2011/06/21 by Francois Monard, Monard, Francois, Guillaume Bal +1
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP
paper · pdf · doi:10.48550/arxiv.1106.4277
25 pages
arxiv created 2012/04/21 · arxiv updated 2012/04/24
This paper concerns the reconstruction of a scalar diffusion coefficient σ(x) from redundant functionals of the form Hi(x)=σ2α(x)|∇ ui|2(x) where α∈\Rm and ui is a solution of the elliptic problem ∇⋅ σ∇ ui=0 for 1≤ i≤ I. The case α=\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) as well as ultrasound modulated optical tomography (UMOT). The case α=1 finds applications in Magnetic Resonance Electrical Impedance Tomography (MREIT). We present two explicit reconstruction procedures of σ for appropriate choices of I and of traces of ui 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 α the results obtained in two and three dimensions in \citeCFGK and \citeBBMT, respectively, in the case α=\frac12. The second procedure consists of solving a system of linear elliptic equations, which we can prove admits a unique solution in specific situations.