Identifying the source term in the potential equation with weighted sparsity regularization

TL;DR

This paper develops a method using weighted sparsity regularization to accurately identify sparse source terms in the potential equation from boundary measurements, applicable to various cases including anisotropic scenarios.

Contribution

It introduces simple criteria for source and sink identification using weighted sparsity regularization, applicable to both isotropic and anisotropic problems, with theoretical guarantees for certain configurations.

Findings

Criteria ensure source/sink identification in specific configurations

Method preserves linearity in discretization, facilitating broad application

Numerical experiments validate the approach's effectiveness

Abstract

We explore the possibility for using boundary measurements to recover a sparse source term f(x) in the potential equation. Employing weighted sparsity regularization and standard results for subgradients, we derive simple-to-check criteria which assure that a number of sinks (f(x) < 0) and sources (f(x) > 0) can be identified. Furthermore, we present two cases for which these criteria always are fulfilled: a) well-separated sources and sinks, and b) many sources or sinks located at the boundary plus one interior source/sink. Our approach is such that the linearity of the associated forward operator is preserved in the discrete formulation. The theory is therefore conveniently developed in terms of Euclidean spaces, and it can be applied to a wide range of problems. In particular, it can be applied to both isotropic and anisotropic cases. We present a series of numerical experiments. This work is motivated by the observation that standard methods typically suggest that internal sinks and sources are located close to the boundary.