Hybrid inverse problems and internal functionals

TL;DR

This paper reviews recent mathematical techniques used to solve the second step of hybrid inverse problems, which combine high contrast and high resolution modalities in medical and geophysical imaging.

Contribution

It summarizes new methods for analyzing internal functionals and ensuring qualitative properties of solutions in hybrid inverse problems.

Findings

Development of mathematical techniques for the second step of hybrid inverse problems

Use of complex geometric optics solutions to verify qualitative properties

Enhanced understanding of coupling effects in multi-wave inverse problems

Abstract

This paper reviews recent results on hybrid inverse problems, which are also called coupled-physics inverse problems of multi-wave inverse problems. Inverse problems tend to be most useful in, e.g., medical and geophysical imaging, when they combine high contrast with high resolution. In some settings, a single modality displays either high contrast or high resolution but not both. In favorable situations, physical effects couple one modality with high contrast with another modality with high resolution. The mathematical analysis of such couplings forms the class of hybrid inverse problems. Hybrid inverse problems typically involve two steps. In a first step, a well-posed problem involving the high-resolution low-contrast modality is solved from knowledge of boundary measurements. In a second step, a quantitative reconstruction of the parameters of interest is performed from knowledge of the point-wise, internal, functionals of the parameters reconstructed during the first step. This paper reviews mathematical techniques that have been developed in recent years to address the second step. Mathematically, many hybrid inverse problems find interpretations in terms of linear and nonlinear (systems of) equations. In the analysis of such equations, one often needs to verify that qualitative properties of solutions to elliptic linear equations are satisfied, for instance the absence of any critical points. This paper reviews several methods to prove that such qualitative properties hold, including the method based on the construction of complex geometric optics solutions.