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

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