天元数学东北中心 Tianyuan Mathematical Center in Northeast China
Workshop on Computational
Methods for Eigenvalue Problems
Jilin University·Changchun
Tianyuan Mathematical Center in Northeast China
2018.8.14 - 8.17
国家自然科学基金数学天元基金资助 吉林大学资助
Contents
Goal of the Workshop ...... 1
Organizing Committee ...... 2
Workshop Sponsors...... 3
Information ...... 4
Invited Speakers ...... 6
Schedule ...... 8
Abstracts ...... 12
Workshop Participants ...... 33
Introduction to Tianyuan Mathematical Center in Northeast China ...... 36
Goal of the Workshop
The purpose of this workshop is to bring together people working on various aspects of the study of practical nonlinear iterative methods, mesh based adaptive methods, scalable computational methods of the discretized problems as well as their corresponding tractable models and highly efficient numerical simulation on computers with more than ten thousand CPU cores, for solving eigenvalue problems arising from modeling the microstructure of matter. This workshop is one of workshop series on Computational Methods for Eigenvalue Problems. The past four workshops were held at Academy of Mathematics and Systems Science, Chinese Academy of Sciences, on July 15-16, 2014, May 30-31, 2015, September 24-25, 2016, and December 9-10, 2017, respectively.
1
Organizing Committee
Xiaoying Dai Academy of Mathematics and Systems Science, Chinese Academy of Sciences Ran Zhang Jilin University Weiying Zheng Academy of Mathematics and Systems Science, Chinese Academy of Sciences Aihui Zhou Academy of Mathematics and Systems Science, Chinese Academy of Sciences
2
Workshop Sponsors
This workshop is sponsored by: ※National Natural Science Foundation Mathematics Tianyuan Fund
※Academy of Mathematics and Systems Science, Chinese Academy of Sciences
※Jilin University.
3
Information
1、Registration Registration Opens: 14:00pm to 20:00pm on August 13, at June Hotel 8:00am to 11:30am on August 14, at the Lecture Hall of the Math Building For other time registration, please contact us. 2、Venue The First Lecture Hall, School of Mathematics of Jilin University
The Math Building
The Lecture Hall 4
3、Dining Breakfast Lunch Dinner 6:30-9:00 12:00-14:00 18:00-20:00
August 13 —— —— June Hotel
August 14 June Hotel June Hotel June Hotel
August 15 June Hotel June Hotel June Hotel
August 16 June Hotel June Hotel Banquet
August 17 June Hotel June Hotel June Hotel
*The Banquet will be held on August 16, starting at 6:00pm.
4、Contact us Xiuping Gao 13596117968 Danhong Wang 13404775711 Xingwu Sun 13504468895
5、Tips
※June Hotel Address: No.811 Xiuzheng Road, Changchun City, Jilin Province, China Tel：+86-431-87058888
※Taking taxi Please do not take a taxi in the night. If you need, Please contact Mr. Lv for picking up service before you come to
Changchun , Tel:13624465812.
5
Invited Speakers
No. Name Institute
1 Zhaojun Bai University of California, Davis, USA
2 Yongyong Cai Beijing Computational Science Research Center, China
3 Huajie Chen Beijing Normal University, China
4 Lizhen Chen Beijing Computational Science Research Center, China
5 Xingqiu Chen Institute of Metal Research, Chinese Academy of Sciences, China
6 Zhengyu Chen University of Waterloo, Canada Academy of Mathematics and Systems Science, Chinese Academy Xiaoying Dai 7 of Sciences, China
8 Jun Fang Institute of Applied Physics and Computational Mathematics,China
9 Weiguo Gao Fudan University, China
10 Xingyu Gao Institute of Applied Physics and Computational Mathematics,China
11 Yali Gao Northwestern Polytechnical University, China Stefano de Scuola Internazionale Superiore di Studi Avanzati (SISSA) and 12 Gironcoli CNR-IOM DEMOCRITOS Simulation Centre, Italy
13 Guanghui Hu University of Macau, China
14 Jun Hu Peking University, China
15 Wenjian Liu Peking University, China Academy of Mathematics and Systems Science, Chinese Academy Xin Liu 16 of Sciences, China
17 Yanming Ma Jilin University, China Laboratoire Jacques-Louis Lions and French Institute for Research Xinran Ruan 18 in Computer Science and Automation, France
19 Tetsuya Sakurai University of Tsukuba, Japan
20 Sihong Shao Peking University, China Tomohiro Nagoya University, Japan 21 SOGABE
22 Yangfeng Su Fudan University, China
6
Michigan Technological University, USA and Beijing Jiguang Sun 23 Computational Science Research Center, China
24 Yanqiu Wang Nanjing Normal University, China
25 Wei Wang Beijing Computational Science Research Center, China Academy of Mathematics and Systems Science, Chinese Academy Hehu Xie 26 of Sciences, China
27 Chao Yang Lawrence Berkeley National Laboratory, USA
28 Xiaobo Yin Central China Normal University, China
29 Zhimin Zhang Beijing Computational Science Research Center, China
30 Yuzhi Zhou Institute of Applied Physics and Computational Mathematics,China
7
Schedule
Workshop on Computational Methods for Eigenvalue Problems
Venue: The First Lecture Hall, School of Mathematics of Jilin University
Date Time Schedule
August 13 14:00－20:00 Registration: June Hotel (Monday)
08:30－08:50 Opening Ceremony and Group Photo
08:50－10:20 Chair: Zhiming Zhang
Zhaojun Bai: Nonlinear Eigenvalue Problems: Recent Advances and 08:50－09:35 Challenges
Tetsuya Sakurai: An Eigen-analysis Engine for Large-scale Simulation and 09:35－10:20 Data Analysis
10:20－10:35 Tea Break
10:35－12:05 Chair: Xingqiu Chen
Stefano de Gironcoli: Comparing Efficiency of Iterative Eigenvalue 10:35－11:20 Solvers: the Quantum ESPRESSO Experience Yuzhi Zhou: Applicability of Kerker Preconditioning Scheme to the Self- 11:25－12:05 consistent Density Functional Theory Calculations August 14 12:05－14:30 Lunch (Tuesday)
14:30－16:00 Chair: Tetsuya Sakurai
Wenjian Liu: iVI: An Iterative Vector Interaction Method for Large 14:30－15:15 Eigensystems Zhiming Zhang: An Efficient Spectral Galerkin Method for Maxwell 15:15－16:00 Transmission Eigenvalue Problems 16:00－16:15 Tea Break
16:15－17:45 Chair: Stefano de Gironcoli
Zhengyu Chen: Eigenvalue Problems in Polymer Theory: Their Physics 16:15－17:00 and Challenges
Huajie Chen: A Discontinuous Galerkin Scheme for Full-potential 17:00－17:45 Electronic Structure Calculations
18:00－20:00 Dinner
8
Date Time Schedule
08:30－10:00 Chair: Zhaojun Bai
Chao Yang: Many-body Localization and Thermalization: A 08:30－09:15 Computational Study
09:15－10:00 Weiguo Gao: TBA
10:00－10:15 Tea Break
10:15－11:45 Chair: Wenjian Liu
10:15－11:00 Xingqiu Chen: TBA
11:00－11:45 Sihong Shao: Computational Quantum Mechanics in Phase Space
11:45－14:30 Lunch August 15 (Wednesday) 14:30－16:00 Chair: Chao Yang
14:30－15:15 Yanming Ma: TBA
Jun Fang: On the Wavefunction Prediction Techniques in Born- 15:15－16:00 Oppenheimer Molecular Dynamics
16:00－16:15 Tea Break
16:15－17:45 Chair: Yanming Ma
Xin Liu: Parallelizable Algorithms for Optimization Problems with 16:15－17:00 Orthogonality Constraints
Tomohiro Sogabe: Recent Progress on Numerical Algorithms for Large 17:00－17:45 Eigenvalue Problems
18:00－20:00 Dinner
9
Date Time Schedule
08:30－10:00 Chair: Ran Zhang
08:30－09:15 Jun Hu: TBA
Hehu Xie: The Construction and Application of Subspace in Solver for 09:15－10:00 Eigenvalue Problems
10:00－10:15 Tea Break
10:15－11:45 Chair: Weiguo Gao
10:15－11:00 Yangfeng Su: Theory and Computation of 2D Eigenvalue Problems
Wei Wang: Domain Decomposition Method for Discrete Elliptic 11:00－11:45 Eigenvalue Problems
August 16 11:45－14:30 Lunch (Thursday) 14:30－16:00 Chair: Xin Liu
Xingyu Gao: The Adaptive Self-consistent Field Iteration for Solving the 14:30－15:15 Kohn-Sham Equation
Guanghui Hu: H-adaptive Finite Element Methods in Kohn-Sham 15:15－16:00 Density Functional Theory
16:00－16:15 Tea Break
16:15－17:45 Chair: Xingyu Gao
16:15－17:00 Yongyong Cai: Ground States of Spinor Bose-Einstein Condensates
Xinran Ruan: Preliminary Studies of the Fundamental Gaps of the 17:00－17:45 Gross-Pitaevskii Equation and the Fractional Schrödinger Operator
18:00－20:00 Banquet
10
Date Time Schedule
08:30－10:00 Chair: Yangfeng Su
08:30－09:15 Jiguang Sun: A Multilevel Memory Efficient Spectral Indicator Method
Yanqiu Wang: Finite Elements on Prisms and Cones with Polygonal 09:15－10:00 Bases
10:00－10:15 Tea Break
10:15－11:45 Chair:Jiguang Sun
Xiaoying Dai: A Conjugate Gradient Method for Electronic Structure 10:15－11:00 Calculations
Yali Gao: Decoupled, Linear, and Energy Stable Finite Element 11:00－11:45 Method for the Two-phase Flows in Karstic Geometry August 17 (Friday) 11:45－14:30 Lunch
14:30－16:00 Chair: Xiaoying Dai
Xiaobo Yin: Acceleration of Stabilized Finite Element Discretizations 14:30－15:15 for the Stokes Eigenvalue Problem
15:15－16:00 Lizhen Chen: Stokes Eigen-modes in 2D Regular Polygons
16:00－16:15 Tea Break
16:15－17:45 Chair: Zhaojun Bai
16:15－17:30 Panel Discussion
18:00－20:00 Dinner
11
Abstracts
Nonlinear Eigenvalue Problems: Recent Advances and Challenges
Zhaojun Bai [email protected], University of California, Davis, USA
Nonlinear eigenvalue problems (NEPs) arise in electronic structure calculations and robust data clustering among many others. The NEPspose intriguing challenges in analysis and computation and are a much less explored topic compared to linear eigenvalue problems. From a linear algebra point of view, I will start this talk with some recent advances in analysis and computation of NEPs and applications. Then I will discuss a number of challenges and open problems.
Ground States of Spinor Bose-Einstein Condensates
Yongyong Cai [email protected], Beijing Computational Science Research Center, China
The remarkable experimental achievement of Bose-Einstein condensation (BEC) in 1995 has drawn significant research interests in understanding the ground states and dynamics of trapped cold atoms. Different from the single component BEC, spinor BEC possesses the spin degree of freedom and exhibits rich phenomenon. In the talk, we will introduce some mathematical results for ground states of spin-1,2 BECs, and a practical imaginary time propagation method for numerical simulation with several different projection strategies.
12
A Discontinuous Galerkin Scheme for Full-potential Electronic Structure Calculations
Huajie Chen
[email protected], Beijing Normal University, China
We construct an efficient numerical scheme for full-potential electronic structure calculations of periodic systems. This scheme uses radial basis functions times spherical harmonics around each nuclei, plane waves away from the nuclei, and patch these parts together by discontinuous Galerkin (DG) method. It has the same philosophy as the widely used (L)APW methods in material sciences, but possesses systematically spectral convergence rate. We provide a rigorous a priori error analysis of the DG approximations for the linear eigenvalue problems. We also present some numerical simulations in electronic sturcture calculations to support our theory. This is a joint work with Xiaoxu Li.
Stokes Eigen-modes in 2D Regular Polygons
Lizhen Chen [email protected], Beijing Computational Science Research Center, China
The Stokes eigen-modes in two-dimensions on regular polygons are computed numerically by spectral element solvers and their convergence toward the eigen-modes on the disc is analyzed.
13
Xingqiu Chen [email protected] Institute of Metal Research, Chinese Academy of Sciences, China TBA
Eigenvalue Problems in Polymer Theory: Their Physics and Challenges
Zhengyu Chen [email protected], University of Waterloo, Canada
The formalism of statistical physics of polymers, either following the Gaussian model or the worm-like chain model, is reviewed with particular attentions paid to the eigenvalue problems. In polymer confinement, under the so-called ground-state dominance approximation, the ground-state eigenvalue and eigenfunction can be directly mapped to polymer confinement free energy and conformational properties, which are experimentally measurable. Recent progress in understanding the DNA confinement problems is discussed together with new numerical challenges.
14
A Conjugate Gradient Method for Electronic Structure Calculations
Xiaoying Dai [email protected] Academy of Mathematics and Systems Science, Chinese Academy of Sciences, China
In this talk, we will introduce a conjugate gradient method for electronic structure calculations. We propose a Hessian based step size strategy, which together with three orthogonality approaches yields three algorithms for computing the ground state energy of atomic and molecular systems. Under some mild assumptions, we prove that our algorithms converge locally. It is shown by our numerical experiments that the conjugate gradient method is efficient. This is a joint work with Zhuang Liu, Liwei Zhang, and Aihui Zhou.
15
On the Wavefunction Prediction Techniques in Born-Oppenheimer Molecular Dynamics
Jun Fang [email protected] Institute of Applied Physics and Computational Mathematics, China
Wavefunction extrapolation greatly reduces the number of self-consistent field (SCF) iterations and thus the overall computational cost of Born-Oppenheimer molecular dynamics (BOMD) that is based on the Kohn–Sham density functional theory. Going against the intuition that the higher order of extrapolation possesses a better accuracy, we demonstrate, from both theoretical and numerical perspectives, that the extrapolation accuracy firstly increases and then decreases with respect to the order, and an optimal extrapolation order in terms of minimal number of SCF iterations always exists. We also prove that the optimal order tends to be larger when using larger MD time steps or more strict SCF convergence criteria. By example BOMD simulations of a solid copper system, we show that the optimal extrapolation order covers a broad range when varying the MD time step or the SCF convergence criterion. Therefore, we suggest the necessity for BOMD simulation packages to open the user interface and to provide more choices on the extrapolation order. Another factor that may influence the extrapolation accuracy is the alignment scheme that eliminates the discontinuity in the wavefunctions with respect to the atomic or cell variables. We prove the equivalence between the two existing schemes, thus the implementation of either of them does not lead to essential difference in the extrapolation accuracy.
16
Weiguo Gao [email protected], Fudan University, China TBA
The Adaptive Self-consistent Field Iteration for Solving the Kohn-Sham Equation
Xingyu Gao [email protected] Institute of Applied Physics and Computational Mathematics, China
We have developed a modified Kerker preconditioning scheme that can be used to improve the self-consistent field (SCF) iteration with a priori knowledge of the system. If we were not sure whether it is metal or insulator, we design a posteriori indicator to monitor if the preconditioner has suppressed charge sloshing during the SCF iteration. Based on the a posteriori indicator, we present two adaptive configuration schemes for the SCF iteration and apply them to Au-MoS2 contacts. Furthermore, we investigate the sensitivity of the a posteriori indicator to the k-point sampling and smearing width.
17
Decoupled, Linear, and Energy Stable Finite Element Method for the Two-phase Flows in Karstic Geometry
Yali Gao [email protected], Northwestern Polytechnical University, China
In this talk, I will consider the numerical approximation for a phase field model of the coupled two-phase free flow and two-phase porous media flow. This model consists of Cahn-Hilliard- Navier-Stokes equations in the free flow region and Cahn-Hilliard- Darcy equations in the porous media region that are coupled by seven interface conditions. The coupled system is decoupled based on the interface conditions and the solution values on the interface from the previous time step. A fully discretized scheme with finite elements for the spatial discretization is developed to solve the decoupled system. In order to deal with the difficulties arising from the interface conditions, the decoupled scheme needs to be constructed appropriately for the interface terms, and a modified discrete energy is introduced with an interface component. Furthermore, the scheme is linearized and energy stable. Hence, at each time step one need only solve a linear elliptic system for each of the two decoupled equations. Stability of the model and the proposed method is rigorously proved. Numerical experiments are presented to illustrate the features of the proposed numerical method and verify the theoretical conclusions.
18
Comparing Efficiency of Iterative Eigenvalue Solvers: the Quantum ESPRESSO Experience
Stefano de Gironcoli [email protected] Scuola Internazionale Superiore di Studi Avanzati (SISSA) and CNR-IOM DEMOCRITOS Simulation Centre, Italy
The iterative determination of a small fraction of the lowest-lying eigenvalue- eigenvector pairs of large sparse matrices is a fundamental task in modern electronic structure applications that needs to be addressed efficiently in terms of time-to-solution and memory requirements. A number of iterative eigensolvers are implemented in the Quantum ESPRESSO [1] suite of codes, including the block Davidson [2] diagonalization, a simple band-by- band conjugate gradient (CG) solver, a Parallel Orbital-updating (ParO) [3] approach, and the Projected Preconditioned Conjugate-Gradient (PPCG) [4] method. I will discussion the merits and problems of the different methods in view of their performance in mixed MPI/OpenMP or hybrid CPU/GPU architectures.
[1] www.quantum-espresso.org [2] ER Davidson, J of Comput Phys 17 (1975) 87-94. [3] Y Pan, XY Dai, S de Gironcoli, XG Gong, GM Rignanese, AH Zhou, J of Comput Phys 348 (2017) 482-492. [4] E Vecharynski, C Yang, JE Pask, J of Comput Phys 290 (2015) 73-89.
19
H-adaptive Finite Element Methods in Kohn-Sham Density Functional Theory
Guanghui Hu [email protected], University of Macau, China
As one of the most successful approximation for the many-body Schrodinger equation, Kohn-Sham density functional theory has been playing a crucial role in a variety of the areas such as quantum computational chemistry, condensed matter physics, nano-optics. In the ground state simulations of a given electronic structure calculation, the study on the nonlinear generalised eigenvalue problem is important, in a class of self- consistent field iteration methods. In this talk, we will describe an h-adaptive finite element framework for Kohn-Sham density functional theory, and introduce the numerical issues caused, Specially, we will introduce the implementation of LOBPCG, which is a popular eigensolver used in the related area, in our algorithm. Numerical experiments will deliver the effectiveness of the our method.
Jun Hu [email protected], Peking University, China TBA
20
iVI: An Iterative Vector Interaction Method for Large Eigensystems
Wenjian Liu [email protected], Peking University, China
Based on the generic “static-dynamic-static” framework for strongly coupled basis vectors [1,2], an iterative Vector Interaction (iVI) method [3,4] is proposed for computing multiple exterior or interior eigenpairs of large generalized eigenvalue problems, with positive-definite or non-positive definite metrics. Although it works with a fixed-dimensional search subspace, iVI can converge quickly and monotonically from above to the exact exterior/interior roots. The algorithms are further specialized to nonrelativistic and relativistic time-dependent density functional theories (TD-DFT) by taking the orbital Hessian as the metric (i.e., the inverse TD-DFT eigenvalue problem) and incorporating explicitly the paired structure into the trial vectors [4]. The efficacy of iVI and iVI-TD-DFT is demonstrated by various examples.
21
Parallelizable Algorithms for Optimization Problems with Orthogonality Constraints
Xin Liu [email protected] Academy of Mathematics and Systems Science, Chinese Academy of Sciences, China
To construct a parallel approach for solving optimization problems with orthogonality constraints is usually regarded as an extremely difficult mission, due to the low scalability of the orthogonalization procedure. However, such demand is particularly huge in some application domains such as material computation. In this talk, we propose two infeasible algorithms, based on augmented Lagrangian penalty function, for solving optimization problems with orthogonality constraints. Different with the classic augmented Lagrangian method, our algorithms update both the prime variables and the dual variables by new strategies. The orthogonalization procedure is only invoked once as the last step of the above mentioned two algorithms. Consequently, the main parts of these two algorithms can be parallelized naturally. We establish global subsequence convergence results for our proposed algorithms. Worst-case complexity and local convergence rate are also studied under some mild assumptions. Numerical experiments, including tests under parallel environment, illustrate that our new algorithms attain good performances and a high scalability in solving discretized Kohn- Sham total energy minimization problems.
22
Yanming Ma [email protected], Jilin University, China TBA
Preliminary Studies of the Fundamental Gaps of the Gross-Pitaevskii Equation and the Fractional Schrödinger Operator
Xinran Ruan [email protected] Laboratoire Jacques-Louis Lions and French Institute for Research in Computer Science and Automation, France
The fundamental gap, the difference between the smallest two eigenvalues, is an important quantity since it relates to the smallest excitation energy of the ground state and, therefore, measures the stability of it. The ‘Fundamental Gap Conjecture’ proposes a lower bound of the fundamental gap of the linear Schrödinger operator via the diameter of the convex domain, regardless of the external potentials, and has been proved recently. In this talk, I will talk about the fundamental gap in two generalized cases: the Gross- Pitaevskii equation (GPE), i.e. the Schrödinger equation with cubic nonlinearity, and the fractional Schrödinger operator. For both cases, the detailed asymptotic results will be shown for two special external potentials: the box potential and the harmonic potential. Based on the asymptotic results, we find new interesting phenomenon of the 23 two cases compared to the original linear Schrödinger operator case: (I) For the GPE with repulsive interactions, degeneracy could lead to a smaller fundamental gap, which is completely different from the linear Schrödinger operator case, where there is no need to distinguish the degenerate case and non-degenerate case. (II) For the fractional Schrödinger operator, the fundamental gap could be arbitrarily close to 0 when the diameter of the domain is fixed. It turns out that a possible lower bound of the fundamental gap in this case depends on not only the diameter of the domain, but also the shape of it. Based on our new findings, new fundamental gap conjectures will be proposed.
An Eigen-analysis Engine for Large-scale Simulation and Data Analysis
Tetsuya Sakurai [email protected] Center for Artificial Intelligence Research, University of Tsukuba, Japan
Large-scale eigenvalue problems arise in wide variety of scientific and engineering applications such as nano-scale materials simulation, vibration analysis of automobile components, data clustering, graph analysis, etc. In such situations, a high performance parallel solver is required in distributed parallel computational environments. In this talk, we present a parallel eigensolver, the Sakurai-Sugiura method (SSM), for large- scale interior eigenvalue problems. This method is derived using numerical quadrature, and has a good parallel scalability. We also show a software package "z-Pares" that is an implementation of SSM. We show some numerical experiments to illustrate the numerical properties of the proposed method.
24
Computational Quantum Mechanics in Phase Space
Sihong Shao
[email protected], Peking University, China
The Wigner function has provided an equivalent and convenient way to render quantum mechanics in phase space. It allows one to express macroscopically measurable quantities, such as currents and heat fluxes, in statistical forms as usually does in classical statistical mechanics, thereby facilitating its applications in nanoelectronics, quantum optics and etc. Distinct from the Schrödinger equation, the most appealing feature of the Wigner equation, which governs the dynamics of the Wigner function, is that it shares many analogies to the classical mechanism and simply reduces to the classical counterpart when the reduced Planck constant vanishes. Despite the theoretical advantages, numerical resolutions for the Wigner equation is notoriously difficult and remains one of the most challenging problems in computational physics, mainly because of the high dimensionality and nonlocal pseudo-differential operator. On one hand, the commonly used finite difference methods fail to capture the highly oscillatory structure accurately. On the other hand, all existing stochastic algorithms, including the affinity-based Wigner Monte Carlo and signed particle Wigner Monte Carlo methods, have been confined to at most 4D phase space. Few results have been reported for higher dimensional simulations. My group has made substantial progress in both aspects. (1) We completed the design and implementation of a highly accurate numerical scheme for the Wigner quantum dynamics in 4D phase space. Our algorithm combines an efficient conservative semi-Lagrangian scheme in the temporal-spatial space with an accurate spectral element method in the momentum space. This accurate Wigner solver has been successfully applied into the investigation of quantum tunneling in double well and quantum double slit interference. Moreover, the Wigner function for a one- dimensional Helium-like system was clearly shown for the first time. (2) We explored the inherent relation between the Wigner equation and a stochastic
25 branching random walk model. With an auxiliary function, we can cast the Wigner equation into a renewal-type integral equation and prove that its solution is equivalent to the first moment of a stochastic branching random walk. In order to realize an efficient, reliable and integrated particle-based scheme to capture complicated quantum features in phase space, we utilized the probabilistic interpretation of the Wigner equation, efficient Monte Carlo strategies and non-parameter density estimation techniques. It should be noted that all proposed numerical schemes fully exploit the mathematical structure of the Wigner equation. Our target is an efficient simulator for analyzing some fundamental issues in many-body quantum mechanics, such as the nuclear quantum effect and dynamical correlation.
Recent Progress on Numerical Algorithms for Large Eigenvalue Problems
Tomohiro SOGABE [email protected], Nagoya University, Japan
Solution of large eigenvalue problems plays a fundamental role in computational science and data science. Many scientific applications have various needs to compute eigenvalues (and the corresponding eigenvectors) of the form - all the eigenvalues - all the eigenvalues in a given region - largest/smallest eigenvalues - some eigenvalues around a desired target point - the kth eigenvalue with a given number k In this talk, scientific needs-oriented algorithms are presented in terms of theory, algorithms, and applications. (This is joint work with: D. Lee and S.-L. Zhang at Nagoya University, Japan) 26
Theory and Computation of 2D Eigenvalue Problems
Yangfeng Su [email protected], Fudan University, China
The 2D eigenvalue problem (2dEVP) is a class of the double eigenvalue problems first studied by Blum and Chang in 1970s. The 2dEVP seeks scalars λ, μ, and a corresponding vector x satisfying the following equations
where A and C are Hermitian and C is indefinite. We show the connections between 2dEVP with well-known numerical linear algebra and optimization problems such as quadratic programming, the distance to instability and H∞ -norm. We will discuss (1) fundamental properties of 2dEVP including well-posedness, types and regularity, (2) numerical algorithms with backward error analysis.
A Multilevel Memory Efficient Spectral Indicator Method
Jiguang Sun [email protected], Michigan Technological University, USA and Beijing Computational Science Research Center, China
Recently a novel family of eigensolvers, called spectral indicator methods (SIMs), was proposed.Given regions of the complex plane, SIMs compute indicators and use them to detect eigenvalues.Regions that contain eigenvalues are subdivided and the procedure is repeated until eigenvalues are isolated with a specified precision. In this talk, by a special way of using Cayley transformation and Krylov subspaces, a memory efficient eigensolver for sparse eigenvalue problems is proposed. The method uses little memory and is particularly suitable for the computation of many eigenvalues of large problems. The eigensolver is realized in Matlab and tested using various matrices.
27
Finite Elements on Prisms and Cones with Polygonal Bases
Yanqiu Wang [email protected], Nanjing Normal University, China
We discuss the construction and efficient implementation of some lowest-order conforming finite elements on prisms and cones with polygonal bases. Note that 3D convex polytopes can be divided into cones by connecting its vertices with the center. Combined with our finite element on cones, this provides a practical discretization on convex polytopal meshes. This is a collaborative work with Prof. Wenbin Chen.
Domain Decomposition Method for Discrete Elliptic Eigenvalue Problems
Wei Wang [email protected], Beijing Computational Science Research Center, China
In this talk, we present a two-level overlapping hybrid domain decomposition method for solving the large scale discrete elliptic eigenvalue problems. In order to eliminate the components in the orthogonal complement space of the eigenspace, we construct a parallel precon-ditioner for the eigenvalue problem in fine space. After one coarse space correction in each iteration,we get the error reduction as γ = c(1 - Cδ/H ), where c,C are constants independent of the mesh size h and the diameter of subdomains H, δ is the overlapping size among the subdomains, and c → 1decreasingly as H → 0. Furthermore, we do not need any assumptions between H and h. Numerical results supporting our theory are given.
28
The Construction and Application of Subspace in Solver for Eigenvalue Problems
Hehu Xie [email protected] Academy of Mathematics and Systems Science, Chinese Academy of Sciences, China
本报告，我们将从求解特征值问题最基本的子空间投影算法开始，介绍其在能
量范数和 L2 范数意义下的误差估计。基于得到的误差估计的结果，我们重新来 讨论构造子空间的方法及其效果，这里将着重讨论利用多重网格方法中的子空间 来构造求解特征值问题的子空间。同时也讨论这种子空间构造方法所带来的在收
敛速度、计算量和并行效率方面的效果。
Many-body Localization and Thermalization: A Computational Study
Chao Yang [email protected], Lawrence Berkeley National Laboratory, USA
One of the intriguing questions in statistical quantum mechanics is concerned with the localization of many-body eigenvectors. Spin Hamiltonians consisting of an interaction and a disordered term are often used to study this phenomenon. When the disorder term is very large, the Hamiltonian behaves like a diagonal matrix, and its eigenvectors are clearly localized. However, many-body localization (MBL) can appear even before the disorder becomes very large due to its random nature. Physically, localization may be interpreted as particles being trapped in a local region or quantum information not being transported elsewhere. On the other hand, when the disorder term 29 is relatively small, eigenvectors tend to spread out more, and time evolution of these eigenvectors may result in an equilibrium state contributed by all eigenstates of the Hamiltonian. This is sometimes known as the eigenstates thermalization hypothesis (ETH). Physicists are particularly interested in the transition between the ETH region and the MBL region as disorder strength changes. One way to determine whether a spin Hamiltonian exhibits MBL or satisfies ETH is to examine, numerically, the eigenvalue level statistics, which is a computationally challenging task. In this talk, I will discuss efficient ways to perform this type of calculation.
Acceleration of Stabilized Finite Element Discretizations for the Stokes Eigenvalue Problem
Xiaobo Yin [email protected], Central China Normal University, China
In this talk, the stabilized finite element method based on local projection stabilization is applied to discretize the Stokes eigenvalue problems, then the corresponding stability and convergence properties are given. Furthermore, we use a postprocessing technique to accelerate the convergence rate of the eigenpair approximations. The postprocessing strategy contains solving an additional Stokes source problem in an augmented finite element space which can be constructed either by refining the mesh, or increasing the order of finite element space. Numerical tests are also provided to confirm the theoretical results.
30
An Efficient Spectral Galerkin Method for Maxwell Transmission Eigenvalue Problems
Zhimin Zhang [email protected], Beijing Computational Science Research Center, China
Maxwell transmission eigenvalue problems have important applications, and yet, their theoretical analysis and numerical computation are very challenging. In this talk, we propose and analyze a spectral Galerkin method under spherical geometry. With some mild conditions, we perform dimension reduction and transfer the problem into the one-dimensional case along the radia direction. By properly select the pole condition and basis functions, we are able to achieve exponential rate of convergence. Numerical tests confirm our theoretical results.
Applicability of Kerker Preconditioning Scheme to the Self-consistent Density Functional Theory Calculations
Yuzhi Zhou [email protected]
Institute of Applied Physics and Computational Mathematics, China
Kerker preconditioner, based on the dielectric function of homogeneous electron gas, is designed to accelerate the self-consistent field (SCF) iteration in the density functional theory (DFT) calculations. However, question still remains regarding its applicability to the inhomogeneous systems. We develop a modified Kerker preconditioning scheme which captures the long-range screening behavior of 31 inhomogeneous systems thus improve the SCF convergence. The effectiveness and efficiency is shown by the tests on long-z slabs of metals, insulators and metal-insulator contacts. For situations without a priori knowledge of the system, we design the a posteriori indicator to monitor if the preconditioner has suppressed charge sloshing during the iterations. Based on the a posteriori indicator, we demonstrate two schemes of the self-adaptive configuration for the SCF iteration.
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Workshop Participants
NO. Name Institute Email University of California, Davis, 1 Zhaojun Bai USA [email protected] Yongyong Beijing Computational Science 2 Cai Research Center, China [email protected]
3 Huajie Chen Beijing Normal University, China [email protected] Beijing Computational Science 4 Lizhen Chen Research Center, China [email protected] Institute of Metal Research, Xingqiu Chinese Academy of Sciences, 5 Chen China [email protected] Zhengyu 6 Chen University of Waterloo, Canada [email protected] Academy of Mathematics and Xiaoying Systems Science, Chinese 7 Dai Academy of Sciences, China [email protected] Dalian University of Technology, 8 Lei Du China [email protected] Institute of Applied Physics and 9 Jun Fang Computational Mathematics,China [email protected]
10 Weiguo Gao Fudan University, China [email protected] Institute of Applied Physics and 11 Xingyu Gao Computational Mathematics,China [email protected] Northwestern Polytechnical 12 Yali Gao University, China [email protected] Scuola Internazionale Superiore di Studi Avanzati (SISSA) and CNR- Stefano de IOM DEMOCRITOS 13 Gironcoli Simulation Centre, Italy [email protected] Academy of Mathematics and Systems Science, Chinese 14 Rui He Academy of Sciences, China [email protected] Guanghui 15 Hu University of Macau, China [email protected]
16 Jun Hu Peking University, China [email protected]
17 Chao Huang Peking University, China [email protected] 33
National University of Defense 18 Shengguo Li Technology, China [email protected]
19 Wenjian Liu Peking University, China [email protected] Academy of Mathematics and Systems Science, Chinese 20 Xin Liu Academy of Sciences, China [email protected] Academy of Mathematics and Systems Science, Chinese 21 Ying Liu Academy of Sciences, China [email protected]
22 Tianyi Lu Fudan University, China [email protected]
23 Yanming Ma Jilin University, China [email protected]
24 Ying Ma Shandong University, China [email protected] Laboratoire Jacques-Louis Lions and French Institute for Research in Computer Science and 25 Xinran Ruan Automation, France [email protected] Tetsuya 26 Sakurai University of Tsukuba, Japan [email protected]
27 Sihong Shao Peking University, China [email protected] Tomohiro sogabe (at)na.nuap.nagoya- 28 Sogabe Nagoya University, Japan u.ac.jp
29 Yangfeng Su Fudan University, China [email protected] Michigan Technological University,USA and Beijing Computational Science Research 30 Jiguang Sun Center, China [email protected]
31 Yanqiu Wang Nanjing Normal University, China [email protected] Pengde Beijing Computational Science 32 Wang Research Center, China [email protected] Academy of Mathematics and Systems Science, Chinese 33 Qiao Wang Academy of Sciences, China [email protected] Beijing Computational Science 34 Wei Wang Research Center, China [email protected], Zhigang 35 Wang Jilin University, China [email protected]
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Academy of Mathematics and Systems Science, Chinese 36 Hehu Xie Academy of Sciences, China [email protected] Lawrence Berkeley National 37 Chao Yang Laboratory, USA [email protected] Central China Normal University, 38 Xiaobo Yin China [email protected]
39 Ran Zhang Jilin University, China [email protected] Xuping Dalian University of Technology, 40 Zhang China [email protected] Zhimin Beijing Computational Science 41 Zhang Research Center, China [email protected] Academy of Mathematics and Systems Science, Chinese 42 Aihui Zhou Academy of Sciences, China [email protected] Institute of Applied Physics and 43 Yuzhi Zhou Computational Mathematics,China [email protected]
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Introduction to Tianyuan Mathematical Center in Northeast China
国家天元数学东北中心是国家自然科学基金委数学天元基金在 2017 年首批 设立的三个平台类项目之一。
数学天元基金是为了推动我国数学尽早实现数学强国目标而在 1990 年设立 的专项基金，以实现老一辈数学家提出的“中国数学要在二十一世纪率先赶上世 界先进水平”的目标。
数学天元基金是源于财政拨款，由国家自然科学基金委员会管理的数学专项 基金，该基金是凝聚数学家集体智慧，探索符合数学特点和发展规律的资助方式， 推动建设数学强国而设立的专项科学基金。数学天元基金项目支持科学技术人员 结合数学学科特点和需求，开展科学研究，培养青年人才，促进学术交流，优化 研究环境，传播数学文化，从而提升中国数学创新能力。
经过近 30 年的发展，在国家自然科学基金委、历届学术领导小组和全国数 学工作者的共同努力下，数学天元基金在学科发展规划、学科方向调整、学科队 伍建设、青年人才培养、研究环境的改善、优秀数学家的培养等方面发挥了重要 作用，为推动我国数学学科迅速发展做出了重要贡献。
国家自然科学基金委数学天元基金领导小组为更好的促进区域数学学科平 衡发展，于 2017 年设立天元数学中心项目，该项目以构建交流平台促进合作和 研究为主旨，针对若干数学及其交叉领域或专题，通过多种形式的学术交流研讨 活动，凝聚相关研究队伍，聚焦科学问题，深化国内外多种领域专家间合作，培 养青年学术骨干，引导年轻人进入学科前沿，促进数学与其他学科、数学各分支 间的交叉融合，提升我国相关领域或专题的整体研究水平，形成优势研究方向， 推动数学学科发展。
首批天元中心包括西北、西南、东北三个中心，其中东北中心主要围绕计算
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数学、大分析和统计学展开活动。国家天元数学东北中心以 17 位国内外有重要 影响的专家学者组成的学术委员会为核心，由吉林大学协同东北师范大学、大连 理工大学、哈尔滨工业大学等 23 所东北共建院校数学学科负责人组成的执行委 员会共建。天元数学东北中心既鼓励自由探索，也主动面向国家重大战略需求， 通过多种形式的学术交流研讨，创造良好的学术交流环境，加强国内外多领域科 学家之间的紧密合作，促进数学与其它学科、数学各分支间的交叉融合，培植新 兴学科增长点，打造在国际上有重要影响的学科方向，力争在相关研究领域取得 重大突破，切实提升我国数学研究的整体地位。同时，培养一批复合型、高素质 的数学后备人才和善于解决实际问题的交叉型人才，力争对中国数学，特别是对
东北地区数学整体水平的提升起到重要推动作用。
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