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:我们探索利用边界数据来查明椭圆形PDE来源的可能性。 尽管相关的前方操作员有很大的空格空间, 但事实证明, 框限制加上加权宽度规范, 能够相当准确地回收恒定大小/强度的源。 此外, 对于强度不一的来源, 反向解决方案的支持将是真实源支持的子集。 我们提出问题分析 和一系列数字实验 。 我们的工作只处理分解的问题 。 引入加权程序的原因是, 标准( 未加权) 孔径规范无法为本文中考虑的源识别任务提供充分的结果 。 本次调查还受到各种应用的驱动, 例如, 从重力场测量和反散射中恢复质量分布。 我们开发了Euclidean空间的方法和分析, 因此, 我们的结果可以应用于许多问题 。 例如, 有关筛选 Pisoson 等式的模型, 以及使用Helmholtz 等式的模型, 包括大波数和小波数模型 。</s>