Consistency of Bayesian inference with Gaussian process priors for a parabolic inverse problem

Abstract

Abstract We consider the statistical non-linear inverse problem of recovering the absorption term f > 0 in the heat equation ∂ t u − 1 2 Δ u + f u = 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