勷勤数学•领军学者报告

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  教授  (邀请人：黎稳)

                                  清华大学

时      间：11月15日  10：00-11：00

地     点：数科院东楼二楼报告厅

报告人简介:

       贾仲孝，1994 年获得德国比勒菲尔德(Bielefeld)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--Leslie Fox 奖获得者，国家“百千万人才工程” 入选者。现任北京数学会第十三届监事会监事长，中国工业与应用数学学会 (CSIAM) 第五、第六届常务理事，第七、第八届中国计算数学学会常务理事，北京数学会第十一和十二届副理事长，中国工业与应用数学学会 (CSIAM) 监事会监事. 主要研究领域：数值线性代数和科学计算。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43个国家和地区的1200多名专家和研究人员在20部经典著作、专著和教材（国外）及954篇论文中他引1601篇次（其中被国际学术界652篇论文他引1033篇次，包括被20本经典著作和专著、教材引用58篇次）。引用者包括美国两院院士Golub、Demmel和Dongarra（2022图灵奖获得者），美国工程院院士Stewart, 英国皇家科学院和美国工程院院士Trefethen, 荷兰工程院院士Van der Vorst, 还有Bjorck、Saad、Sorensen等许多著名学者。

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.

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