报告题目：A FEAST SVDsolver based on Chebyshev-Jackson series for computing partial singular triplets of large matrices

报 告 人：贾仲孝（清华大学）

报告时间：2024年7月16日  16:00-17:00

报告地点：格物楼528

报告摘要：

The FEAST eigensolver is extended to the computation of the singular triplets of a large matrix A with 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 shown to be 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 of interest. We present error bounds for the approximate spectral projector and reliable estimates for the number of desired singular triplets, prove the convergence of the resulting FEAST SVDsolver, and propose practical selection strategies for determining the series degree and 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 the FEAST SVDsolver is substantially more efficient than the contour integral-based FEAST SVDsolver, and it is also more robust and stable 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）。