Inverse anisotropic diffusion from power density measurements in two dimensions

Details Inverse anisotropic diffusion from power density measurements in two dimensions Inverse anisotropic diffusion from power density measurements in two dimensions

2011-10-20

arxiv.org/abs/1110.4606v2

Mathematics

Analysis of PDEs

27 pages, 6 figures

Scientific paper

This paper concerns the reconstruction of an anisotropic diffusion tensor $\gamma=(\gamma_{ij})_{1\leq i,j\leq 2}$ from knowledge of internal functionals of the form $\gamma\nabla u_i\cdot\nabla u_j$ with $u_i$ for $1\leq i\leq I$ 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.

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