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

Speaker
Maximilian Siebel , Freie Univ. Berlin

Abstract
We consider the statistical inverse problem of recovering a parameter \(\theta\in H^\alpha\) from data arising from a Gaussian regression model given by \(Y=\mathscr{G}(\theta)(Z)+\varepsilon\), where \(\mathscr{G}:\mathbb{L}^2\to\mathbb{L}^2\) is a nonlinear forward map, \(Z\) represents random design points, and \(\varepsilon\) denotes Gaussian noise. Our estimation strategy is based on a least squares approach with \(\Vert\cdot\Vert_{H^\alpha}\)-constraints. We establish the existence of a least squares estimator \(\hat{\theta}\) as a maximizer of a specified functional under Lipschitz-type assumptions on the forward map \(\mathscr{G}\). A general concentration result is shown, which is used to prove consistency of \(\hat{\theta}\) and establish 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 this general model to the Darcy problem, where the recovery of an unknown coefficient function \(f\) is the primary focus. For this example, we also provide specific rates of convergence for both the prediction and estimation errors. Additionally, we briefly discuss the applicability of the general model to other problems.