Exponential relaxation data analysis by parametrized regularization of severely ill-posed Fredholm integral equations of the first kind

TL;DR

This paper introduces a new method for solving severely ill-posed Fredholm integral equations of the first kind by using parametrized discretization and simultaneous parameter optimization, demonstrated on Laplace transform inversion and NMR data deconvolution.

Contribution

It proposes a novel regularization approach with parametrized discretization and joint parameter selection for severely ill-posed problems, improving solution stability.

Findings

Effective in noisy Laplace transform inversion

Successful deconvolution of NMR relaxation data

Demonstrates improved regularization parameter selection

Abstract

This paper presents a novel approach to construct regularizing operators for severely ill-posed Fredholm integral equations of the first kind by introducing parametrized discretization. The optimal values of discretization and regularization parameters are computed simultaneously by solving a minimization problem formulated based on a regularization parameter search criterion. The effectiveness of the proposed approach is demonstrated through examples of noisy Laplace transform inversions and the deconvolution of nuclear magnetic resonance relaxation data.