报告题目:Theoretical and computable optimal subspace expansions for matrix eigenvalue problems
报 告 人:贾仲孝 清华大学
报告时间:2025年01月12日10:00—11:00
报告地点:格物楼数学研究中心528报告厅
报告摘要:Consider the optimal subspace expansion problem for the matrix eigenvalue problem $Ax=\lambda x$: {\em Which vector $w$ in the current subspace $\mathcal{V}$, after multiplied by $A$, provides an optimal subspace expansion for approximating a desired eigenvector $x$ in the sense that $x$ has the smallest angle with the expandedsubspace $\mathcal{V}_w=\mathcal{V}+{\rm span}\{Aw\}$, i.e., $w_{opt}=\arg\max_{w\in\mathcal{V}}\cos\angle(\mathcal{V}_w,x)$}?This problem is important as many iterative methods construct nested subspaces that successively expand $\mathcal{V}$ to $\mathcal{V}_w$. Ye ({\em Linear Algebra Appl.}, 428 (2008), p. 911--918) derives an expression of $w_{opt}$ for $A$ general, but it could not be exploited to construct a computable (nearly) optimally expanded subspace. He turns to deriving a maximization characterization of $\cos\angle(\mathcal{V}_w,x)$ for a {\em given} $w\in \mathcal{V}$ when $A$ is Hermitian, but his proof and analysis cannot extend to the non-Hermitian case. We generalize Ye's maximization characterization to the general case and find its maximizer. Our main contributions consist of explicit expressions of $w_{opt}$, $(I-P_V)Aw_{opt}$ and the optimally expanded subspace $\mathcal{V}_{w_{opt}}$ for $A$ general, where $P_V$ is the orthogonal projector onto $\mathcal{V}$, and a key result is that $\mathcal{V}_{w_{opt}}=\mathcal{V}+{\rm span}\{RR^{\dagger}x\}$ with $R=AV-V(V^HAV)$ and $\dagger$ the Moore--Penrose generalized inverse. These results are fully exploited to obtain computable optimally expanded subspaces $\mathcal{V}_{\widetilde{w}_{opt}}$ within the framework of the standard, harmonic, refined, and refined harmonic Rayleigh--Ritz methods. How to efficiently achieve the proposed subspace expansion approaches is considered. Numerical experiments demonstrate the effectiveness of our computable optimal expansions.
报告人简介:贾仲孝,1994 年获得德国比勒菲尔德(Bielefeld)大学博士学位,清华大学数学科学系二级教授,第六届国际青年数值分析家--Leslie Fox 奖获得者 (1993),国家“百千万人才工程” 入选者 (1999)。现任北京数学会第十三届监事会监事长(2021.12—2026.12),曾任清华大学数学科学系学术委员会副主任 (2009—2021),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 (CSIAM) 第五、第六届常务理事 (2008.9—2016.8),第七、第八届中国计算数学学会常务理事(2006.10—2014.10),北京数学会第十一和十二届副理事长(2013.12—2021.12),中国工业与应用数学学会 (CSIAM) 监事会监事(2020.1—2021.10). 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, Mathematics of Computation, Numerische Mathematik, SIAM Journal on Matrix Analysis and Applications, SIAM Journal on Optimization, SIAM Journal on Scientific Computing 等国际著名杂志上发表论文70余篇,研究工作被 43个国家和地区的1200多名专家和研究人员在国外20部经典著作、专著和教材、国内1部专著及965篇论文中他引1623篇次,其中被国际学术界660篇论文他引1050篇次,包括被20本经典著作和专著、教材引用58篇次。引用者包括美国两院院士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。引用的书目包括 Bai、Demmel、Dongarra、Ruhe、van der Vorst 等五人编辑的 “Templates for the Solution of Algebraic Eigenvalue Problems: a Practical Guide ”(2000),Golub & van Loan 的经典著作“Matrix Computations” 第三、第四版 (1996,2013),Stewart 的经典著作“Matrix Algorithms II: Eigensystems ”(2001),Bjorck 的专著 Numerical Methods in Matrix Computations (2015),van der Vorst 的专著 “Computational Methods for Large Eigenvalue Problems” (2002),Trefethen & Embree 的专著“Spectra and Pseudospectra, The Behavior of Nonnormal Matrices and Operators” (2005),Meurant & Tebbens 的专著 Krylov Methods for Nonsymmetric Linear Systems ”(2020),Quarteroni、Sacco & Saleri 的专著 Numerical Mathematics (2000),Brezinski、Meurant & Revido-Zaglia 的著作 “A Journey Through the History of Numerical Linear Algebra” (2022), Bjorck的专著Numerical Methods for Least Squares Problems: Second Edition (2024),等等.