Quasi-solution of linear inverse problems in non-reflexive Banach spaces
This work provides a theoretical and numerical framework for regularizing inverse problems in non-reflexive spaces, which is important for applications like PDE-constrained optimization, but the results are incremental extensions of existing theory.
The paper extends the method of quasi-solutions (Ivanov regularization) to linear ill-posed problems in non-reflexive Banach spaces, proving regularization properties and convergence rates under the Morozov discrepancy principle, and demonstrates numerical computation for inverse source problems in L^∞.
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 $L^\infty(Ω)$ 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.