Bayesian Nonparametric Inference in Elliptic PDEs: Convergence Rates and Implementation

Abstract

Parameter identification problems in partial differential equations (PDEs) consist in determining one or more functional coefficient in a PDE. In this article, the Bayesian nonparametric approach to such problems is considered. Focusing on the representative example of inferring the diffusivity function in an elliptic PDE from noisy observations of the PDE solution, the performance of Bayesian procedures based on Gaussian process priors is investigated. Building on recent developments in the literature, we derive novel asymptotic theoretical guarantees that establish posterior consistency and convergence rates for methodologically attractive Gaussian series priors based on the Dirichlet-Laplacian eigenbasis. An implementation of the associated posterior-based inference is provided and illustrated via a numerical simulation study, where excellent agreement with the theory is obtained.