Convergence Rates for the Maximum A Posteriori Estimator in PDE-Regression Models with Random Design

We consider the statistical inverse problem of recovering a parameter $\theta\in H^\alpha$ from data arising from the Gaussian regression problem \begin{equation*} Y = \mathscr{G}(\theta)(Z)+\varepsilon \end{equation*} with nonlinear forward map $\mathscr{G}:\mathbb{L}^2\to\mathbb{L}^2$, random design points $Z$ and Gaussian noise $\varepsilon$. The estimation strategy is based on a least squares approach under $\Vert\cdot\Vert_{H^\alpha}$-constraints. We establish the existence of a least squares estimator $\hat{\theta}$ as a maximizer for a given functional under Lipschitz-type assumptions on the forward map $\mathscr{G}$. A general concentration result is shown, which is used to prove consistency and upper bounds for the prediction error. The corresponding rates of convergence reflect not only the smoothness of the parameter of interest but also the ill-posedness of the underlying inverse problem. We apply the general model to the Darcy problem, where the recovery of an unknown coefficient function $f$ of a PDE is of interest. For this example, we also provide corresponding rates of convergence for the prediction and estimation errors. Additionally, we briefly discuss the applicability of the general model to other problems.