  This investigation is motivated by PDE-constrained optimization problems
arising in connection with electrocardiograms (ECGs) and electroencephalography
(EEG). Standard sparsity regularization does not necessarily produce adequate
results for these applications because only boundary data/observations are
available for the identification of the unknown source, which may be interior.
We therefore study a weighted $\ell^1$-regularization technique for solving
inverse problems when the forward operator has a significant null space. In
particular, we prove that a sparse source, regardless of whether it is interior
or located at the boundary, can be exactly recovered with this weighting
procedure as the regularization parameter $\alpha$ tends to zero. Our analysis
is supported by numerical experiments for cases with one and several local
sources. The theory is developed in terms of Euclidean spaces, and our results
can therefore be applied to many problems.
