A multiscale finite element method for elliptic problems in composite materials and porous media
Journal of Computational Physics
Prediction and the quantification of uncertainty
Physica D - Special issue originating from the 18th Annual International Conference of the Center for Nonlinear Studies, Los Alamos, NM, May 11&mdash ;15, 1998
Monte Carlo Statistical Methods (Springer Texts in Statistics)
Monte Carlo Statistical Methods (Springer Texts in Statistics)
Preconditioning Markov Chain Monte Carlo Simulations Using Coarse-Scale Models
SIAM Journal on Scientific Computing
Markov Chains and Stochastic Stability
Markov Chains and Stochastic Stability
Monte Carlo Strategies in Scientific Computing
Monte Carlo Strategies in Scientific Computing
Multiscale finite element methods for porous media flows and their applications
Applied Numerical Mathematics
Journal of Computational Physics
Adaptive langevin sampler for separation of t-distribution modelled astrophysical maps
IEEE Transactions on Image Processing
Journal of Computational Physics
Journal of Computational Physics
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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.