Monotonicity Principle in Tomography of Nonlinear Conducting Materials

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.

arXiv comment28 pages, 6 figures

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• Reconstruction of singular and degenerate inclusions in Calderón's problem
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• The p-Laplace "Signature" for Quasilinear Inverse Problems with Large Boundary Data
• Piecewise nonlinear materials and Monotonicity Principle
• The $p_0$-Laplace "Signature" for Quasilinear Inverse Problems
• The inverse obstacle problem for nonlinear inclusions
• Tomography of nonlinear materials via the Monotonicity Principle
• Monotonicity of the Laplace Transform for Tomography in Dissipative Systems