報 告 人：賈仲孝 教授
報告題目：A cross-product free Jacobi--Davidson type method for computing a partial generalized singular value decomposition of a large matrix pair
報告時間：2023年06月23日（周五）上午9:30—10:30
報告地點：靜遠樓204學術報告廳
主辦單位：數(shù)學與統(tǒng)計學院、數(shù)學研究院、科學技術研究院
報告人簡介：
賈仲孝，1993年獲得德國比勒菲爾德大學博士學位，清華大學數(shù)學科學系二級教授，第六屆國際青年數(shù)值分析家--Leslie Fox獎獲得者（年齡不超過31歲），國家“百千萬人才工程”入選者（1999）,清華大學數(shù)學科學系學術委員會副主任(2009—2021)，2010年度“何梁何利獎”數(shù)學力學專業(yè)組評委，中國工業(yè)與應用數(shù)學學會(CSIAM)第五和第六屆常務理事(2008.9—2012.8，2012.8—2016.8)，第七和第八屆中國計算數(shù)學學會常務理事(2006.10—2014.10)，北京數(shù)學會第十一和十二屆副理事長（2013.12—2021.12），中國工業(yè)與應用數(shù)學學會(CSIAM)監(jiān)事會監(jiān)事(2020.1—2021.10)，北京數(shù)學會第十三屆監(jiān)事會監(jiān)事長（2021.12—2026.12）。主要研究領域：數(shù)值線性代數(shù)和科學計算。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)值解法等領域做出了系統(tǒng)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(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。對于非對稱情形的特征值問題，首次建立了這三類方法的普適性收斂性理論。國際計算數(shù)學界權威Stewart的經(jīng)典專著“Matrix Algorithms: Vol. II Eigensystems, SIAM, Philadelphia, 2001”（470頁）和國際著名計算數(shù)學家van der Vorst的專著“Computational Methods for Large Eigenvalue Problems, North-Holland (Elsevier), 2002”（177頁）分別用10頁多和4頁多的篇幅系統(tǒng)描述和討論賈仲孝的精化投影方法。在Inverse Problems,Mathematics of Computation, Numerische Mathematik, SIAM Journal on Matrix Analysis and Applications, SIAM Journal on Optimization, SIAM Journal on Scientific Computing等國際頂尖和著名知名雜志上發(fā)表論文70篇，研究工作被廣泛引用，引發(fā)了大量的后續(xù)研究。論文被40個國家和地區(qū)的700多名專家和研究人員在17部經(jīng)典著作、專著和教材，包括Golub & van Loan的Matrix Computations第三、第四版等，及600余篇論文中引用逾1200余篇次。
報告摘要：
A cross-product free (CPF) Jacobi--Davidson (JD) type method is proposed to compute a partial generalized singular value decomposition (GSVD) of a large regular matrix pair $\{A,B\}$, called CPF-JDGSVD. It implicitly solves the mathematically equivalent generalized eigenvalue problem of the cross-product matrix pair $\{A^TA,B^TB\}$ using the Rayleigh--Ritz projection method but does not form the cross-product matrices explicitly, and thus avoids the possible accuracy loss of the computed generalized singular values and generalized singular vectors. The method is an inner-outer iteration method, where the expansion of the right searching subspace forms the inner iterations that approximately solve the correction equations involved and the outer iterations extract approximate GSVD components with respect to the subspaces. A convergence result is established for the outer iterations, compact bounds are derived for the condition numbers of the correction equations, and the least solution accuracy requirements on the inner iterations are found, which can maximize the overall efficiency of CPF-JDGSVD as much as possible. Based on them, practical stopping criteria are designed for the inner iterations. A thick-restart CPF-JDGSVD algorithm with deflation and purgation is developed to compute several GSVD components of $\{A,B\}$ associated with the generalized singular values closest to a given target $\tau$. Numerical experiments illustrate the efficiency of the algorithm.