报告题目：An augmented matrix-based CJ-FEAST SVDsolver for computing a partial singular value decomposition with the singular values in a given interval

报 告 人：贾仲孝（清华大学）

报告时间：2024年7月19日  16:00-17:00

报告地点：格物楼528

报告摘要：

The cross-product matrix-based CJ-FEAST SVDsolver proposed previously by the authors is shown to compute the left singular vector possibly much less accurately than the right singular vector and may be numerically backward unstable when a desired singular value is small. In this talk, an alternative augmented matrix-based CJ-FEAST SVDsolver is proposed to compute the singular triplets of a large matrix A with the singular values in an interval [a, b] contained in the singular spectrum. The new CJ-FEAST SVDsolver is a subspace iteration applied to an approximate spectral projector of the augmented matrix[0, A^T; A, 0]associated with the eigenvalues in [a, b], and it constructs approximate left and right singular subspaces independently, onto which A is projected to obtain the Ritz approximations to the desired singular triplets. Compact estimates are given for the accuracy of the approximate spectral projector constructed by the Chebyshev-Jackson (CJ) series expansion in terms of series degree, and a number of convergence results are established. The new solver is proved to be always numerically backward stable. A convergence comparison of the cross-product matrix-based and augmented matrix-based CJ-FEAST SVDsolvers is made, and a general-purpose choice strategy between the two solvers is proposed for the robustness and overall efficiency. Numerical experiments confirm all the results and meanwhile demonstrate that the proposed solver is more robust and substantially more efficient than the corresponding contour integral-based versions that exploit the trapezoidal rule and the Gauss-Legendre quadrature to construct an approximate spectral projector.

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、有国际影响的研究成果。在Inverse Problems, Mathematics of Computation, Numerische Mathematik, SlAM系列等杂志上发表论文70篇，研究工作被引用逾1300篇次，引用的专著和教材17部，包括Bai、Demmel、Dongarra等五人编辑的指南Templates for the Solution of Algebraic Eigenvalue Problems: a Practical Guide（2000），Golub & van Loan的Matrix Computations（1996, 2013），Stewart的Matrix Algorithms II: Eigensystems（2001），Bjorck的Numerical Methods in Matrix Computations（2015），van der Vorst的Computational Mathods for Large Eigenvalue Problems（2002），Trefethen & M. Embree的Spectra and Pseudospectra, The Behavior of Nonnormal Matrices and Operators（2005），Meurant & Tebbens的Krylov Methods for Nonsymmetric Linear Systems（2020），Quarteroni、Sacco & Saleri的Numerical Mathematics（2000）。