# zbMATH — the first resource for mathematics

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
Full Text: