Multi-Parameter Tikhonov Regularization -- An Augmented Approach

TL;DR

This paper introduces a new parameter choice strategy called the balanced discrepancy principle for multi-parameter Tikhonov regularization, improving the solution of inverse problems with multiple features.

Contribution

It develops a novel balanced discrepancy principle for selecting regularization parameters in multi-parameter Tikhonov regularization, with theoretical justification and efficient algorithms.

Findings

Numerical results demonstrate the effectiveness of the balanced discrepancy principle.

Theoretical error estimates support the proposed parameter choice strategy.

The approach successfully promotes distinct features in inverse problem solutions.

Abstract

We study multi-parameter regularization (multiple penalties) for solving linear inverse problems to promote simultaneously distinct features of the sought-for objects. We revisit a balancing principle for choosing regularization parameters from the viewpoint of augmented Tikhonov regularization, and derive a new parameter choice strategy called the \textit{balanced discrepancy principle}. A priori and a posteriori error estimates are provided to theoretically justify the principles, and numerical algorithms for efficiently implementing the principles are also provided. Numerical results on denoising are presented to illustrate the feasibility of the balanced discrepancy principle.