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

Abstract

We explore the possibility for using boundary measurements to recover a sparse source term f ( x ) 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 f(x)>0 ) and sources ( f ( x ) > 0 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.