IPI
Inverse diffusion problems with redundant internal information
François Monard Guillaume Bal
This paper concerns the reconstruction of a scalar diffusion coefficient $\sigma(x)$ from redundant functionals of the form $H_i(x)=\sigma^{2\alpha}(x)|\nabla u_i|^2(x)$ where $\alpha\in\mathbb{R}$ and $u_i$ is a solution of the elliptic problem $\nabla\cdot \sigma \nabla u_i=0$ for $1\leq i\leq I$. The case $\alpha=\frac12$ is used to model measurements obtained from modulating a domain of interest by ultrasound and finds applications in ultrasound modulated electrical impedance tomography (UMEIT), ultrasound modulated optical tomography (UMOT) as well as impedance acoustic computerized tomography (ImpACT). The case $\alpha=1$ finds applications in Magnetic Resonance Electrical Impedance Tomography (MREIT).
    We present two explicit reconstruction procedures of $\sigma$ for appropriate choices of $I$ and of traces of $u_i$ at the boundary of a domain of interest. The first procedure involves the solution of an over-determined system of ordinary differential equations and generalizes to the multi-dimensional case and to (almost) arbitrary values of $\alpha$ the results obtained in two and three dimensions in [10] and [5], respectively, in the case $\alpha=\frac12$. The second procedure consists of solving a system of linear elliptic equations, which we can prove admits a unique solution in specific situations.
keywords: hybrid methods power density measurements differential geometry. Inverse conductivity strongly coupled elliptic systems Calderón's problem
IPI
Inverse diffusion from knowledge of power densities
Guillaume Bal Eric Bonnetier François Monard Faouzi Triki
This paper concerns the reconstruction of a diffusion coefficient in an elliptic equation from knowledge of several power densities. The power density is the product of the diffusion coefficient with the square of the modulus of the gradient of the elliptic solution. The derivation of such internal functionals comes from perturbing the medium of interest by acoustic (plane) waves, which results in small changes in the diffusion coefficient. After appropriate asymptotic expansions and (Fourier) transformation, this allow us to construct the power density of the equation point-wise inside the domain. Such a setting finds applications in ultrasound modulated electrical impedance tomography and ultrasound modulated optical tomography.
    We show that the diffusion coefficient can be uniquely and stably reconstructed from knowledge of a sufficient large number of power densities. Explicit expressions for the reconstruction of the diffusion coefficient are also provided. Such results hold for a large class of boundary conditions for the elliptic equation in the two-dimensional setting. In three dimensions, the results are proved for a more restrictive class of boundary conditions constructed by means of complex geometrical optics solutions.
keywords: Calderón's problem inverse diffusion hybrid methods Inverse conductivity power density measurements optical tomography. complex geometrical optics solutions
IPI
Inverse transport with isotropic sources and angularly averaged measurements
Guillaume Bal Ian Langmore François Monard
We consider the reconstruction of a spatially-dependent scattering coefficient in a linear transport equation from diffusion-type measurements. In this setup, the contribution to the measurement is an integral of the scattering kernel against a product of harmonic functions, plus an additional term that is small when absorption and scattering are small. The linearized problem is severely ill-posed. We construct a regularized inverse that allows for reconstruction of the low frequency content of the scattering kernel, up to quadratic error, from the nonlinear map. An iterative scheme is used to improve this error so that it is small when the high frequency content of the scattering kernel is small.
keywords: Inverse Transport Theory Averaged Measurements Optical Tomography Complex Geometrical Optics Solutions.

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