Convergence Rates for Exponentially Ill-Posed Inverse Problems with Impulsive Noise

TL;DR

This paper analyzes the convergence rates of Tikhonov regularization with $L^1$ fidelity for exponentially ill-posed inverse problems affected by impulsive noise, showing improved polynomial rates over traditional logarithmic ones.

Contribution

It extends the analysis of regularization performance to infinitely smoothing operators with impulsive noise, demonstrating achievable polynomial convergence rates.

Findings

High order polynomial convergence rates are possible with impulsive noise.

Banach spaces of analytic functions are effective tools for analysis.

Applications include the backwards heat equation and gradiometry inverse problems.

Abstract

This paper is concerned with exponentially ill-posed operator equations with additive impulsive noise on the right hand side, i.e. the noise is large on a small part of the domain and small or zero outside. It is well known that Tikhonov regularization with an $L^1$ data fidelity term outperforms Tikhonov regularization with an $L^2$ fidelity term in this case. This effect has recently been explained and quantified for the case of finitely smoothing operators. Here we extend this analysis to the case of infinitely smoothing forward operators under standard Sobolev smoothness assumptions on the solution, i.e. exponentially ill-posed inverse problems. It turns out that high order polynomial rates of convergence in the size of the support of large noise can be achieved rather than the poor logarithmic convergence rates typical for exponentially ill-posed problems. The main tools of our analysis are Banach spaces of analytic functions and interpolation-type inequalities for such spaces. We discuss two examples, the (periodic) backwards heat equation and an inverse problem in gradiometry.