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.

翻译：本文探索利用边界数据识别椭圆型偏微分方程中源项的可能性。尽管相关正向算子具有较大的零空间，但研究表明，结合盒约束与加权稀疏正则化方法，能够较为精确地恢复具有恒定幅值/强度的源项。此外，对于强度变化的源项，逆解支撑集将是真实源支撑集的子集。本文同时给出问题分析及系列数值实验，仅针对离散化问题进行研究。引入加权过程的必要性在于：标准（非加权）稀疏正则化无法为本文所考虑的源识别任务提供充分结果。本研究亦受应用场景驱动，例如通过引力场测量反演质量分布及逆散射问题。我们基于欧几里得空间发展方法论与分析框架，所得结果可适用于多种问题。例如，该结果既适用于含屏蔽泊松方程的模型，也适用于含亥姆霍兹方程（涵盖大波数与小波数情形）的模型。