Convergence Rates for Inverse Problems with Impulsive Noise

TL;DR

This paper analyzes the convergence behavior of Tikhonov regularization with L^1 fidelity for inverse problems affected by impulsive noise, demonstrating improved error estimates and validating findings with numerical experiments.

Contribution

It provides new convergence rate analysis for L^1 fidelity regularization in impulsive noise scenarios, improving upon previous estimates.

Findings

L^1 fidelity regularization yields more accurate solutions than L^2 in impulsive noise cases.

The paper presents improved error bounds for Tikhonov regularization with L^1 fidelity.

Numerical results confirm the theoretical convergence rates and advantages of L^1-based methods.

Abstract

We study inverse problems F(f) = g with perturbed right hand side g^{obs} corrupted by so-called impulsive noise, i.e. noise which is concentrated on a small subset of the domain of definition of g. It is well known that Tikhonov-type regularization with an L^1 data fidelity term yields significantly more accurate results than Tikhonov regularization with classical L^2 data fidelity terms for this type of noise. The purpose of this paper is to provide a convergence analysis explaining this remarkable difference in accuracy. Our error estimates significantly improve previous error estimates for Tikhonov regularization with L^1-fidelity term in the case of impulsive noise. We present numerical results which are in good agreement with the predictions of our analysis.