A probabilistic finite element method based on random meshes: Error estimators and Bayesian inverse problems

Abstract

We present a novel probabilistic finite element method (FEM) for the solution and uncertainty quantification of elliptic partial differential equations based on random meshes, which we call random mesh FEM (RM-FEM). Our methodology allows to introduce a probability measure on standard piecewise linear FEM. We present a posteriori error estimators based uniquely on probabilistic information. A series of numerical experiments illustrates the potential of the RM-FEM for error estimation and validates our analysis. We furthermore demonstrate how employing the RM-FEM enhances the quality of the solution of Bayesian inverse problems, thus allowing a better quantification of numerical errors in pipelines of computations.

References in corpus (3)

• Inferring solutions of differential equations using noisy multi-fidelity data
• Probabilistic ODE Solvers with Runge-Kutta Means
• Complexity bounds on supermesh construction for quasi-uniform meshes

Cited by in corpus (4)

• Variational Bayesian Approximation of Inverse Problems using Sparse Precision Matrices
• Randomised one-step time integration methods for deterministic operator differential equations
• Probabilistic Numerical Method of Lines for Time-Dependent Partial Differential Equations
• Sampling Methods for Bayesian Inference Involving Convergent Noisy Approximations of Forward Maps