论文标题
通过加权稀疏正规化在电势方程中识别源项
Identifying the source term in the potential equation with weighted sparsity regularization
论文作者
论文摘要
我们探讨了使用边界测量结果恢复电位方程中稀疏源项F（x）的可能性。我们采用加权稀疏性正规化和亚级别的标准结果，我们得出了简单的检查标准，这些标准确保可以识别许多接收器（F（x）<0）和来源（F（x）> 0）。此外，我们提出了两个始终满足这些标准的情况：a）分离良好的来源和水槽，b）位于边界上的许多来源或水槽以及一个内部源/水槽。我们的方法是，在离散配方中保留了相关的前向操作员的线性。因此，该理论是在欧几里得空间方面方便地发展的，并且可以应用于广泛的问题。特别是，它可以应用于各向同性和各向异性病例。我们提出了一系列数值实验。这项工作是由于观察到标准方法通常表明内部水槽和来源位于边界附近的动机。
We explore the possibility for using boundary measurements to recover a sparse source term f(x) in the potential equation. Employing weighted sparsity regularization and standard results for subgradients, we derive simple-to-check criteria which assure that a number of sinks (f(x) < 0) and sources (f(x) > 0) can be identified. Furthermore, we present two cases for which these criteria always are fulfilled: a) well-separated sources and sinks, and b) many sources or sinks located at the boundary plus one interior source/sink. Our approach is such that the linearity of the associated forward operator is preserved in the discrete formulation. The theory is therefore conveniently developed in terms of Euclidean spaces, and it can be applied to a wide range of problems. In particular, it can be applied to both isotropic and anisotropic cases. We present a series of numerical experiments. This work is motivated by the observation that standard methods typically suggest that internal sinks and sources are located close to the boundary.
