A new framework for multi-parameter regularization

Abstract

This paper proposes a new approach for choosing the regularization parameters in multi-parameter regularization methods when applied to approximate the solution of linear discrete ill-posed problems. We consider both direct methods, such as Tikhonov regularization with two or more regularization terms, and iterative methods based on the projection of a Tikhonov-regularized problem onto Krylov subspaces of increasing dimension. The latter methods regularize by choosing appropriate regularization terms and the dimension of the Krylov subspace. Our investigation focuses on selecting a proper set of regularization parameters that satisfies the discrepancy principle and maximizes a suitable quantity, whose size reflects the quality of the computed approximate solution. Theoretical results are shown and illustrated by numerical experiments.