報(bào) 告 人：賈仲孝 教授
報(bào)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(bào)告時(shí)間：2023年06月23日（周五）上午9:30—10:30
報(bào)告地點(diǎn)：靜遠(yuǎn)樓204學(xué)術(shù)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：
賈仲孝，1993年獲得德國(guó)比勒菲爾德大學(xué)博士學(xué)位，清華大學(xué)數(shù)學(xué)科學(xué)系二級(jí)教授，第六屆國(guó)際青年數(shù)值分析家--Leslie Fox獎(jiǎng)獲得者（年齡不超過(guò)31歲），國(guó)家“百千萬(wàn)人才工程”入選者（1999）,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(zhuān)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)第五和第六屆常務(wù)理事(2008.9—2012.8，2012.8—2016.8)，第七和第八屆中國(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)，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）。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(wèn)題、奇異值分解和廣義奇異值分解問(wèn)題、離散不適定問(wèn)題和反問(wèn)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(wèn)題和奇異值分解問(wèn)題的三類(lèi)投影方法之一。對(duì)于非對(duì)稱(chēng)情形的特征值問(wèn)題，首次建立了這三類(lèi)方法的普適性收斂性理論。國(guó)際計(jì)算數(shù)學(xué)界權(quán)威Stewart的經(jīng)典專(zhuān)著“Matrix Algorithms: Vol. II Eigensystems, SIAM, Philadelphia, 2001”（470頁(yè)）和國(guó)際著名計(jì)算數(shù)學(xué)家van der Vorst的專(zhuān)著“Computational Methods for Large Eigenvalue Problems, North-Holland (Elsevier), 2002”（177頁(yè)）分別用10頁(yè)多和4頁(yè)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(guó)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(gè)國(guó)家和地區(qū)的700多名專(zhuān)家和研究人員在17部經(jīng)典著作、專(zhuān)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(bào)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.