Inverse anisotropic diffusion from power density measurements in two dimensions

TL;DR

This paper demonstrates that a 2D anisotropic diffusion tensor can be uniquely and stably reconstructed from internal measurements using four solutions, with explicit methods and numerical validation.

Contribution

It provides a novel explicit reconstruction method for anisotropic diffusion tensors from internal functionals in two dimensions, with stability analysis and numerical implementation.

Findings

Unique and stable reconstruction of the diffusion tensor with four measurements.

Explicit reconstruction procedures are developed and numerically validated.

Applicable to coupled-physics inverse problems involving ultrasound modulation.

Abstract

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.