A Bayesian approach to modeling finite element discretization error

Anne Poot*, Pierre Kerfriden, Iuri Rocha, Frans van der Meer

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

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Abstract

In this work, the uncertainty associated with the finite element discretization error is modeled following the Bayesian paradigm. First, a continuous formulation is derived, where a Gaussian process prior over the solution space is updated based on observations from a finite element discretization. To avoid the computation of intractable integrals, a second, finer, discretization is introduced that is assumed sufficiently dense to represent the true solution field. A prior distribution is assumed over the fine discretization, which is then updated based on observations from the coarse discretization. This yields a posterior distribution with a mean that serves as an estimate of the solution, and a covariance that models the uncertainty associated with this estimate. Two particular choices of prior are investigated: a prior defined implicitly by assigning a white noise distribution to the right-hand side term, and a prior whose covariance function is equal to the Green’s function of the partial differential equation. The former yields a posterior distribution with a mean close to the reference solution, but a covariance that contains little information regarding the finite element discretization error. The latter, on the other hand, yields posterior distribution with a mean equal to the coarse finite element solution, and a covariance with a close connection to the discretization error. For both choices of prior a contradiction arises, since the discretization error depends on the right-hand side term, but the posterior covariance does not. We demonstrate how, by rescaling the eigenvalues of the posterior covariance, this independence can be avoided.

Original languageEnglish
Article number167
Number of pages17
JournalStatistics and Computing
Volume34
Issue number5
DOIs
Publication statusPublished - 2024

Keywords

  • Bayesian inference
  • Error estimation
  • Finite element method
  • Probabilistic numerics
  • Uncertainty quantification

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