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Inverse diffusion problems with redundant internal information. (English) Zbl 1302.35449
Summary: 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 [Y. Capdeboscq et al., SIAM J. Imaging Sci. 2, No. 4, 1003–1030 (2009; Zbl 1180.35549)] and [G. Bal et al., Inverse Probl. Imaging 7, No. 2, 353–375 (2013; Zbl 1267.35249)], respectively, in the case \(\alpha=\frac{1}{2}\). The second procedure consists of solving a system of linear elliptic equations, which we can prove admits a unique solution in specific situations.

MSC:
35R30 Inverse problems for PDEs
65N21 Numerical methods for inverse problems for boundary value problems involving PDEs
35J15 Second-order elliptic equations
53B21 Methods of local Riemannian geometry
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