An efficient and flexible Abel-inversion method for noisy data

TL;DR

The paper introduces a flexible, efficient Abel-inversion method for noisy data that combines regularization techniques, ensuring uniform convergence and applicability across various scientific fields.

Contribution

It presents a novel Abel-inversion approach using Tikhonov regularization and compact function sets, offering improved flexibility and convergence for noisy data problems.

Findings

Method achieves uniform convergence as data errors decrease.

Applicable to simulated models with known solutions.

Demonstrated on an astrophysical data example.

Abstract

We propose an efficient and flexible method for solving Abel integral equation of the first kind, frequently appearing in many fields of astrophysics, physics, chemistry, and applied sciences. This equation represents an ill-posed problem, thus solving it requires some kind of regularization. Our method is based on solving the equation on a so-called compact set of functions and/or using Tikhonov's regularization. A priori constraints on the unknown function, defining a compact set, are very loose and can be set using simple physical considerations. Tikhonov's regularization on itself does not require any explicit a priori constraints on the unknown function and can be used independently of such constraints or in combination with them. Various target degrees of smoothness of the unknown function may be set, as required by the problem at hand. The advantage of the method, apart from its flexibility, is that it gives uniform convergence of the approximate solution to the exact solution, as the errors of input data tend to zero. The method is illustrated on several simulated models with known solutions. An example of astrophysical application of the method is also given.