Coarse-gradient Langevin algorithms for dynamic data integration and uncertainty quantification

P. Dostert, Y. Efendiev, T. Y. Hou*, W. Luo

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

38 Scopus citations


The main goal of this paper is to design an efficient sampling technique for dynamic data integration using the Langevin algorithms. Based on a coarse-scale model of the problem, we compute the proposals of the Langevin algorithms using the coarse-scale gradient of the target distribution. To guarantee a correct and efficient sampling, each proposal is first tested by a Metropolis acceptance-rejection step with a coarse-scale distribution. If the proposal is accepted in the first stage, then a fine-scale simulation is performed at the second stage to determine the acceptance probability. Comparing with the direct Langevin algorithm, the new method generates a modified Markov chain by incorporating the coarse-scale information of the problem. Under some mild technical conditions we prove that the modified Markov chain converges to the correct posterior distribution. We would like to note that the coarse-scale models used in the simulations need to be inexpensive, but not necessarily very accurate, as our analysis and numerical simulations demonstrate. We present numerical examples for sampling permeability fields using two-point geostatistics. Karhunen-Loève expansion is used to represent the realizations of the permeability field conditioned to the dynamic data, such as the production data, as well as the static data. The numerical examples show that the coarse-gradient Langevin algorithms are much faster than the direct Langevin algorithms but have similar acceptance rates.

Original languageEnglish (US)
Pages (from-to)123-142
Number of pages20
JournalJournal of Computational Physics
Issue number1
StatePublished - Sep 1 2006
Externally publishedYes


  • History matching
  • Langevin
  • MCMC
  • Multiscale
  • Porous media
  • Two-Phase flow
  • Uncertainty

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics
  • Numerical Analysis
  • General Physics and Astronomy
  • Computer Science Applications
  • Modeling and Simulation
  • Physics and Astronomy (miscellaneous)


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