An efficient and flexible Abel-inversion method for noisy data

We propose an efficient and flexible method for solving the 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. Ou...

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Bibliographic Details
Published in:Monthly notices of the Royal Astronomical Society 2016-12, Vol.463 (2), p.2079-2079
Main Author: Antokhin, Igor I
Format: Article
Language:English
Subjects:
Approximation
Astronomy
Convergence
Efficiency
Exact solutions
Flexibility
Integral equations
Mathematical analysis
Mathematical models
Regularization
Solutions
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Summary:We propose an efficient and flexible method for solving the 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 in 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.
ISSN:0035-8711
1365-2966
DOI:10.1093/mnras/stw2111