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

TL;DR

This paper analyzes the consistency and convergence rates of Bayesian Gaussian process methods for recovering the absorption coefficient in a heat equation inverse problem, demonstrating optimality of certain priors.

Contribution

It establishes posterior contraction rates and proves the optimality of Gaussian process priors in a nonlinear parabolic inverse problem setting.

Findings

Posterior distributions concentrate around the true parameter as data increases.

Derived convergence rates for the posterior mean reconstruction.

Showed that truncated Gaussian priors achieve minimax optimal rates.

Abstract

We consider the statistical nonlinear inverse problem of recovering the absorption term $f>0$ in the heat equation $$ \partial_tu-\frac{1}{2}\Delta u+fu=0 \quad \text{on $\mathcal{O}\times(0,\textbf{T})$}\quad u = g \quad \text{on $\partial\mathcal{O}\times(0,\textbf{T})$}\quad u(\cdot,0)=u_0 \quad \text{on $\mathcal{O}$}, $$ where $\mathcal{O}\in\mathbb{R}^d$ is a bounded domain, $\textbf{T}<\infty$ 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 $\mathcal{O}\times(0,\textbf{T})$. 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.