On convergence and convergence rates for Ivanov and Morozov regularization and application to some parameter identification problems in elliptic PDEs
Provides theoretical guarantees for alternative regularization methods in inverse problems, relevant for the mathematical community working on ill-posed problems.
This paper proves convergence and convergence rates for Ivanov and Morozov regularization methods, showing they are not equivalent to Tikhonov regularization, and applies the results to parameter identification in elliptic PDEs.
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.