We study a weighted $\ell^1$-regularization technique for solving inverse problems when the forward operator has a significant nullspace. In particular, we prove that a sparse source can be exactly recovered as the regularization parameter $\alpha$ tends to zero. Furthermore, for positive values of $\alpha$, we show that the regularized inverse solution equals the true source multiplied by a scalar $\gamma$, where $\gamma = 1 - c\alpha$. Our analysis is supported by numerical experiments for cases with one and several local sources. This investigation is motivated by PDE-constrained optimization problems arising in connection with ECG and EEG recordings, but the theory is developed in terms of Euclidean spaces. Our results can therefore be applied to many problems.

翻译：当前方操作员有重大空域时,我们研究一种加权的1美元-1美元-正规化技术,以解决反向问题。特别是,我们证明,随着正规化参数为1美元-阿尔法元的趋向为零,稀有来源可以完全恢复。此外,对于正值美元-阿尔法元,我们表明,正规化的逆向解决办法等于真实来源乘以1美元-伽马元,即$-伽玛=1-c\阿尔法元。我们的分析得到一个和几个本地来源案件的数字实验的支持。这一调查的动机是ECG和EEG录音中出现的受PDE限制的优化问题,但理论是从Euclidean空间发展出来的。因此,我们的结果可以应用于许多问题。