On statistical Calderón problems
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For D a bounded domain in R^d, d > 3, with smooth boundary ∂ D, the non-linear inverse problem of recovering the unknown conductivity γ determining solutions u=u_γ, f of the partial differential equation ∇·(γ∇ u)&=0 in D, u&=f on ∂ D, from noisy observations Y of the Dirichlet-to-Neumann map f Λ_γ(f) = γ∂ u_γ,f/∂ν|_∂ D, with ∂/∂ν denoting the outward normal derivative, is considered. The data Y consist of Λ_γ corrupted by additive Gaussian noise at noise level ε>0, and a statistical algorithm γ̂(Y) is constructed which is shown to recover γ in supremum-norm loss at a statistical convergence rate of the order (1/ε)^-δ as ε→ 0. It is further shown that this convergence rate is optimal, possibly up to the precise value of the exponent δ>0, in an information theoretic sense. The estimator γ̂(Y) has a Bayesian interpretation as the posterior mean of a suitable Gaussian process prior for γ and can be computed by MCMC methods.
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