报告人: 贾仲孝 教授（清华大学）

 

报告时间: 2022年11月3日 (周四) 15:30—17:00

 

腾讯会议: 694-553-790

 

报告摘要: For the large-scale linear discrete ill-posed problem min||Ax-b|| or Ax=b with b contaminated by Gaussian white noise, the Lanczos bidiagonalization based Krylov solver LSQR and its mathematically equivalent CGLS, the Conjugate Gradient (CG) method implicitly applied to ATAx=ATb, are most commonly used, and CGME, the CG method applied to min||AATy-b|| or AATy=b with x=ATy, and LSMR, which is equivalent to the minimal residual (MINRES) method applied to ATAx=ATb, have also been choices. These methods exhibit typical semi-convergence feature, and the iteration number k plays the role of the regularization parameter. However, there has been no definitive answer to the long-standing fundamental question: Can LSQR and CGLS find 2-norm filtering best possible regularized solutions? The same question is for CGME and LSMR too. At iteration k, LSQR, CGME and LSMR compute different iterates from the same k dimensional Krylov subspace. A first and fundamental step towards to answering the above question is to accurately estimate the accuracy of the underlying k dimensional Krylov subspace approximating the k dimensional dominant right singular subspace of A. Assuming that the singular values of A are simple, we present a general sinΘtheorem for the 2-norm distances between these two subspaces and derive accurate estimates on them for severely, moderately and mildly ill-posed problems. We also establish some relationships between the smallest Ritz values and these distances. Numerical experiments justify the sharpness of our results.

 

报告人简介: 贾仲孝，清华大学数学科学系二级教授，博士生导师，1994年于德国Bielefeld大学获得理学博士学位，主要从事数学线性代数和大规模科学计算领域的研究，在矩阵（广义）特征值问题和（广义）奇异值分解问题的数值解法的理论和算法领域、离散不适定问题和反问题的正则化理论和数值解法领域等做出了系统的、有重要国际影响的研究成果。所提出的精化Rayleigh-Ritz方法与传统的标准Rayleigh-Ritz方法和调和Rayleigh-Ritz方法一道，自2000年以来被公认为是求解这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。对于非对称情形的特征值问题，首次建立了这三类方法的普适性收敛性理论。此外，在线性最小二乘和总体最小二乘问题的扰动理论、稀疏线性方程组的迭代法和有效预处理技术、信頼域子问题的数值求解方法研究等领域均做出了国际水平的研究成果。1995—2021年期间，在Mathematics of Computation, Numerische Mathematik, SIAM Journal on Matrix Analysis and Applications, SIAM Journal on Optimization, SIAM Journal on Scientific Computing, Inverse Problems等国际顶尖和著名知名杂志上发表论文60余篇。研究成果被广泛引用，引发了大量的后续研究，被40个国家和地区的900多名专家和研究人员在国外17部经典著作、专著、手册和教材及近700篇论文（其中国外500余篇）中引用逾1200篇次，其中国际上的引用830篇次。1993年于英国牛津获Leslie Fox奖，后入选1999年“国家百千万人工程”、2001年清华大学“百人计划”特聘教授等。曾任中国工业与应用数学学会(CSIAM)常务理事、监事会监事, 中国计算数学学会常务理事，北京数学会副理事长, 清华大学数学科学系学术委员会副主任。

 

邀请人：黄金枝