报告人: 贾仲孝教授（清华大学）

报告时间: 2023年9月6日(周三) 15:00—16:30

报告地点: 维格堂 319

报告摘要: Abstract: LSQR and its mathematically equivalent CGLS have been popularly used over the decades for large-scale linear discrete ill-posed problems, where the iteration number $k$ plays the role of the regularization parameter. It has been long known that if the Ritz values in LSQR converge to the large singular values of $A$ in natural order, that is, they interlace the first $k+1$ large singular values of $A$, until the semi-convergence of LSQR occurs then LSQR must have the same the regularization ability as the truncated singular value decomposition (TSVD) method and can compute a 2-norm filtering best possible regularized solution. However, hitherto there has been no definitive rigorous result on the approximation behavior of the Ritz values in the context of ill-posed problems. In this talk, for severely, moderately and mildly ill-posed problems, we give accurate solutions of the two closely related fundamental and highly challenging problems on the regularization of LSQR: (i) How accurate are the low rank approximations generated by Golub-Kahan-Lanczos bidiagonalization? (ii) Whether or not the Ritz values involved in LSQR approximate the large singular values of $A$ in natural order? We also show how to reliably judge the accuracy of low rank approximations cheaply. Numerical experiments confirm our results.

报告人简介: 贾仲孝，1994年获得德国比勒菲尔德大学博士学位，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--L. 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。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41个国家和地区的1000名专家与研究人员在19部经典著作、专著和教材（国外）及760多篇论文中他引1360多篇次（其中被国际学术界581篇论文引用941篇次，包括被书目引用54篇次）。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和Revido-Zaglia的A Journey Through the History of Numerical Linear Algebra (2022).

邀请人：黄金枝