Inverse medium problems, saddle point formulation

TL;DR

The work addresses inverse medium problems by formulating a saddle point system that links the unknown medium to an induced source through $\lambda=F(u,\theta)$ and recovers the medium via $\mu(x)=\lambda(x)/u(x)$. It advances a direct sampling method built on probing indices and ADI-like iterative probing to efficiently image the medium from boundary Cauchy data, with extensions to general linear and nonlinear operators, time-dependent problems, and high-order or moving-potential formulations. Theoretical results guarantee convergence of the probing iterations to a saddle point and, in the vanishing-noise limit, recovery of the target state under appropriate boundary constraints; the framework unifies LS, constrained, high-order, and time-dependent variants. Practically, the approach offers a fast, robust alternative to conventional optimization-based inverse problem solvers and provides a flexible toolkit for DOT/EIT-type imaging, including DSM-based localization and direct back-projection of the induced source.

Abstract

In this paper we discuss inverse medium problems. We develop the direct sampling method based on probing indices using the saddle point formulation. The medium is constructed by solutions of saddle point problems. The method improves the probing functions for the direct sampling method and directly images the medium. The method is very efficient and can be applied to a general class of inverse medium problems.