  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.
