Choosing Regularization Parameters in Iterative Methods for Ill-Posed Problems 论文

摘要

. Numerical solution of ill-posed problems is often accomplished by discretization (projection onto a finite dimensional subspace) followed by regularization. If the discrete problem has high dimension, though, typically we compute an approximate solution by projection onto an even smaller dimensional space, via iterative methods based on Krylov subspaces. In this work we present efficient algorithms that regularize after this second projection rather than before it. We prove some results on the approximate equivalence of this approach to other forms of regularization and we present numerical examples. Key words. ill-posed problems, regularization, discrepancy principle, iterative methods, Lcurve, Tikhonov, TSVD, projection, Krylov subspace 65R30,65F20 Running Title: Choosing Regularization Parameters 1. Introduction. Linear, discrete ill-posed problems of the form Ax = b (1) or min x kAx \\Gamma bk 2 ; or equivalently, A Ax = A b (2) arise, for example, from the discretizat...