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

Daniel Sanz-Alonso, Nathan Waniorek

TL;DR

This work develops a unified, weighted inner product framework for solving infinite-dimensional Bayesian inverse problems via finite-dimensional discretizations. It provides operator-norm error bounds for finite element and graph discretizations of Matérn priors and deconvolution forward models, and embeds these discretizations into ensemble Kalman updates and MAP estimation for nonlinear problems. The results show that discretization errors in priors and forward maps propagate in a controlled way, with bounded effective dimension ensuring scalability under mesh refinement, and establish Γ-convergence of MAP estimators to the continuum problem. The framework is illustrated with function-space problems and an Eulerian Navier–Stokes data-assimilation example, highlighting practical impact for PDE-constrained Bayesian inference and scalable inference pipelines.

Abstract

This paper analyzes a popular computational framework to solve 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 to characterize the error in the approximation to the posterior. We also embed the computational framework into ensemble Kalman methods and MAP estimators for nonlinear inverse problems. Our operator-norm bounds for prior discretizations guarantee the scalability and accuracy of these algorithms under mesh refinement.

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

TL;DR

This work develops a unified, weighted inner product framework for solving infinite-dimensional Bayesian inverse problems via finite-dimensional discretizations. It provides operator-norm error bounds for finite element and graph discretizations of Matérn priors and deconvolution forward models, and embeds these discretizations into ensemble Kalman updates and MAP estimation for nonlinear problems. The results show that discretization errors in priors and forward maps propagate in a controlled way, with bounded effective dimension ensuring scalability under mesh refinement, and establish Γ-convergence of MAP estimators to the continuum problem. The framework is illustrated with function-space problems and an Eulerian Navier–Stokes data-assimilation example, highlighting practical impact for PDE-constrained Bayesian inference and scalable inference pipelines.

Abstract

This paper analyzes a popular computational framework to solve 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 to characterize the error in the approximation to the posterior. We also embed the computational framework into ensemble Kalman methods and MAP estimators for nonlinear inverse problems. Our operator-norm bounds for prior discretizations guarantee the scalability and accuracy of these algorithms under mesh refinement.
Paper Structure (29 sections, 19 theorems, 175 equations, 1 table)

This paper contains 29 sections, 19 theorems, 175 equations, 1 table.

Table of Contents

  1. Introduction
  2. Outline and Main Contributions
  3. Notation
  4. Problem Setting and Computational Framework
  5. Inverse Problem in Hilbert Space
  6. Computational Framework
  7. Inverse Problems in Function Space
  8. Finite Element Discretizations
  9. Graph-Based Discretizations
  10. Error Analysis
  11. Error Analysis: General Computational Framework
  12. Error Analysis: Finite Element Discretizations
  13. Finite Element Approximation of Matérn-Type Prior Covariance
  14. Finite Element Approximation of Heat Forward Model
  15. Finite Element Orthogonal Projection
  16. ...and 14 more sections

Key Result

Proposition 2.5

\newlabelthm:transportMaps0 There is a constant $c$ such that, with probability one, there exists a sequence of transport maps $T_n:\mathcal{M}\to \mathcal{M}_n$ such that $\gamma_n=T_{n\sharp}\gamma$ and where $c_d=3/4$ if $d=2$ and $c_d=1/d$ otherwise.

Theorems & Definitions (43)

  • Example 2.1: Matérn-Type Prior
  • Example 2.2: Deconvolution Forward Model
  • Example 2.3: Finite Element Approximation of Matérn Covariance
  • Example 2.4: Finite Element Approximation of Heat Forward Model
  • Proposition 2.5: Existence of Transport Maps
  • Example 2.6: Graph Approximation of Matérn Covariance
  • Example 2.7: Graph Approximation of Heat Forward Model
  • Lemma 3.2: Continuity of Kalman Gain Update
  • Proof 1
  • Theorem 3.3: Mean Approximation Error
  • ...and 33 more