Positive Jacobian constraints for elliptic boundary value problems with piecewise-regular coefficients arising from multi-wave inverse problems

TL;DR

This paper introduces a method to impose positive Jacobian constraints in elliptic boundary value problems with piecewise-regular coefficients, improving the analysis of multi-wave inverse problems involving discontinuous media.

Contribution

It extends Jacobian constraint techniques to piecewise-regular coefficients in elliptic PDEs, addressing practical issues with discontinuous media in inverse problems.

Findings

Established conditions for positive Jacobian constraints in piecewise-regular settings

Demonstrated applicability to multi-wave inverse problems with embedded discontinuities

Provided theoretical framework for stability in inverse reconstructions

Abstract

Multi-wave inverse problems are indirect imaging methods using the interaction of two different imaging modalities. One brings spatial accuracy, and the other contrast sensitivity. The inversion method typically involve two steps. The first step is devoted to accessing internal datum of quantities related to the unknown parameters being observed. The second step involves recovering the parameters themselves from the internal data. To perform that inversion, a typical requirement is that the Jacobian of fields involved does not vanish. A number of authors have considered this problem in the past two decades, and a variety of methods have been developed. Existing techniques require H{\"o}lder continuity of the parameters to be reconstructed. In practical applications, the medium may present embedded elements, with distinct physical properties, leading to discontinuous coefficients. In this article we explain how a Jacobian constraint can imposed in the piecewise regular case, when the physical model is a divergence form second order linear elliptic boundary value problem.