A new approach to nonlinear constrained Tikhonov regularization

TL;DR

This paper introduces a new optimization-based method for nonlinear constrained Tikhonov regularization, providing convergence analysis and application to parameter identification problems, including elliptic PDEs.

Contribution

It proposes a second-order optimality condition as a nonlinearity condition and derives convergence rates for various regularization parameter choice rules.

Findings

Convergence rates established for a priori and a posteriori rules.

Application to parameter identification in elliptic PDEs demonstrated.

New source and nonlinearity conditions derived for the class of problems.

Abstract

We present a novel approach to nonlinear constrained Tikhonov regularization from the viewpoint of optimization theory. A second-order sufficient optimality condition is suggested as a nonlinearity condition to handle the nonlinearity of the forward operator. The approach is exploited to derive convergence rates results for a priori as well as a posteriori choice rules, e.g., discrepancy principle and balancing principle, for selecting the regularization parameter. The idea is further illustrated on a general class of parameter identification problems, for which (new) source and nonlinearity conditions are derived and the structural property of the nonlinearity term is revealed. A number of examples including identifying distributed parameters in elliptic differential equations are presented.