報告人：賈仲孝 教授
報告題目：A CJ-FEAST GSVDsolver for computing a partial GSVD of a large matrix pair with the generalized singular values in a given interval
報告時間：2026年1月25日（周日）16：00-17：00
報告地點：云龍校區6號樓304報告廳
主辦單位：數學與統計學院、數學研究院、科學技術研究院
報告人簡介：
清華大學數學科學系二級教授，1994年獲得德國比勒菲爾德(Bielefeld)大學博士學位，第六屆國際青年數值分析家--Leslie Fox 獎獲得者(1993)，國家“百千萬人才工程”入選者(1999)?，F任北京數學會第十三屆監事會監事長（2021.12—2026.12），曾任清華大學數學科學系學術委員會副主任(2009—2021)，2010 年度“何梁何利獎”數學力學專業組評委，中國工業與應用數學學會(CSIAM)第五、第六屆常務理事(2008.9—2016.8)，第七、第八屆中國計算數學學會常務理事(2006.10—2014.10)，北京數學會第十一和十二屆副理事長（2013.12—2021.12），中國工業與應用數學學會(CSIAM) 監事會監事(2020.1—2021.10)。主要研究領域：數值線性代數和科學計算。在代數特征值問題、奇異值分解和廣義奇異值分解問題、離散不適定問題和反問題的正則化理論和數值解法等領域做出了系統性的、有國際影響的重要研究成果，所提出的精化投影方法被公認為是求解大規模矩陣特征值問題和奇異值分解問題的三類投影方法之一（注：后來發展為標準RR投影方法、精化RR投影方法、調和RR投影方法、精化調和RR投影方法共四類投影方法）。在Inverse Problems, Mathematics of Computation, Numerische Mathematik, SIAM Journal on Matrix Analysis and Applications, SIAM Journal on Optimization, SIAM Journal on Scientific Computing 等國際著名雜志上發表論文70余篇。
報告摘要：
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.