報(bào)告人：賈仲孝 教授
報(bào)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(bào)告時(shí)間：2026年1月25日（周日）16：00-17：00
報(bào)告地點(diǎn)：云龍校區(qū)6號(hào)樓304報(bào)告廳
主辦單位：數(shù)學(xué)與統(tǒng)計(jì)學(xué)院、數(shù)學(xué)研究院、科學(xué)技術(shù)研究院
報(bào)告人簡(jiǎn)介：
清華大學(xué)數(shù)學(xué)科學(xué)系二級(jí)教授，1994年獲得德國(guó)比勒菲爾德(Bielefeld)大學(xué)博士學(xué)位，第六屆國(guó)際青年數(shù)值分析家--Leslie Fox 獎(jiǎng)獲得者(1993)，國(guó)家“百千萬人才工程”入選者(1999)。現(xiàn)任北京數(shù)學(xué)會(huì)第十三屆監(jiān)事會(huì)監(jiān)事長(zhǎng)（2021.12—2026.12），曾任清華大學(xué)數(shù)學(xué)科學(xué)系學(xué)術(shù)委員會(huì)副主任(2009—2021)，2010 年度“何梁何利獎(jiǎng)”數(shù)學(xué)力學(xué)專業(yè)組評(píng)委，中國(guó)工業(yè)與應(yīng)用數(shù)學(xué)學(xué)會(huì)(CSIAM)第五、第六屆常務(wù)理事(2008.9—2016.8)，第七、第八屆中國(guó)計(jì)算數(shù)學(xué)學(xué)會(huì)常務(wù)理事(2006.10—2014.10)，北京數(shù)學(xué)會(huì)第十一和十二屆副理事長(zhǎng)（2013.12—2021.12），中國(guó)工業(yè)與應(yīng)用數(shù)學(xué)學(xué)會(huì)(CSIAM) 監(jiān)事會(huì)監(jiān)事(2020.1—2021.10)。主要研究領(lǐng)域：數(shù)值線性代數(shù)和科學(xué)計(jì)算。在代數(shù)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(shù)值解法等領(lǐng)域做出了系統(tǒng)性的、有國(guó)際影響的重要研究成果，所提出的精化投影方法被公認(rèn)為是求解大規(guī)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(fā)展為標(biāo)準(zhǔn)RR投影方法、精化RR投影方法、調(diào)和RR投影方法、精化調(diào)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(guó)際著名雜志上發(fā)表論文70余篇。
報(bào)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.