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 = parallel-to Au - z parallel-to2 + 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.