Truncated Singular Value Decomposition Solutions to Discrete Ill-Posed Problems with Ill-Determined Numerical Rank

Tikhonov regularization is a standard method for obtaining smooth solutions to discrete ill-posed problems. A more recent method, based on the singular value decomposition (SVD), is the truncated SVD method. The purpose of this paper is to show, under mild conditions, that the success of both truncated SVD and Tikhonov regularization depends on satisfaction of a discrete Picard condition, involving both the matrix and the right-hand side. When this condition is satisfied, then both methods are guaranteed to produce smooth solutions that are very similar.