Analysis of a Computational Framework for Bayesian Inverse Problems: Ensemble Kalman Updates and MAP Estimators under Mesh Refinement

Abstract

SIAM/ASA Journal on Uncertainty Quantification, Volume 12, Issue 1, Page 30-68, March 2024.
Abstract. This paper analyzes a popular computational framework for solving infinite-dimensional Bayesian inverse problems, discretizing the prior and the forward model in a finite-dimensional weighted inner product space. We demonstrate the benefit of working on a weighted space by establishing operator-norm bounds for finite element and graph-based discretizations of Matérn-type priors and deconvolution forward models. For linear-Gaussian inverse problems, we develop a general theory for characterizing the error in the approximation to the posterior. We also embed the computational framework into ensemble Kalman methods and maximum a posteriori (MAP) estimators for nonlinear inverse problems. Our operator-norm bounds for prior discretizations guarantee the scalability and accuracy of these algorithms under mesh refinement.