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This paper analyzes a popular computational framework to solveinfinite-dimensional Bayesian inverse problems, discretizing the prior and theforward model in a finite-dimensional weighted inner product space. Wedemonstrate the benefit of working on a weighted space by establishingoperator-norm bounds for finite element and graph-based discretizations ofMat\'ern-type priors and deconvolution forward models. For linear-Gaussianinverse problems, we develop a general theory to characterize the error in theapproximation to the posterior. We also embed the computational framework intoensemble Kalman methods and MAP estimators for nonlinear inverse problems. Ouroperator-norm bounds for prior discretizations guarantee the scalability andaccuracy of these algorithms under mesh refinement.

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