2,008 research outputs found
Preconditioning Markov Chain Monte Carlo Simulations Using Coarse-Scale Models
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 Karhunen--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
A constrained pressure-temperature residual (CPTR) method for non-isothermal multiphase flow in porous media
For both isothermal and thermal petroleum reservoir simulation, the
Constrained Pressure Residual (CPR) method is the industry-standard
preconditioner. This method is a two-stage process involving the solution of a
restricted pressure system. While initially designed for the isothermal case,
CPR is also the standard for thermal cases. However, its treatment of the
energy conservation equation does not incorporate heat diffusion, which is
often dominant in thermal cases. In this paper, we present an extension of CPR:
the Constrained Pressure-Temperature Residual (CPTR) method, where a restricted
pressure-temperature system is solved in the first stage. In previous work, we
introduced a block preconditioner with an efficient Schur complement
approximation for a pressure-temperature system. Here, we extend this method
for multiphase flow as the first stage of CPTR. The algorithmic performance of
different two-stage preconditioners is evaluated for reservoir simulation test
cases.Comment: 28 pages, 2 figures. Sources/sinks description in arXiv:1902.0009
A robust adaptive algebraic multigrid linear solver for structural mechanics
The numerical simulation of structural mechanics applications via finite
elements usually requires the solution of large-size and ill-conditioned linear
systems, especially when accurate results are sought for derived variables
interpolated with lower order functions, like stress or deformation fields.
Such task represents the most time-consuming kernel in commercial simulators;
thus, it is of significant interest the development of robust and efficient
linear solvers for such applications. In this context, direct solvers, which
are based on LU factorization techniques, are often used due to their
robustness and easy setup; however, they can reach only superlinear complexity,
in the best case, thus, have limited applicability depending on the problem
size. On the other hand, iterative solvers based on algebraic multigrid (AMG)
preconditioners can reach up to linear complexity for sufficiently regular
problems but do not always converge and require more knowledge from the user
for an efficient setup. In this work, we present an adaptive AMG method
specifically designed to improve its usability and efficiency in the solution
of structural problems. We show numerical results for several practical
applications with millions of unknowns and compare our method with two
state-of-the-art linear solvers proving its efficiency and robustness.Comment: 50 pages, 16 figures, submitted to CMAM
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