Monotonicity Principle in Tomography of Nonlinear Conducting Materials

TL;DR

This paper investigates the Monotonicity Principle in nonlinear electrical conductivity tomography, proving its validity for Dirichlet Energy and introducing an Average DtN operator to extend its applicability beyond linear cases.

Contribution

It establishes the Monotonicity Principle for Dirichlet Energy in nonlinear problems and introduces a new boundary operator to extend the principle's applicability.

Findings

Monotonicity Principle holds for Dirichlet Energy under mild assumptions.

Impossible to transfer the Monotonicity result from Dirichlet Energy to DtN operator in general cases.

Introduction of an Average DtN operator to overcome limitations in nonlinear conductivity tomography.

Abstract

We treat an inverse electrical conductivity problem which deals with the reconstruction of nonlinear electrical conductivity starting from boundary measurements in steady currents operations. In this framework, a key role is played by the Monotonicity Principle, which establishes a monotonic relation connecting the unknown material property to the (measured) Dirichlet-to-Neumann operator (DtN). Monotonicity Principles are the foundation for a class of non-iterative and real-time imaging methods and algorithms. In this article, we prove that the Monotonicity Principle for the Dirichlet Energy in nonlinear problems holds under mild assumptions. Then, we show that apart from linear and $p$-Laplacian cases, it is impossible to transfer this Monotonicity result from the Dirichlet Energy to the DtN operator. To overcome this issue, we introduce a new boundary operator, identified as an Average DtN operator.