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Recent Advances of Numerical Linear Algebra: Theory, Algorithms
and Applications |
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Bai Zhao-Jun,
University of California at Davis, USA (bai@cs.ucdavis.edu) |
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Abstract.
Optimization of
large scale numerical linear algebra computations is a
long-standing problem in computational science and engineering.
Significant progress has been made both in general procedures
and also in more focused situations which exploit specific
matrix structure and sparsity patterns, or other special
characteristics of the problem at hand. However, challenges of
the advancement of modern complex and multiscale mathematical
modeling and simulation to matrix theory and algorithms have not
been widely addressed in our research community.Most solvers are
not designed in ways that are robust and efficient for
underlying ``multiscale matrices''. This is an important
emerging research area since a broad range of scientific and
engineering modeling and simulation problems involve multiple
scales for which traditional monoscale approaches have proven to
be inadequate, even with the largest supercomputers.
In this lecture series, we will focus on the study of recent
advances of numerical linear algebra theory and algorithms for
large linear system of equations arising from multiscale
material science simulations. They include bilinear form
computing, self-adapting direct solvers, and iterative solvers
with robust preconditioning techniques.
Depending the availability of lecture hours, we also plan to
discuss a couple of selected topics, such as large scale linear
and nonlinear eigenvalue problems and model order reduction of
dynamical systems.
It is expected that the students have a solid knowledge of
undergraduate-level linear algebra, some knowledge of numerical
analysis and matrix computations, in addition, have some
experience with writing computer programs in Matlab, Fortran
and/or C.
Lecture notes will be made available prior to the lectures. The
most part of lecture materials are based on the joint work with
a large of collaborators I fortunately have had over the past
years, including Chinese scholars Professors Yangfeng Su, Wenbin
Chen and Weiguo Gao of Fudan University. |
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Krylov Subspace Methods -Their
Application to Singular Systems and Least Squares Problems |
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Ken Hayami, National Institute
of Informatics, Tokyo, Japan (hayami@nii.ac.jp) |
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Abstract.
Krylov subspace methods are robust and
efficient methods for solving large sparse systems of linear
equations occurring in science and engineering.
Among these methods, in this lecture, we will study the
Generalized Minimum Residual (GMRES) method and the Generalized
Conjugate Residual (GCR) method.
Then, I will introduce our recent work on the analysis of the
behaviour of the GMRES and GCR methods on singular linear
systems, and the application of the GMRES method to over- and
under-determined linear least squares problems. |
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Multigrid and Domain Decomposition Methods |
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Xu Xue-Jun, Lsec,
Institute of Computational Mathematics, Academy of Mathematics
and System Sciences, Chinese Academy of Sciences, Beijing, China
(xxj@lsec.cc.ac.cn) |
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Abstract.
Multigrid and domain decomposition methods are the most
efficient modern techniques for solving large scale algebraic
systems arising from the discretization of partial differential
equations. In this course, I shall introduce the basic idea and
convergence theory of multigrid and domain decomposition methods
by considering their applications to the second order and fourth
order elliptic problems. |
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Multigrid Methods |
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Shi Zhong-Ci,
Institute of Computational Mathematics, Chinese Academy of
Sciences, Beijing, China (shi@lsec.cc.ac.cn) |
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Abstract.
Multigrid method is considered as one of the most
efficient algorithms that solves sparse linear systems.of O(N)
unknowns with O(N) computational complexity for large classes of
problems.
In this talk we will briefly describe the method
and some new developments. |
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Organization of the Summer School |
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Zhang Guo-Feng,
Lanzhou University, China (gf_zhang@lzu.edu.cn) |
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Abstract. |
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Computation of Matrix Permanent and Its Application in
Nanoscience |
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Bai Feng-Shan,
Tsinghua University, Beijing, China (fbai@math.tsinghua.edu.cn) |
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Abstract. |
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Recent Advances for Solving Saddle Point Problems |
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Bai Zhong-Zhi,
Institute of Computational Mathematics, Academy of Mathematics
and System Sciences, Chinese Academy of Sciences, Beijing, China
(bzz@lsec.cc.ac.cn) |
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Abstract. The
Hermitian and skew-Hermitian splitting (HSS) iteration scheme is
an efficient and practical method for solving large sparse non-Hermitian
system of linear equations. In this talk, after reviewing the
HSS iteration method and its basic convergence theory for non-Hermitian
positive definite matrices, we give a sufficient and necessary
condition for guaranteeing its convergence for nonsingular and
non-Hermitian positive semidefinite matrices.
We further generalize this method to solve
the saddle-point problems, obtaining the accelerated Hermitian
and skew-Hermitian splitting (AHSS) iteration method.
The optimal iteration parameters involved
in the AHSS iteration method are exactly computed, and some
numerical results are used to examine the advantages of HSS over
AHSS when the exact optimal iteration parameters are employed. |
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Inverse Eigenvalue Problems in Structural Dynamic Model Updating |
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Dai Hua,
Department of Mathematics, Nanjing University of Aeronautics &
Astronautics, Nanjing 210016, P R China, (hdai@nuaa.edu.cn) |
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Abstract.
Structural dynamic model updating studies the relationship
between the dynamic behavior and spatial properties of a
structure. Structural dynamic analysis for the direct problem
predicts the dynamic behavior changes from nominated spatial
property changes. Structural dynamic model updating does the
opposite, and is concerned with the correction of finite element
models by processing records of dynamic behavior from vibration
test. Model updating is a rapidly developing technology. There
are a lots of numerical algebra problems in the area. This talk
will provide a review of the state of the art for inverse
eigenvalue problems arising in structural dynamic model
updating, and outline the main problems faced by model updating. |
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Backward Errors for Structured Eigenvalue Problems |
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Li Wen,
School of Mathematical Sciences, South China Normal University,
Guangzhou, 510631, China (liwen@scnu.edu.cn) |
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Abstract.
In this talk we consider some open problems
proposed by Tisseur in SIMAX 03, the computable backward errors
for approximate eigenpairs of some structured matrices such as
conjugate symplectic matrices, conjugate symplectic and
Hermitian doubly structured matrices and conjugate symplectic
and Hamiltonian doubly structured matrices are presented.
* Joint work with Weiwei Xu |
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迭代算法研究及其应用 |
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Liu
Xing-Ping, Beijing Institute of Applied Physics and
Computational Mathematics, China (lxp@mail.iapcm.ac.cn) |
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Abstract.
本报告涉及的内容有：当前国内外某些科学工程计算领域对代数方程组求解算法的需求及其重要性，代数方程组求解算法在国内外某些科学工程计算领域的应用情况，代数方程组迭代求解算法今后的发展趋势介绍。 |
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Preconditioned Lanczos Method for Generalized Toeplitz
Eigenvalue Problems |
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Lu Lin-Zhang,
Xiamen University, China (lzlu@xmu.edu.cn) |
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Abstract. |
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Su Yang-Feng,
Fudan University, China (yfsu@fudan.edu.cn) |
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Abstract. |
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Multilevel Wavelet-like
Incremental Unknowns: θ-Scheme and Weighted Semi-Implicit Scheme |
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Wu Yu-Jiang,
Lanzhou University, Lanzhou 730000, China (myjaw@lzu.edu.cn) |
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Abstract.
* Joint work with Jia Xing-Xing and She An-Long |
