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

Abstract

We consider the statistical nonlinear inverse problem of recovering the absorption term $f>0$ in the heat equation $${\partial }_{t}u-\frac{1}{2}\Delta u+fu=0\text{on O×(0,T)}u=g\text{on ∂O×(0,T)}u(\cdot ,0)=u_0\text{on O},$$ where $\mathcal{O}\in {\mathbb{R}}^d$ is a bounded domain, $\text{T}<\infty$ 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 $\mathcal{O}\times (0,\text{T})$. 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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• On polynomial-time computation of high-dimensional posterior measures by Langevin-type algorithms
• On log-concave approximations of high-dimensional posterior measures and stability properties in non-linear inverse problems
• Parameter estimation for SPDEs based on discrete observations in time and space