Regularization Algorithms Based on Total Least Squares

Abstract

Discretizations of inverse problems lead to systems of linear equations with a highly ill-conditioned coefficient matrix, and in order to compute stable solutions to these systems it is necessary to apply regularization methods. Classical regularization methods, such as Tikhonov's method or truncated {\em SVD}, are not designed for problems in which both the coefficient matrix and the right-hand side are known only approximately. For this reason, we develop {\em TLS}\/-based regularization methods that take this situation into account.

Here, we survey two different approaches to incorporation of regularization, or stabilization, into the {\em TLS} setting. The two methods are similar in spirit to Tikhonov regularization and truncated {\em SVD}, respectively. We analyze the regularizing properties of the methods and demonstrate by numerical examples that in certain cases with large perturbations, these new methods are able to yield more accurate regularized solutions than those produced by the standard methods. (Also cross-referenced as UMIACS-TR-96-65)