报告题目: A FEA ces ST SVDsolver based on Chebyshev--Jackson series for computing partial singular triplets of large matrices
报告人: 贾仲孝教授（清华大学）
报告时间: 2023年6月28日 (周三) 10:00—11:30
报告地点: 维格堂 319
报告摘要: The FEAST eigensolver is extended to the computation of the singular triplets of a large matrix Awith the singular values in a given interval. The resulting FEAST SVDsolver is subspace iteration applied to an approximate spectral projector of A^TA corresponding to the desired singular values in a given interval, and constructs approximate left and right singular subspaces corresponding to the desired singular values, onto which A is projected to obtain Ritz approximations. Differently from a commonly used contour integral-based FEAST solver, we propose a robust alternative that constructs approximate spectral projectors by using the Chebyshev--Jackson polynomial series, which are symmetric positive semi-definite with the eigenvalues in [0,1]. We prove the pointwise convergence of this series and give compact estimates for pointwise errors of it and the step function that corresponds to the exact spectral projector. We present error bounds for the approximate spectral projector and reliable estimates for the number of desired singular triplets, establish numerous convergence results on the resulting FEAST SVDsolver, and propose practical selection strategies for determining the series degree and for reliably determining the subspace dimension. The solver and results on it are directly applicable or adaptable to the real symmetric and complex Hermitian eigenvalue problem. Numerical experiments illustrate that our FEAST SVDsolver is at least competitive with and is much more efficient than the contour integral-based FEAST SVDsolver when the desired singular values are extreme and interior ones, respectively, and it is also more robust than the latter.
报告人简介: 贾仲孝，1994年获得德国比勒菲尔德大学博士学位，清华大学数学科学系二级教授，第六届国际青年数值分析家--L. Fox奖获得者(1993)，国家“百千万人才工程”入选者 (1999)。现任北京数学会第十三届监事会监事长（2021.12—2026.12），曾任清华大学数学科学系学术委员会副主任 (2009—2021)，2010年度“何梁何利奖”数学力学专业组评委，中国工业与应用数学学会 (CSIAM) 第五和第六届常务理事(2008.9—2016.8)，第七和第八届中国计算数学学会常务理事(2006.10—2014.10)，北京数学会第十一和十二届副理事长（2013.12—2021.12），中国工业与应用数学学会 (CSIAM) 监事会监事(2020.1—2021.10)。主要研究领域：数值线性代数和科学计算。在代数特征值问题、奇异值分解和广义奇异值分解问题、离散不适定问题和反问题的正则化理论和数值解法等领域做出了系统性的、有国际影响的重要研究成果，所提出的精化投影方法被公认为是求解大规模矩阵特征值问题和奇异值分解问题的三类投影方法之一。在Inverse Problems, Mathematics of Computation, Numerische Mathematik, SIAM Journal on Matrix Analysis and Applications, SIAM Journal on Optimization, SIAM Journal on Scientific Computing等国际著名杂志上发表论文70余篇，研究工作被41个国家和地区的900多名专家与研究人员在19部经典著作、专著和教材（国外）及760多篇论文中他引1350篇次（其中被国际学术界575篇论文引用935篇次）。引用的书目包括Bai、Demmel、Dongarra、Ruhe、van der Vorst等五人编辑的Templates for the Solution of Algebraic Eigenvalue Problems: a Practical Guide (2000)，Golub & van Loan的经典著作Matrix Computations第三、第四版(1996，2013)，Stewart的经典著作Matrix Algorithms II: Eigensystems (2001)，Bjorck的专著Numerical Methods in Matrix Computations (2015)，van der Vorst的专著“Computational Methods for Large Eigenvalue Problems (2002)，Trefethen & Embree的专著Spectra and Pseudospectra, The Behavior of Nonnormal Matrices and Operators (2005)，Meurant & Tebbens的专著 Krylov Methods for Nonsymmetric Linear Systems (2020)，Quarteroni、Sacco & Saleri的专著 Numerical Mathematics (2000)，Brezinski、Meurant和Revido-Zaglia的A Journey Through the History of Numerical Linear Algebra (2022).
邀请人：黄金枝