Discretization of variational regularization in Banach spaces

TL;DR

This paper analyzes finite dimensional variational regularization for nonlinear ill-posed operator equations in Banach spaces, establishing convergence and rates considering operator approximations and noisy data, including complex nonseparable spaces.

Contribution

It provides a comprehensive convergence analysis and rate results for variational regularization in Banach spaces, especially addressing nonseparable spaces like total variation and bounded deformation.

Findings

Convergence of regularized solutions with operator approximation and noise

Establishment of convergence rates via Bregman distances

Detailed analysis of nonseparable Banach spaces like TV and BD spaces

Abstract

Consider a nonlinear ill-posed operator equation $F(u)=y$ where $F$ is defined on a Banach space $X$. In general, for solving this equation numerically, a finite dimensional approximation of $X$ and an approximation of $F$ are required. Moreover, in general the given data $\yd$ of $y$ are noisy. In this paper we analyze finite dimensional variational regularization, which takes into account operator approximations and noisy data: We show (semi-)convergence of the regularized solution of the finite dimensional problems and establish convergence rates in terms of Bregman distances under appropriate sourcewise representation of a solution of the equation. The more involved case of regularization in nonseparable Banach spaces is discussed in detail. In particular we consider the space of finite total variation functions, the space of functions of finite bounded deformation, and the $L^\infty$--space.