In this paper we extend a recent idea of formulating and regularizing inverse problems as minimization problems, so without using a forward operator, thus avoiding explicit evaluation of a parameter-to-state map. We do so by rephrasing three application examples in this minimization form, namely (a) electrical impedance tomography with the complete electrode model (b) identification of a nonlinear magnetic permeability from magnetic flux measurements (c) localization of sound sources from microphone array measurements. To establish convergence of the proposed regularization approach for these problems, we first of all extend the existing theory. In particular, we take advantage of the fact that observations are finite dimensional here, so that inversion of the noisy data can to some extent be done separately, using a right inverse of the observation operator. This new approach is actually applicable to a wide range of real world problems.

中文翻译：

---

基于最小化反问题公式及其正则化的一些应用示例

在本文中，我们扩展了将反问题公式化和正则化为最小化问题的最新思想，因此无需使用前向运算符，从而避免了对参数到状态图的显式评估。为此，我们以最小化形式重述了三个应用示例，即（a）具有完整电极模型的电阻抗层析成像（b）通过磁通量测量识别非线性磁导率（c）从麦克风阵列测量中确定声源的位置。为了建立针对这些问题的拟议正则化方法的收敛性，我们首先扩展现有理论。特别是，我们利用了这里的观测是有限维的这一事实，因此可以在某种程度上单独进行嘈杂数据的反演，使用观察算子的右逆。这种新方法实际上适用于各种各样的现实世界中的问题。