A numerical study of iterative refinement schemes for weakly singular integral equations

A Fredholm integral equation of the second kind in L1([a, b], C) with a weakly singular kernel is considered. Sufficient conditions are given for the existence and uniqueness of the solution. We adapt the product integration method proposed in C0 ([a, b], C) to apply it in L1 ([a, b], C), and discretize the equation. To improve the accuracy of the approximate solution, we use different iterative refinement schemes which we compare one to each other. Numerical evidence is given with an application in Astrophysics.

show abstract

“…We note that Scheme C is the fastest one to reach the tolerance. This confirms the results obtained in [9]. Remark 3.…”

Section: An Application In Astrophysicssupporting

confidence: 91%

An Extension of the Product Integration Method to L1 With Applications in Astrophysics

Grammont

Ahués

Kaboul

2016

Mathematical Modelling and Analysis

View full text Add to dashboard Cite

show abstract

Eigenvalue computations in the context of data-sparse approximations of integral operators

Román

Vasconcelos

Nunes

2013

Journal of Computational and Applied Mathematics

View full text Add to dashboard Cite

Finite Rank Approximation based method for solving the RTE in stellar atmospheres and application to an inverse problem

Osses

Titaud

2006

EAS Publications Series

View full text Add to dashboard Cite

The Finite Rank Approximation (FRA) based method is well known in operator approximation theory but it is also useful for suggesting numerical methods for solving integral equations. In this document we describe two FRA methods for the numerical resolution of the integral formulation of the 1D Radiative Transfer Equation (RTE) posed in a static slab; we browse some advantages of them (especially the possibility to control the error) and we give some reduction of computation technics (iterative refinement schemes) which can be used with these methods. Numerical results obtained for a realistic Sun atmosphere model are given. In the last section we give an example of an inverse problem associated to the RTE: we show the iterative recovering of the albedo from the measurement of the outgoing specific intensity at a surface of the considered domain.

show abstract

A numerical study of iterative refinement schemes for weakly singular integral equations

Cited by 8 publications

References 6 publications

An Extension of the Product Integration Method to L1 With Applications in Astrophysics

An Extension of the Product Integration Method to L1 With Applications in Astrophysics

Eigenvalue computations in the context of data-sparse approximations of integral operators

Finite Rank Approximation based method for solving the RTE in stellar atmospheres and application to an inverse problem

Contact Info

Product

Resources

About