Calculus of Variations and Geometric Measure Theory

A. Corbo Esposito - L. Faella - G. Piscitelli - R. Prakash - A. Tamburrino

Monotonicity Principle in Tomography of Nonlinear Conducting Materials

created by piscitelli on 29 Sep 2022

[BibTeX]

preprint

Inserted: 29 sep 2022

Year: 2020

ArXiv: 2007.15431 PDF

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.