Inverse anisotropic conductivity from internal current densities

Abstract

This paper concerns the reconstruction of an anisotropic conductivity tensor γ from internal current densities of the form J = γ∇u, where u solves a second-order elliptic equation ∇ · (γ∇u) = 0 on a bounded domain X with prescribed boundary conditions. A minimum number of such functionals equal to n + 2, where n is the spatial dimension, is sufficient to guarantee a local reconstruction. We show that γ can be uniquely reconstructed with a loss of one derivative compared to errors in the measurement of J. In the special case where γ is scalar, it can be reconstructed with no loss of derivatives. We provide a precise statement of what components may be reconstructed with a loss of zero or one derivatives.

Reconstruction of a Fully Anisotropic Elasticity Tensor from Knowledge of Displacement Fields

“…A discussion then follows on what regularity or property (e.g., the Runge approximation property) is required a priori on the unknown parameters so that the hypotheses of reconstructibility may be fulfilled. The idea of constructing local solutions fulfilling certain maximality conditions, which are then controlled from the boundary of the domain via Runge approximation, was also used in the context of reconstruction of conductivity tensors from knowledge of so-called power density functionals [29] or current density functionals [9].…”

A regularized weighted least gradient problem for conductivity imaging

“…Continuous dependence on σ on a (for a given unperturbed Dirichlet data) can be found in [18], and, for partial data in [19]. For further references on determining the isotropic conductivity based on measurements of current densities see [35,11,13,14,10,15,12], and for reconstructions on anisotropic conductivities from multiple measurements see [9,6,1,2].…”

Orthotropic conductivity reconstruction with virtual‐resistive network and Faraday's law

“…Then, the voltage u satisfies Ohm's law: The uniqueness of isotropic conductivity with such an internal measurement has been studied by several authors (). Whereas studies on the uniqueness in anisotropic conductivity reconstruction have been further conducted more recently ().…”

“…The uniqueness of isotropic conductivity with such an internal measurement has been studied by several authors ( [10][11][12][13][14][15][16]). Whereas studies on the uniqueness in anisotropic conductivity reconstruction have been further conducted more recently ( [17][18][19][20][21]).…”