Weighted sparsity regularization for source identification for elliptic PDEs

TL;DR

This paper introduces a weighted -regularization method for accurately identifying sparse sources in elliptic PDE inverse problems, especially when only boundary data is available, with proven exact recovery as regularization diminishes.

Contribution

It develops a weighted -regularization approach that guarantees exact sparse source recovery in elliptic PDE inverse problems, even with interior sources and limited boundary data.

Findings

Exact recovery of sparse sources as regularization parameter approaches zero

Method effective for interior and boundary sources

Numerical experiments validate theoretical results

Abstract

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.