity analysis shows that, in low dimen- sions,C it is slower than the 1-Lohner algorithm by the factor 9/8 only. This lack of performanceC is compensated by a significantly smaller truncation error of the method. This allows to take larger time steps when computing the trajectories and thus our algorithm appears to be slightly faster than the 1-Lohner in real applications. C References:
[1] Alefeld, G., Inclusion methods for systems of nonlinear equations— the interval Newton method and modifications. Topics in validated computations (Oldenburg, 1993). Vol. 5 of Stud. Comput. Math. North-Holland, Amsterdam, pp. 7–26.
[2] Zgliczynski,´ P.,C1-Lohner algorithm, Foundations of Computa- tional Mathematics, (2002), 2 (4), 429–465.
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The properties of negation and zero in ringoids as defined by Kulisch Ronald van Nooijen and Alla Kolechkina Delft University of Technology [email protected]
Keywords: ringoid, algebra, computer arithmetic
In [1,2] Kulisch defines (ordered) ringoids and vectoids to provide a theoretical basis for computer arithmetic and interval arithmetic. One interesting aspect of his treatment is the search for necessary and sufficient conditions for a meaningful notion of negation and zero. In this paper we consider this both from the point of view of functions on the underlying set and from a category theoretical standpoint. It turns out that the conditions provided by Kulisch can be restated in other forms, but that the original form is probably both necessary and sufficient for the intended purpose.
References:
[1] U.W. Kulisch and W.L. Miranker, The arithmetic of the digital computer: a new approach, SIAM Rev., 28 (1986), No 1, pp. 1–40.
[2] U.W. Kulisch, Computer arithmetic and validity: Theory, imple- mentation, and applications, de Gruyter, Berlin, 2nd revised and extended edition, 2013.
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Validated constructive error estimatations for bi-harmonic problems Yoshitaka Watanabe, Takehiko Kinoshita and Mitsuhiro T. Nakao Kyushu University 6-10-1 Hakozaki, Higashi-ku, Fukuoka 812-8518, Japan [email protected]
Keywords: PDEs, bi-harmonic problem, error estimation This talk presents some constructive error estimates for the approx- imate solutions of two-dimensional bi-harmonic equations: ∆2u = f in Ω, ∂u (1) u = = 0 in ∂Ω ∂n by using verified computational techniques. Here Ω is a bounded con- vex polygonal domain, f L2(Ω), and ∂u/∂n stands for the outer normal derivative of u. The∈ error estimations for (1) are expected to provide invaluable information for computer-assisted proofs of nonlin- ear bi-harmonic problems [1]. Several numerical examples based on Legendre polynomials [2] confirm the effectiveness of our proposed ap- proach. References: [1] K. Nagatou, K. Hashimoto, M.T. Nakao, Numerical verifi- cation of stationary solutions for Navier-Stokes problems, Journal of Computational and Applied Mathematics, 199 (2007), pp. 445– 451. [2] T. Kinoshita, M.T. Nakao, On very accurate enclosure of the 2 optimal constant in the a priori error estimates for H0 -projection, Journal of Computational and Applied Mathematics, 234 (2010), pp. 526–537.
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Computing the Worst-Case Peak Gain of Digital Filter in Interval Arithmetic Anastasia Volkova, Christoph Lauter and Thibault Hilaire Sorbonne Universit´es, UPMC Univ Paris 06, UMR 7606, LIP6 4, place Jussieu 75005 Paris, France first name.last [email protected]
Keywords:digital filters, interval arithmetic, worst-case peak gain The Worst-Case Peak Gain (WCPG) of a Linear Time Invariant (LTI) filter is used to determine the output interval of a filter and in error propagation analysis [5]. Consider a stable LTI filter in state-space representation: H x(k + 1) = Ax(k) + Bu(k) (1) H y(k) = Cx(k) + Du(k) where u(k) is the input vector, y(k) is the output vector, x(k) is the state vector and matrices A, B, C, D contain the filter coefficients. The WCPG of a linear filter can be computed [1] as the infinite k sum W := D + k∞=0 CA B . In [6] the authors have proposed an algorithm for| the| reliable evaluation of the WCPG matrix in multiple P precision. However, usually the filter coefficients are rounded prior to imple- mentation, changing A, B, C and D by rounding. To provide a re- liable filter implementation, these rounding errors must be taken into account in the WCPG computation. We represent the rounded coef- ficients as interval [2] matrices with small radii. Let MI := M , M h c ri to be an interval matrix centered in Mc with radius Mr. Then, the WCPG matrix of a filter = AI, BI, CI, DI is an interval H I I I Ik I W := D + k∞=0 C A B . In this work we adapt the algorithm presented in [6] to obtain a reli- P able evaluation of the WCPG interval. The WCPG is computed in two stages: the reliable truncation of the infinite sum and the summation.
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We determine the truncation order only for the center matrices but add a correction term after the final step. This step requires to perform an eigenvalue decomposition. To obtain trusted error bounds on the computed eigenvalues we use the Theory of Verified Inclusions developed by S. Rump [4]. The summation is done using Interval Arithmetic in midpoint- radius form. However, powering a dense interval matrix can lead to an interval explosion. Instead of powering AI we power an almost diag- I I onal matrix T , for which T 2 < 1 is true. We use an analogue of Gershgorin circle theorem [3]|| to|| verify a spectral norm condition that needs to be satisfied for the WCPG sum to converge. It is obvious that we cannot guarantee an a priori given bound on the WCPG matrix radius Wr because the radii of the input matrices are the limiting factors. However, when given point coefficient matrices (intervals with zero radii) and an absolute error bound ε we guarantee that the output WCPG interval in not larger than ε in width.
References [1] V. Balakrishnan and S. Boyd. On computing the worst-case peak gain of linear systems. Systems & Control Letters, 19:265–269, 1992. [2] H. Dawood. Theories of Interval Arithmetic: Mathematical Foundations and Applications. LAP Lambert Academic Publishing, 2011. [3] S. Gershgorin. Uber¨ die Abgrenzung der Eigenwerte einer Matrix. Bull. Acad. Sci. URSS, 1931(6):749–754, 1931. [4] S. M. Rump. New results on verified inclusions. In Accurate Scientific Computa- tions, Symposium, Proceedings, 1985. [5] A. Volkova, T. Hilaire, and C. Lauter. Determining fixed-point formats for a digital filter implementation using the worst-case peak gain measure. In 2015 49th Asilomar Conference on Signals, Systems and Computers, Nov 2015. [6] A. Volkova, T. Hilaire, and C. Lauter. Reliable evaluation of the worst-case peak gain matrix in multiple precision. In Computer Arithmetic (ARITH), 2015 IEEE 22nd Symposium on, pages 96–103, June 2015.
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Computer-assisted existence proofs for one-dimensional Schr¨odinger-Poisson systems Jonathan Wunderlich Karlsruhe Institute of Technology Department of Mathematics Englerstraße 2, 76131 Karlsruhe, Germany [email protected]
Keywords: Computer-assisted proof, Schr¨odinger-Poisson, existence, enclosure
We are aiming at non-trivial solutions of the one-dimensional time- independent Schr¨odinger-Poisson system
u00 + V u + Φuu = f(u) − 2 on R, Φ00 + cΦ = u − u u ) lim Φu = 0 x →±∞ where c > 0 is an additional parameter needed in the one-dimensional 1 case, V L∞(R) is a positve potential and f C (R). This∈ one-dimensional Schr¨odinger-Poisson∈ system is a simplified model for the three-dimensional time-dependent version
~2 i~∂tψ ∆ψ + qeWeψ = f(u) 3 − − 2m on [0, ) R ∞ × ε∆W = q ψ 2 − e e| | lim We = 0 x | |→∞ which (for f 0) plays an important role in today’s semiconductor technology. ψ≡represents the wavefunction of a particle, in our case of an electron, and m is its mass. We describes the electric potential
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which depends on the wavefunction ψ by the above Poisson equation with Dirichlet boundary conditions. If the nonlinearity satisfies the covariance property
iϕ iϕ f(e z) = e f(z)(z C, ϕ R), ∈ ∈ we can eliminate the time dependency by the standing wave ansatz
iωt ψ(x, t) = u(x)e (t [0, ), x R) ∈ ∞ ∈ with a parameter ω > 0. To prove non-trivial solutions of the one-dimensional Schr¨odinger- Poisson system we first “solve” the second equation using the corre- sponding Green’s function Γ, and insert the result into the first one:
∞ 2 u00 + V + Γ( , t)u(t) dt u = f(u) on R. − · Z−∞ Furthermore u should be a solution with finite energy level for physical 1 reasons, i.e. we look for a solution u H (R). Applying computer-assistance to∈ the above equation, we are able to prove the existence of a non-trivial solution of the one-dimensional Schr¨odinger-Poisson system for the case c = 50, constant potential V 1, and the nonlinearity f chosen as f(y) = y3 (y R). ≡Starting from a numerical approximate solution,∈ we compute a bound for its defect, and a norm bound for the inverse of the lineariza- tion at the approximate solution. For the latter, eigenvalue bounds play a crucial role, especially the eigenvalues “close to” zero. There- for we use the Rayleigh-Ritz method and a corollary of the Temple- Lehmann theorem to get enclosures of the eigenvalues of the lineariza- tion below the essential spectrum. With these data in hand, we can use a fixed-point argument to ob- tain the desired existence of a non-trivial solution “nearby” the approx- imate one. In addition to the pure existence result, the used methods also provide an enclosure of the exact solution.
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Numerical verification of existence of homoclinic orbits in dynamical systems Nobito Yamamoto1, Kaname Matsue2 and Shun Yamano1 1The University of Electro-Communications, Chofu-ga-oka, Chofu, 182-8585, Tokyo, Japan 2The Institute of Statistical Mathematics, 10-3, Midori-Cho, Tachikawa, Tokyo, 190-8562, Japan [email protected]
Keywords: verified numerics, dynamical systems, ODEs, homoclinic orbits We propose numerical verification methods to prove existence of homoclinic orbits in dynamical systems described by ODEs. We treat an example problem with a saddle equilibrium in R3 which has 1- dimensional stable manifold and 2-dimensional unstable manifold. Whereas a previous research [1] adopts time-reverse computation for homoclinic orbits of hyperbolic equilibria with 1-dimensional un- stable manifolds, it is so hard for us to follow the flows (time dependent solutions of ODEs) by time-reverse computation because of numerical instability of our problem. This means that we cannot use naive meth- ods, e.g. intermediate value theorem, in order to capture homoclinic orbits of hyperbolic equilibria with 2-dimensional unstable manifolds. Our methods are based on theories of mapping degree as well as Lyapunov functions constructed explicitly by verified computation. Brouwer coincidence theorem, which leads to Brouwer fixed point the- orem, is used for specifying parameters to have a homoclinic orbit. Consider an autonomous system of ODEs :
d n x = f(x; p), x, f R , (1) dt ∈ where p R2 is a pair of parameters p = (a, b). We suppose that the system∈ (1) may have a hyperbolic equilibrium with 2-dimensional
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unstable manifold admitting a homoclinic orbit, and our aim is to prove the existence of the homoclinic orbit and to specify a narrow area which contains the parameters attaining the existence of the homoclinic orbit. Computing an approximate flow to be homoclinic-like, we construct a mapping from a small rectangular domain of the parameters (a, b) to a plane including a family of equilibria along with flows starting from specified points on the unstable manifolds in a neighborhood of the equilibria with respect to p = (a, b). The starting points are chosen from a small area where the approximate flow runs through, and the flows are expected to come back to a neighborhood of the equilibria which contains the stable manifolds. Lyapunov functions are com- posed by verified computation in order to specify the starting points on the unstable manifolds and the neighborhood of the stable mani- folds, where we use our previous works [2]. The degree of this mapping is estimated by our new theorem so called Interval Simplex Theorem, which counts the degree using interval arithmetic and verified compu- tation. Finally, we apply Brouwer coincidence theorem to proving that there is an equilibrium as an image of the mapping, which means that the rectangular domain of p contains a pair of parameters that admits a homoclinic orbit.
References:
[1] D. Wilczak, The Existence of Shilnikov Homoclinic Orbits in theMichelson System: A Computer Assisted Proof, Foundations of Computational Mathematics, 6(4), 2006, pp. 495–535.
[2] K. Matsue, T. Hiwaki and N. Yamamoto, On the construc- tion of Lyapunov functions with computer assistance,, arXiv preprint, arXiv:1604.05953
[3] S. Yamano, Numerical verification methods for homoclinic or- bits in continuous dynamical systems, Master’s thesis for graduate school of The University of Electro-Communications, in Japanese, 2016.
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Verification method for system of linear equations by QR factorization 1 Yuka Yanagisawa†, Shin’ichi Oishi‡ and Fumi Noda‡ Research Institute for Science and Engineering, Waseda University † Department of Applied Mathematics, Faculty of Science and ‡ Engineering, Waseda University 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan [email protected]
Keywords: linear equation, QR factorization, accurate numerical al- gorithm We are concerned with the matrix equation Ax = b where A is an n n real matrix and x and b be n-vectors. If A = QR is a QR factor- ization,× then we can write QRx = b where Q is n n with orthonormal columns and R is n n and upper triangular. This× equation is easy solve because R is triangular.× The principal method for computing QR factorization is Householder triangularization, which is excellent numerical stability. Nevertheless, QR factorization is not the standard method for computing the approximate solution to Ax = b in prac- tice, since it requires two times computational cost of LU factorization with partial pivoting, which is unstable for matrices with large growth factors2 [1]. Assume that an approximate solutionx ˜ is given with an approxi- mate QR factorization. In this talk, we will present an accurate and fast method using QR factorization for proving nonsingularity of A 1 and for calculating rigorous error bounds for A− b x˜ . Specifi- k − k2 cally, we focus a method of calculating an upper bound of I BA 2 where B is an approximate inverse of A and I is the nk −n iden-k × tity matrix. It is well-know that If I BA 2 is satisfied, then A is 1 k − kB(Ax˜ b) 2 proved to be nonsingular and A− b x˜ 2 1k I −BAk . In a previous k − k ≤ −k − k2 1NTT Communications Corporation 2In practice, such matrices are very rare in applications.
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work, Oishi-Rump proposed a method utilizing an approximate LU factors [2]. This method satisfies I BA nu κ2(A) where κ2(A) is the 2-norm condition numberk of −A andk∞u≈is the· unit roundoff. It 16 3 4 3 requires 3 n flops (3n flops for calculating an approximate inverse B using the LU factors and 4n3 flops for calculating the upper bound of I BA ). Our method based on the Yomoda’s approach [3] utilize an | − | 17 3 4 3 approximate QR factors. Our method which requires 3 n flops (3n 1 3 flops for calculating the QR factors Q, R such that A QR, 3n flops for calculating an approximate inverse of R, 2n3 flops≈ for calculating 3 the upper bound of I Q>Q and 2n flops for calculating the upper | − | bound of Q> XA ) gives more accurate result than the Oishi-Rump method [2].| Moreover,− | our algorithm can treat the case where A is 1 ill-conditioned such that κ2(A) u− , if we use an accurate dot prod- uct algorithm [4]. We also present≈ detailed analysis of our method and some numerical results.
References:
[1] Lloyd N. Trefethen, David Bau III, Numerical Linear alge- bra, SIAM, Philadelphia, 1997.
[2] S. Oishi and S. M. Rump, Fast Verification of Solutions of matrix equations, Numer. Math, 90-4 (2002), pp. 755–773.
[3] E. Yomoda, Studies on the numerical verification of regularity of matrices, Unpublished thesis for master degree, Tokyo Woman’s Christian University, (2012), (in Japanese).
[4] K. Ozaki, T. Ogita, S. Oishi, Tight and Efficient Enclosure of Matrix Multiplication by Using Optimized BLAS, Numerical Linear Algebra with Applications, 18-2 (2011), pp. 237–248.
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An Accurate and Efficient Solution of Ill-conditioned Linear Systems by Preconditioning Methods 1, 1 2 Yuka Kobayashi ∗, Takeshi Ogita and Katsuhisa Ozaki 1Tokyo Woman’s Christian University 167–8585 Tokyo, Japan 2Shibaura Institute of Technology 337–8570 Saitama, Japan ∗ [email protected]
Keywords: accurate numerical algorithm, solution of linear systems, preconditioning technique We are concerned with an accurate numerical solution of linear systems n n n Ax = b, A R × , b R (1) ∈ ∈ by using floating-point arithmetic. There are two standard meth- ods using an LU factorization to solve (1) accurately. One of the standard methods is using an LU factorization with the iterative re- finement method. If the problem is not ill-conditioned, it can work well. However, this method is not effective for ill-conditioned problems. The other method is using an LU factorization with multiple-precision arithmetic [1, 2]. Although it can work for ill-conditioned problems, it takes significant computing time regardless of the condition number of A. To remedy the defects of the standard methods, we propose an algorithm based on the preconditioning method [3] using a result of an LU factorization by floating-point arithmetic. The proposed algorithm can provide accurate numerical solutions for ill-conditioned problems beyond the limit of the working precision. Moreover, it requires less computational cost than the previous preconditioning method [4, 5] using an approximate inverse of A as a preconditioner. We conducted numerical experiments using the proposed algorithm on MATLAB. For n = 10000, we can obtain approximate solutions of Ax = b with the
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condition number up to 1030. If we use accurate computations using BLAS [6], it takes only 7 9.5 times as long as an LU factoriza- tion. Results of numerical∼ experiments are presented for confirming the effectiveness of the proposed algorithm.
References:
[1] The GNU MPFR Library, http://www.mpfr.org
[2] The GNU Multiple Precision Arithmetic Library, https://gmplib.org
[3] T. Ogita, Accurate matrix factorization: inverse LU and inverse QR factorizations, SIAM J. Matrix Anal. Appl., 31:5 (2010), pp. 2477– 2497.
[4] S. M. Rump, Inversion of extremely ill-conditioned matrices in floating-point, Japan J. Indust. Appl. Math., 26 (2009), pp. 249– 277.
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152 Author index
Afravi Mahdokht, 73 Grodet Aymeric, 53 Alsed`ai Soler Llu´ıs,122 Aschemann Harald, 110, 112 Hilaire Thibault, 143 Awala Hussein, 37 Hlad´ıkMilan, 55 H¨usken Matthias, 57 B´anhelyiBal´azs,21 Iakymchuk Roman, 59 Bag´oczkiZsolt , 21 Baker Kearfott Ralph, 69 Jaulin Luc, 44, 61, 116 Barragan Pedro, 38 Jeangoudoux Clothilde, 63 Benet Luis, 123 Jeannerod Claude-Pierre, 98 Bouwmeester Henricus, 31 Jiang Hao, 65 Br´ehardFlorent, 40 Joldes Mioara, 26, 40, 100 Brisebarre Nicolas, 40 J´ez´ekielFabi´enne,97
Capi´nskiiMaciej, 23 Kashiwagi Masahide, 67, 90 Ceberio Martine, 24 Kimura Takuma, 71, 120 Chapoutot Alexandre, 42 Kinoshita Takehiko, 142 Costa Tiago M., 31 Klenke Cornelia, 108 Csendes Tibor, 21 Kobayashi Kenta, 137 Kobayashi Ryo, 120 Defour David, 59 Kobayashi Yuka, 151 Desrochers Benoit, 44 Kokubu Hiroshi, 28 Domes Ferenc, 92 Kolechkina Alla, 141 Du Peibing, 65 Krˇc´alMarek, 48 Kreinovich V.Ya., 127 F´evotte Fran¸cois,46 Kreinovich Vladik, 38, 73 Franek Peter, 48, 106 Kryukov Nikolay, 123 F´evotte Fran¸cois,97 Kubica Bart lomiej, 75 Gaidashev Denis, 50 Kubo Takayuki, 133 Garloff J¨urgen,51 Kuˇr´atko Jan, 106 Graillat Stef, 59, 63, 97 Lathuili`ereBruno, 46
153 SCAN 2016
Lathuili`ereBruno, 97 Ozaki Katsuhisa, 93, 151 Lauter Christoph, 63, 143 Lavor Carlile, 31 Paulen Radoslav, 95 Le Bars Fabrice, 116 Peng Lin, 65 Lessard Jean-Philippe, 30 Picot Romain, 97 Li Kuan, 65 Plet Antoine, 98 List Ivo, 77 Plum Michael, 33 LIU Xuefeng, 79 Popescu Valentina, 100 Lodwick Weldon, 31 Popova Evgenija D., 102 Lodwick Weldon A., 81 Purohit Harsh, 104 Louvet Nicolas, 98 Ratschan Stefan, 106 Markot Mihaly, 92 Rauh Andreas, 108, 110, 112 Matsuda Nozomu, 82 Reddy Anwesh, 95 Matsue Kaname, 84, 131, 147 Revol Nathalie, 114 McKenna P. Joseph, 33 Rohou Simon, 116 Mihaylova Lyudmila, 116 Romero i S`anchez David, 122 Minamoto Teruya, 71 Rump Siegfried M., 118, 119 Miyajima Shinya, 86, 88 Mizuguchi Makoto, 131, 133 Sanders David P., 123 Montanher Tiago, 92 Sandretto Julien Alexandre dit, Morikura Yusuke, 90 42 Mukkula Gottu, 95 Sasaki Takiko, 131 Muller Jean-Michel, 98, 100 Schichl Hermann, 92 Sekine Kouta, 90, 125, 135 Nagatou Kaori, 33 Semenov K.K., 127 Nakao Mitsuhiro T., 71, 142 Solopchenko G.N., 127 Nataraj P.S.V., 104 Stadtherr Mark A., 35 Neumaier Arnold, 92 Noda Fumi, 149 Taher Lotfi, 81 Nozawa Yusuke, 90 Takayasu Akitoshi, 131, 133 Tanaka Kazuaki, 125, 131, 135 Ogita Takeshi, 93, 151 Tiede Susann, 108 Oishi Shi’nichi, 90 Titi Jihad, 51 Oishi Shin’ichi, 120, 125, 131, Tsuchiya Takuya, 53, 137 133, 135, 149
154 SCAN 2016 van Nooijen Ronald, 141 Veiseh Hana, 81 Veres Sandor M., 116 Volkova Anastasia, 143
Wagner Hubert, 48 Walawska Irmina, 139 Watanabe Yoshitaka, 142 Wilczak Daniel, 139 Wunderlich Jonathan, 145
Yamamoto Nobito, 82, 147 Yamano Shun, 147 Yampolsky Michael, 50 Yanagisawa Yuka, 149
Zerr Benoˆıt,61 Zhao Yao, 35
155