L^1 data fitting for Inverse Problems yields optimal rates of convergence in case of discretized white Gaussian noise

TL;DR

This paper analyzes L^1 data fitting in inverse problems, demonstrating that it achieves optimal convergence rates under discretized white Gaussian noise, and compares its robustness and performance to traditional methods.

Contribution

It provides a theoretical analysis of L^1 data fidelity in inverse problems, establishing order-optimal convergence rates in the presence of discretized Gaussian noise.

Findings

L^1 data fitting yields optimal convergence rates under Gaussian noise.

The approach is robust to measurement errors and operator inaccuracies.

Numerical simulations confirm theoretical results.

Abstract

It is well-known in practice, that L^1 data fitting leads to improved robustness compared to standard L^2 data fitting. However, it is unclear whether resulting algorithms will perform as well in case of regular data without outliers. In this paper, we therefore analyze generalized Tikhonov regularization with L^1 data fidelity for Inverse Problems F(u) = g in a general setting, including general measurement errors and errors in the forward operator. The derived results are then applied to the situation of discretized Gaussian white noise, and we show that the resulting error bounds allow for order-optimal rates of convergence. These findings are also investigated in numerical simulations.