Inverse anisotropic diffusion from power density measurements in two dimensions

Abstract

This paper concerns the reconstruction of an anisotropic diffusion tensor $\gamma =({\gamma }_{ij}{)}_{1\le i,j\le 2}$ from knowledge of internal functionals of the form $\gamma \nabla u_i\cdot \nabla u_j$ with $u_i$ for $1\le i\le I$ solutions of the elliptic equation $\nabla \cdot \gamma \nabla u_i=0$ on a two dimensional bounded domain with appropriate boundary conditions. We show that for I=4 and appropriately chosen boundary conditions, $\gamma$ may uniquely and stably be reconstructed from such internal functionals, which appear in coupled-physics inverse problems involving the ultrasound modulation of electrical or optical coefficients. Explicit reconstruction procedures for the diffusion tensor are presented and implemented numerically.

27 pages, 6 figures

References in corpus (4)

• 2D and 3D reconstructions in acousto-electric tomography
• Stabilizing Inverse Problems by Internal Data
• Hybrid inverse problems and internal functionals
• Inverse diffusion problems with redundant internal information

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• Stability of Coupled-Physics Inverse Problems with internal measurements
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• Inverse anisotropic conductivity from power densities in dimension $n\ge 3$
• On Multiple Frequency Power Density Measurements
• Imaging of isotropic and anisotropic conductivities from power densities in three dimensions
• Combining the Runge approximation and the Whitney embedding theorem in hybrid imaging
• Sound speed uncertainty in Acousto-Electric Tomography
• Jacobian of solutions to the conductivity equation in limited view