Preconditioning Markov chain Monte Carlo simulations using coarse-scale models

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

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

139 Scopus citations


We study the preconditioning of Markov chain Monte Carlo (MCMC) methods using coarse-scale models with applications to subsurface characterization. The purpose of preconditioning is to reduce the fine-scale computational cost and increase the acceptance rate in the MCMC sampling. This goal is achieved by generating Markov chains based on two-stage computations. In the first stage, a new proposal is first tested by the coarse-scale model based on multiscale finite volume methods. The full fine-scale computation will be conducted only if the proposal passes the coarse-scale screening. For more efficient simulations, an approximation of the full fine-scale computation using precomputed multiscale basis functions can also be used. Comparing with the regular MCMC method, the preconditioned MCMC method generates a modified Markov chain by incorporating the coarse-scale information of the problem. The conditions under which the modified Markov chain will converge to the correct posterior distribution are stated in the paper. The validity of these assumptions for our application and the conditions which would guarantee a high acceptance rate are also discussed. We would like to note that 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. The Karhurien-Loève expansion is used to represent the realizations of the permeability field conditioned to the dynamic data, such as production data, as well as some static data. Our numerical examples show that the acceptance rate can be increased by more than 10 times if MCMC simulations are preconditioned using coarse-scale models.

Original languageEnglish (US)
Pages (from-to)776-803
Number of pages28
JournalSIAM Journal on Scientific Computing
Issue number2
StatePublished - 2006
Externally publishedYes


  • Markov chain Monte Carlo
  • Multiscale
  • Porous media
  • Preconditioning

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics


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