A new regularization method for linear exponentially ill-posed problems

TL;DR

This paper introduces a novel regularization technique tailored for linear exponentially ill-posed problems, demonstrating improved performance over Tikhonov regularization through theoretical analysis and numerical experiments.

Contribution

A new regularization method specifically designed for exponential ill-posedness, with theoretical convergence analysis and empirical validation.

Findings

The new method achieves order optimal convergence rates.

It outperforms Tikhonov regularization in numerical tests.

The approach is effective under logarithmic source conditions.

Abstract

This paper provides a new regularization method which is particularly suitable for linear exponentially ill-posed problems. Under logarithmic source conditions (which have a natural interpretation in terms of Sobolev spaces in the aforementioned context), concepts of qualifications as well as order optimal rates of convergence are presented. Optimality results under general source conditions expressed in terms of index functions are also studied. Finally, numerical experiments on three test problems attest the better performance of the new method compared to the well known Tikhonov method in instances of exponentially ill-posed problems.