An efficient and flexible Abel-inversion method for noisy data

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.

Temperature and pressure profiles of an ablation-controlled arc plasma in air

“…It is however important to keep in mind that any analysis of spatially-resolved spectroscopic measurements is a challenging task since it is inherently an inverse problem. Solutions to such problems are strongly sensitive to small measurement perturbations (noise) and therefore are generally not unique [28]. Despite of this fact, the proposed method shows a positive convergence until the error in the evaluation of the optimization function falls towards zero.…”

On the nature of the single eclipse per 80d orbit of the H-rich luminous WN star WR22

“…Another method for solving light curves of WR+O binaries was proposed by Cherepashchuk et al (1984) and further developed by Antokhin (2012Antokhin ( , 2016. By directly solving integral equations describing the light curve via regularizing algorithms, this method has the advantage of not requiring a parametric description of the limb darkening, the wind velocity law, etc.…”