The [math]-Laplace “Signature” for Quasilinear Inverse Problems

SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 351-388, March 2024.
Abstract. This paper refers to an imaging problem in the presence of nonlinear materials. Specifically, the problem we address falls within the framework of Electrical Resistance Tomography and involves two different materials, one or both of which are nonlinear. Tomography with nonlinear materials is in the early stages of development, although breakthroughs are expected in the not-too-distant future. The original contribution this work makes is that the nonlinear problem can be approximated by a weighted [math]-Laplace problem. From the perspective of tomography, this is a significant result because it highlights the central role played by the [math]-Laplacian in inverse problems with nonlinear materials. Moreover, when [math], this result allows all the imaging methods and algorithms developed for linear materials to be brought into the arena of problems with nonlinear materials. The main result of this work is that for “small” Dirichlet data, (i) one material can be replaced by a perfect electric conductor and (ii) the other material can be replaced by a material giving rise to a weighted [math]-Laplace problem.