An $L^\infty$ regularisation strategy to the inverse source identification problem for elliptic equations

TL;DR

This paper introduces an $L^ Infty$ regularisation method for the inverse source problem in elliptic equations, offering uniform closeness to noisy measurements, which improves upon classical $L^2$ regularisation.

Contribution

It develops a novel $L^ Infty$ regularisation approach using Calculus of Variations to better handle inverse source identification in elliptic PDEs.

Findings

The $L^ Infty$ regularisation provides uniform approximation to noisy data.

The method improves stability over classical $L^2$ regularisation.

The approach is applicable to non-homogeneous linear elliptic equations.

Abstract

In this paper we utilise new methods of Calculus of Variations in $L^\infty$ to provide a regularisation strategy to the ill-posed inverse problem of identifying the source of a non-homogeneous linear elliptic equation, satisfying Dirichlet data on a domain. One of the advantages over the classical Tykhonov regularisation in $L^2$ is that the approximated solution of the PDE is uniformly close to the noisy measurements taken on a compact subset of the domain.