Analysis of bounded variation penalty methods for ill-posed problems

This paper presents an abstract analysis of bounded variation (BV) methods for ill-posed operator equations Au = z. Let T (u) def = kAu \ zk 2 + ffJ (u) ; where the penalty, or regularization, parameter ff? 0 and the functional J (u) is the BV norm or seminorm 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 ff, and the functional J (u). In addition, convergence results are obtained which apply when these perturbations vanish and the regularization parameter is chosen appropriately.