On convergence and convergence rates for Ivanov and Morozov regularization and application to some parameter identification problems in elliptic PDEs

TL;DR

This paper analyzes the convergence and rates of Ivanov and Morozov regularization methods, providing theoretical guarantees and applying them to parameter identification in elliptic PDEs, offering alternatives to Tikhonov regularization.

Contribution

It offers a convergence analysis and rates for Ivanov and Morozov regularization, including their application to elliptic PDE parameter identification, highlighting their differences from Tikhonov.

Findings

Proved well-definedness of Ivanov and Morozov methods.

Established convergence and convergence rates under source conditions.

Applied methods successfully to linear and nonlinear elliptic PDE problems.

Abstract

In this paper we provide a convergence analysis of some variational methods alternative to the classical Tikhonov regularization, namely Ivanov regularization (also called method of quasi solutions) with some versions of the discrepancy principle for choosing the regularization parameter, and Morozov regularization (also called method of the residuals). After motivating nonequivalence with Tikhonov regularization by means of an example, we prove well-definedness of the Ivanov and the Morozov method, convergence in the sense of regularization, as well as convergence rates under variational source conditions. Finally, we apply these results to some linear and nonlinear parameter identification problems in elliptic boundary value problems.