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.

翻译：本研究源于与心电图（ECG）和脑电图（EEG）相关的偏微分方程约束优化问题。标准稀疏正则化在这些应用中未必能产生理想结果，因为只有边界数据/观测值可用于识别可能位于内部的未知源。为此，我们研究了一种加权$\ell^1$正则化技术，用于解决前向算子具有显著零空间时的反问题。特别地，我们证明了无论稀疏源位于内部还是边界，当正则化参数$\alpha$趋于零时，该加权过程都能精确恢复该源。我们的分析得到了针对单个及多个局部源数值实验的支持。该理论基于欧几里得空间展开，因此其结果可适用于许多问题。