报告题目：A CJ-FEAST GSVDsolver for computing a partial GSVD of a large matrix pair with the generalized singular values in a given interval

报 告 人：贾仲孝 教授 （清华大学）

报告时间：2025年6月13日（星期五） 9:00-10:00

报告地点：数学科学学院115  

校内联系人： 杜磊 副教授   联系方式：84708351-8606

报告摘要：For the large generalized singular value decomposition (GSVD) computation, given three left and right searching subspaces, we propose a class of general projection methods that works on (A, B) directly, and computes approximations to the desired GSVD components. Based on it, we propose a CJ-FEAST GSVDsolver to compute a partial generalized singular value decomposition (GSVD) of a large matrix pair (A, B) with the generalized singular values in any given interval. The solver itself is a highly nontrivial extension of the FEAST eigensolver for the standard or generalized eigenvalue problem and the CJ-FEAST SVDsolvers for the singular value decomposition (SVD) problem. We exploit the Chebyshev–Jackson (CJ) series to construct an approximate spectral projector of the matrix pair (A^T A, B^T B) associated with the generalized singular values of interest, use subspace iteration on it to generate a right subspace, and premultiply it with A and B to obtain two left subspaces. The spectral projector andits approximations are unsymmetric, and the convergence problems and algorithmic implementations on the CJ-FEAST GSVDsolver are far more difficult and complicated than those on the two available CJ-FEAST SVDsolvers. We derive accuracy estimates for the approximate spectral projector and its eigenvalues, and establish a number of convergence results on the underlying subspaces and the approximate GSVD components obtained by the CJ-FEAST GSVDsolver. We propose general purpose choice strategies for the series degree and subspace dimension. Numerical experiments illustrate that (1) the CJ-FEAST GSVDsolver is practical and it is much more robust and accurate than its contour integral-based variant with the trapezoidal rule and the Gauss–Legendre quadrature and speeds up the latter several dozen to hundred times, and (2) it is competitive with and has huge advantage over a very best Jacobi–Davidson GSVDsolver when the number of desired GSVD components is no more than dozens and is more than one hundred, respectively. The CJ-FEAST GSVDsolver is directly adaptable to the generalized eigenvalue problem of a large symmetric positive definite pair.

报告人简介：贾仲孝,清华大学二级教授,德国Bielefeld大学博士毕业,第六届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.曾任清华大学数学科学系学术委员会副主任,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监事会监事.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（注:后来发展为标准RR投影方法、精化RR投影方法、调和RR投影方法、精化调和RR投影方法共四类投影方法）.在Inverse Problems, Math. Comput., Numer. Math., SIMAX, SIAM J. Optim., SISC等国际著名杂志上发表论文70余篇.研究成果 被国内外学者广泛引用,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.