Bayesian nonparametric inference in PDE models: asymptotic theory and implementation

Abstract

Parameter identification problems in partial differential equations (PDEs) consist in determining one or more unknown functional parameters in a PDE. Here, 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. Recent asymptotic theoretical guarantees establishing posterior consistency and convergence rates are reviewed and expanded upon. An implementation of the associated posterior-based inference is provided, and illustrated via a numerical simulation study where two different discretisation strategies are devised. The reproducible code is available at: https://github.com/MattGiord.