学术报告:贾仲孝 教授 清华大学
发布时间: 2016-06-04  浏览次数:

报 告 人:贾仲孝 教授 (清华大学)

报告题目:The Regularization Theory of the Krylov Iterative Solvers LSQR, CGLS,  LSMR and CGME For Linear Discrete Ill-Posed Problems

报告时间:2016年6月5日(周日)上午 9:00

报告地点:静远楼二楼重点实验室204

报告摘要:For the large-scale linear discrete ill-posed problem $\min\|Ax-b\|$ or  Ax=b with a noisy b, LSQR, which is a Krylov iterative solver based on Lanczos  bidiagonalization, and its mathematically equivalent CGLS are most commonly  used. They have intrinsic regularizing effects, where the number of iterations  plays the role of regularization parameter. However, there has been no answer to  the long-standing fundamental concern: for which kinds of problems LSQR and CGLS  can find best possible regularized solutions. The concern was actually raised  foresightedly by Bj\{o}rck and Eld\'{e}n in 1979. Here a best possible  regularized solution means that it is at least as accurate as the best  regularised solution obtained by the TSVD method, which and the best possible  solution of standard-form Tikhonov regularisation are both of the same order of  the worst-case error and cannot be improved under the assumption that the  solution to an underlying linear compact operator equation is continuous or its  derivative squares integrable. In this paper we make a complete analysis on the  regularization of LSQR for severely, moderately and mildly ill-posed problems.  We first consider the case that the singular values of A are simple. We  establish accurate $\sin\Theta$ theorems for the 2-norm distance between the  underlying k-dimensional Krylov subspace and the k-dimensional dominant right  singular subspace of A. Based on them and some follow-up results, for the first  two kinds of problems, we prove the following results: (i) the k-step Lanczos  bidiagonalization always generates a near best rank k approximation to A; (ii)  the k Ritz values always approximate the first k large singular values in  natural order; (iii) the k-step LSQR always captures the k dominant SVD  components of A, so that LSQR can find a best possible regularized solution;  (iv) the diagonals and subdiagonals of the bidiagonal matrices generated by  Lanczos bidiagonalization decay as fast as the singular values of A. However,  for the third kind of problem, the above results do not hold generally. The  decay rates of diagonals and subdiagonals of the bidiagonal matrices can be used  to decide if LSQR can find a best possible regularization solution. We also  analyze the regularisation of other two Krylov solvers LSMR and CGME that amount  to MINRES and the CG method and MINRES applied to $A^TAx=A^Tb$ and  $\min\|AA^Ty-b\|$ with $x=A^Ty$, respectively, proving that LSMR has similar  regularizing effects to LSQR for each kind of problem and both are superior to  CGME. We extend all the results to the case that A has multiple singular values.  Numerical experiments confirm our theory on LSQR.

报告人简介:贾仲孝,清华大学“百人计划”特聘教授,二级教授,博士生导师。1994年于德国Bielefeld大学获得博士学位  (数学)。主要研究领域包括代数特征值问题和奇异值分解问题的数值方法及应用,大规模线性方程组的迭代法和预处理技术,最小二乘问题和总体最小二乘问题的理论分析和数值解法,不适定问题和反问题的正则化理论和数值解法等。第五、六届中国工业与应用数学学会(CSIAM)常务理事(2008.9—2012.8,2012.8—2016.8);第七、八届中国计算数学学会常务理事(2006.10—2014.10);第十一届北京数学会副理事长(2013.12—2017.12);清华大学数学科学系学术委员会副主任(2009—)。1993年6月在英国牛津大学被授予“第六届国际青年数值分析家奖-Leslie  Fox奖”(数值分析最佳研究论文奖),是六名获奖者之一;1997年9月被国家自然科学基金委员会数理科学部选为49名优秀中青年数学家之一;1999年4月起享受国务院政府专家特殊津贴;1999年入选辽宁省首批“百千万人才工程”百人层次;1999度“国家百千万人才工程”第一、二层次;2000年两篇论文被美国科学信息研究所(ISI)授予在国际上有高影响力论文(High  Impact Papers)的“经典引文奖(Citation Classic  Award)”;2001年1月入选清华大学“百人计划”。曾承担国家攀登计划“大规模科学与工程计算的方法和理论”,“国家重点基础研究发展规划项目(973)”“大规模科学计算研究”和  承担“国家重点基础研究发展规划项目(973)”“数学机械化方法及其在数字化设计制造中的应用”中的子课题;主持多项国家自然科学基金。多次受邀在国际学术会议上做大会邀请报告和专题邀请报告。        据不完全统计,  在国际上,贾仲孝的44篇论文和博士学位论文被国际学术界391人(含中国学者125人)在12本专著和教材及280篇论文中他引507篇次。除了中国大陆外,国际上引用的学者来自美、英、法、德、意、荷兰、瑞士、瑞典、加拿大等33个国家与地区,包括美国科学院院士Golub、Demmel、Dongarra、美国工程院院士Stewart、荷兰工程院院士van  der Vorst、英国皇家学会会员Trefethen,  还有Saad等许多著名学者。自2000年起,他提出的精化投影方法已经被国际上三本权威专著列为求解大规模矩阵特征问题的三类投影方法之一。由国际权威计算数学家Demmel、Dongarra、van  der Vorst等五人编辑的“Templates for the Solution of Algebraic Eigenvalue Problems: a  Practical Guide, SIAM, Philadelphia,  2000”将贾仲孝的精化投影类方法作为解大规模矩阵特征问题的三类方法之一。Stewart的经典专著“Matrix Algorithms: Vol. II  Eigensystems, SIAM, Philadelphia, 2001”(475页)和van der Vorst的专著“Computational  Methods for Large Eigenvalue Problems, North-Holland (Elsevier),  2002”(177页)分别用10页多和4页多的篇幅系统描述和讨论贾仲孝的精化投影方法。美国、德国、法国等多个国家的专家将精化投影方法作为研究生教材的内容。


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