Monotonicity Principle for Tomography in Nonlinear Conducting Materials

Abstract

Abstract In this paper, we deal with an inverse electrical conductivity problem which considers the reconstruction of nonlinear electrical conductivity in steady currents operations using boundary measurements. In the current set up, we establish a monotonic relation between the unknown material property to the (measured) Dirichlet-to-Neumann operator (DtN). It is in fact the Monotonicity Principle which is the base of a class of non-iterative and real-time imaging methods and algorithms. To be more precise, we indicate the issues appear in our nonlinear case to transfer this Monotonicity result from the Dirichlet Energy to the DtN operator which is the fundamental huddle in comparison to linear and p-Laplacian cases. Finally, we introduce a new Average DtN operator which is different from the existing ones and resolves complications produced by non-linearity in our problem.