  We explore the possibility for using boundary data to identify sources in
elliptic PDEs. Even though the associated forward operator has a large null
space, it turns out that box constraints, combined with weighted sparsity
regularization, can enable rather accurate recovery of sources with constant
magnitude/strength. In addition, for sources with varying strength, the support
of the inverse solution will be a subset of the support of the true source. We
present both an analysis of the problem and a series of numerical experiments.
Our work only addresses discretized problems.
  The reason for introducing the weighting procedure is that standard
(unweighted) sparsity regularization fails to provide adequate results for the
source identification task considered in this paper. This investigation is also
motivated by applications, e.g., recovering mass distributions from
measurements of gravitational fields and inverse scattering. We develop the
methodology and the analysis in terms of Euclidean spaces, and our results can
therefore be applied to many problems. For example, the results are equally
applicable to models involving the screened Poisson equation as to models using
the Helmholtz equation, with both large and small wave numbers.

Numerical Analysis;Numerical Analysis;Optimization and Control