Parameter Choice Strategies for Multipenalty Regularization

Abstract

The widespread applicability of the multipenalty regularization is limited by the fact that theoretically optimal rate of reconstruction for a given problem can be realized by a one-parameter counterpart, provided that relevant information on the problem is available and taken into account in the regularization. In this paper, we explore the situation where no such information is given, but still accuracy of optimal order can be guaranteed by employing multipenalty regularization. Our focus is on the analysis and the justification of an a posteriori parameter choice rule for such a regularization scheme. First we present a modified version of the discrepancy principle within the multipenalty regularization framework. As a consequence we provide a theoretical justification to the multipenalty regularization scheme equipped with the a posteriori parameter choice rule. We then establish a fast numerical realization of the proposed discrepancy principle based on a model function approximation. Finally, we provide extensive numerical results which confirm and support the theoretical estimates and illustrate the robustness and the superiority of the proposed scheme compared to the “classical” regularization methods.