Analysis of bounded variation penalty methods for ill-posed problems

Abstract

This paper presents an abstract analysis of bounded variation (BV) methods for ill-posed operator equations Au=z. Let T(u) def =//Au-z// 2 + alpha J(u) where the penalty, or 'regularization parameter alpha >0 and the functional J(u) is the BV norm or semi-norm of u, also known as the total variation of u. Under mild restrictions on the operator A and the functional J(u), it is shown that the functional T(u) has a unique minimizer which is stable with respect to certain perturbations in the data z, the operator A, the parameter alpha , and the functional J(u). In addition, convergence results are obtained which apply when these perturbations vanish and the regularization parameter is chosen appropriately.