论文标题
加权稀疏性正规化用于椭圆pdes的来源识别
Weighted sparsity regularization for source identification for elliptic PDEs
论文作者
论文摘要
这项研究是由与心电图（ECG）和脑电图（EEG）有关的PDE构成的优化问题所激发的。标准的稀疏性正则化不一定会为这些应用产生足够的结果，因为只有边界数据/观测值可用于识别未知来源，这可能是内部的。因此，我们研究了一种加权$ \ ell^1 $ regolarization技术，用于解决远期操作员的空空间时解决反问题。特别是，我们证明，无论是内部还是位于边界，稀疏的来源都可以通过此加权过程精确地恢复，因为正则化参数$α$趋于零。我们的分析得到了一个和几个局部来源的病例的数值实验的支持。该理论是根据欧几里得空间开发的，因此我们的结果可以应用于许多问题。
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 $α$ 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.
