Consistency of Bayesian inference with Gaussian process priors for a parabolic inverse problem
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We consider the statistical nonlinear inverse problem of recovering the absorption term f>0 in the heat equation ∂_tu-1/2Δ u+fu=0 on 𝒪×(0,T) u = g on ∂𝒪×(0,T) u(·,0)=u_0 on 𝒪, where 𝒪∈ℝ^d is a bounded domain, T<∞ is a fixed time, and g,u_0 are given sufficiently smooth functions describing boundary and initial values respectively. The data consists of N discrete noisy point evaluations of the solution u_f on 𝒪×(0,T). We study the statistical performance of Bayesian nonparametric procedures based on a large class of Gaussian process priors. We show that, as the number of measurements increases, the resulting posterior distributions concentrate around the true parameter generating the data, and derive a convergence rate for the reconstruction error of the associated posterior means. We also consider the optimality of the contraction rates and prove a lower bound for the minimax convergence rate for inferring f from the data, and show that optimal rates can be achieved with truncated Gaussian priors.
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