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Mesh size in computational grids significantly affects probabilistic inverse problems. This study introduces goal-oriented mesh refinement for finite-element Bayesian inference, ensuring accuracy without user-defined tolerances for improved model identification.

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Area of Science:

  • Computational Mathematics
  • Engineering & Applied Sciences

Background:

  • Computational grid size critically influences numerical outcomes in partial differential equation (PDE)-based probabilistic inverse problems.
  • Inaccurate, mesh-dependent results are common in structural and material engineering model identification.
  • Existing methods lack robust strategies for adaptive mesh refinement in Bayesian inference.

Purpose of the Study:

  • To bridge adaptive methods for deterministic/probabilistic simulations with finite-element (FE)-based Bayesian inference.
  • To develop a goal-oriented mesh refinement strategy for exact inference using Markov Chain Monte Carlo (MCMC) algorithms.
  • To ensure accurate posterior distributions of quantities of interest (QoI) by controlling spatial discretization errors.

Main Methods:

  • Proposed goal-oriented mesh refinement based on a finite subset of quantities of interest (QoI).
  • Employed parallel MCMC chains: an approximate chain and an enhanced chain with corrected likelihood function.
  • Utilized efficient deterministic error estimation to correct for spatial discretization errors in PDEs.

Main Results:

  • Demonstrated that mesh refinement can be performed adaptively and goal-oriented for Bayesian inference.
  • Showcased a method that does not require user-defined tolerances for QoI accuracy, relying instead on prior information.
  • Introduced a technique to control MCMC sampler errors, validating a combined mesh and algorithmic quality control strategy.

Conclusions:

  • The proposed approach effectively controls errors from both spatial discretization and MCMC sampling in FE-based Bayesian inference.
  • Goal-oriented adaptive mesh refinement ensures the reliability of numerical results for engineering applications.
  • This strategy enhances the accuracy and robustness of probabilistic inverse problem solutions.