Quasi-solution of linear inverse problems in non-reflexive Banach spaces

TL;DR

This paper investigates the use of Ivanov regularization for linear ill-posed problems in non-reflexive Banach spaces, establishing convergence, parameter choice rules, and numerical methods with practical examples.

Contribution

It extends the theory of quasi-solutions to non-reflexive Banach spaces, providing convergence analysis and a numerical approach for inverse source problems.

Findings

Regularization properties are established for quasi-solutions.

Parameter choice rules like the Morozov discrepancy principle are characterized.

Numerical methods successfully compute quasi-solutions with illustrative examples.

Abstract

We consider the method of quasi-solutions (also referred to as Ivanov regularization) for the regularization of linear ill-posed problems in non-reflexive Banach spaces. Using the equivalence to a metric projection onto the image of the forward operator, it is possible to show regularization properties and to characterize parameter choice rules that lead to a convergent regularization method, which includes the Morozov discrepancy principle. Convergence rates in a suitably chosen Bregman distance can be obtained as well. We also address the numerical computation of quasi-solutions to inverse source problems for partial differential equations in $L^\infty(\Omega)$ using a semismooth Newton method and a backtracking line search for the parameter choice according to the discrepancy principle. Numerical examples illustrate the behavior of quasi-solutions in this setting.