Abstract
We consider the regularization of the inverse conductivity problem with discontinuous conductivities, like for example the so-called inclusion problem. We theoretically validate the use of some of the most widely adopted regularization operators, like for instancetotal variation and the Mumford-Shah functional, byproving a convergence result for the solutions to the regularized minimum problems.
Highlights
A conducting body is contained in a bounded domain Ω ⊂ RN,N ≥ 2, with Lipschitz boundary ∂Ω
If we prescribe a current density f on the boundary, where f ∈ L2(∂Ω) with zero mean, the electrostatic potential u in Ω is the solution to the Neumann boundary value problem div(σ∇u) = 0 in Ω σ∇u · ν = f on ∂Ω
A correct choice of the regularization operator and of its coefficient should guarantee that (1.2) admits a solution, that is there exists a minimizer σε, for any ε > 0, and that, as ε → 0+, σε converges, in a suitable norm, to the looked for conductivity σ0
Summary
A conducting body is contained in a bounded domain Ω ⊂ RN ,N ≥ 2, with Lipschitz boundary ∂Ω. A correct choice of the regularization operator and of its coefficient should guarantee that (1.2) admits a solution, that is there exists a minimizer σε, for any ε > 0, and that, as ε → 0+, σε converges, in a suitable norm, to the looked for conductivity σ0.
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