Monotonicity Principle for Tomography in Nonlinear Conducting Materials

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.

Imaging of nonlinear materials via the Monotonicity Principle

“…In order to guarantee the well-posedness of the direct problem (1.1), suitable assumptions on γ A are required. These can be found in [20], where the authors considered a more general problem, with both A and B = Ω \ A filled by nonlinear materials. In the following we specialize the assumptions in [20] to the case of interest consisting of a bounded nonlinear material property γ nl and a bounded linear material property γ bg .…”

“…where u ∈ H 1 (Ω) and u| ∂Ω = f ∈ X ⋄ (∂Ω). Furthermore (see [20]), the unique weak solution u of problem (2.3) can be variationally characterized as…”

“…This work complements [19][20][21] and provides the imaging method and related algorithm for treating nonlinear materials. To the best of our knowledge, this is the first ad-hoc imaging method for nonlinear problems.…”

“…In [17,18] the Monotonicity Principle (MP) is generalized to the case of monomial conductivity. The most comprehensive results are those of [19][20][21] where the authors discover a MP for arbitrary nonlinear materials.…”

“…Then, [22] recognizes its relevance in inverse problems and a new class of imaging methods and algorithms is proposed. In [19][20][21], the MP is extended to nonlinear materials, under very general assumptions. In order to treat nonlinear materials, the authors introduce a new proper boundary operator, called average DtN, which reflects the monotonicity of the Dirichlet energy, with respect to the material property.…”