贾仲孝 教授学术报告
Title: The State of the Art of Krylov Iterative Solvers for Large Linear Discrete Ill-posed Problems (3)
Speaker: Prof. Zhongxiao Jia（贾仲孝）
Affiliation: Tsinghua University（清华大学）
Time: 19:30, Wednesday, 26th October, 2022
Venue: 腾讯会议/VooV Meeting (ID: 355-8075-5066)
入会链接：https://meeting.tencent.com/dm/Jf5Gtf2X6vdO
Inviter: Tiexiang Li（李铁香）
Abstract
Starting with the TSVD method and Tikhonov regularization, we review the state of the art of Krylov iterative solvers for large linear discrete ill-posed problems. We consider the regularization of Krylov solvers LSQR, CGME, LSMR, MINRES, MR-II, GMRES and RRGMRES. We survey the history of this theme, and discuss several long prevailing misconceptions and misrepresentations of these solvers and their hybrid variants. We reveal (i) why GMRES and RRGMRES generally fail to work for nonsymmetric ill-posed problems and (ii) why the hybrid solvers may be mathematically problematic and unstable. We list several fundamental problems on accurate regularizing effects of a Krylov solver, and summarize our affirmative results on them. Finally, we pose a conjecture on the regularization ability of the most important and popular Krylov solvers LSQR and LSMR.
About the Speaker
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)第五和第六届常务理事 (2008.9—2016.8)，第七和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、有国际影响的研究成果。在Inverse Problems, Mathematics of Computation, Numerische Mathematik, SIAM系列等杂志上发表论文70篇，研究工作被引用逾1300篇次，引用的专著和教材17部，包括Bai、Demmel、Dongarra等五人编辑的指南Templates for the Solution of Algebraic Eigenvalue Problems: a Practical Guide (2000)，Golub & van Loan的Matrix Computations (1996，2013)，Stewart的Matrix Algorithms II: Eigensystems (2001)，Bjorck的Numerical Methods in Matrix Computations (2015)，van der Vorst的Computational Methods for Large Eigenvalue Problems (2002)，Trefethen & M. Embree的Spectra and Pseudospectra, The Behavior of Nonnormal Matrices and Operators (2005)，Meurant & Tebbens的Krylov Methods for Nonsymmetric Linear Systems (2020)，Quarteroni、Sacco & Saleri的Numerical Mathematics (2000).