On Regularizing Effects of MINRES and MR-II for Large-Scale Symmetric Discrete Ill-Posed Problems

摘要

For large scale symmetric discrete ill-posed problems, MINRES and MR-II areoften used iterative regularization solvers. We call a regularized solutionbest possible if it is at least as accurate as the best regularized solutionobtained by the truncated singular value decomposition (TSVD) method. In thispaper, we analyze their regularizing effects and establish the followingresults: (i) the filtered SVD expression are derived for the regularizedsolutions by MINRES; (ii) a hybrid MINRES that uses explicit regularizationwithin projected problems is needed to compute a best possible regularizedsolution to a given ill-posed problem; (iii) the $k$th iterate by MINRES ismore accurate than the $(k-1)$th iterate by MR-II until the semi-convergence ofMINRES, but MR-II has globally better regularizing effects than MINRES; (iv)bounds are obtained for the 2-norm distance between an underlying$k$-dimensional Krylov subspace and the $k$-dimensional dominant eigenspace.They show that MR-II has better regularizing effects for severely andmoderately ill-posed problems than for mildly ill-posed problems, and a hybridMR-II is needed to get a best possible regularized solution for mildlyill-posed problems; (v) bounds are derived for the entries generated by thesymmetric Lanczos process that MR-II is based on, showing how fast they decay.Numerical experiments confirm our assertions. Stronger than our theory, theregularizing effects of MR-II are experimentally shown to be good enough toobtain best possible regularized solutions for severely and moderatelyill-posed problems.