Multilevel Monte Carlo methods using ensemble level mixed MsFEM for two-phase flow and transport simulations

In this paper, we propose multilevel Monte Carlo (MLMC) methods that use ensemble level mixed multiscale methods in the simulations of multiphase flow and transport. The contribution of this paper is twofold: (1) a design of ensemble level mixed multiscale finite element methods and (2) a novel use of mixed multiscale finite element methods within multilevel Monte Carlo techniques to speed up the computations. The main idea of ensemble level multiscale methods is to construct local multiscale basis functions that can be used for any member of the ensemble. In this paper, we consider two ensemble level mixed multiscale finite element methods: (1) the no-local-solve-online ensemble level method (NLSO); and (2) the local-solve-online ensemble level method (LSO). The first approach was proposed in Aarnes and Efendiev (SIAM J. Sci. Comput. 30(5):2319-2339, 2008) while the second approach is new. Both mixed multiscale methods use a number of snapshots of the permeability media in generating multiscale basis functions. As a result, in the off-line stage, we construct multiple basis functions for each coarse region where basis functions correspond to different realizations. In the no-local-solve-online ensemble level method, one uses the whole set of precomputed basis functions to approximate the solution for an arbitrary realization. In the local-solve-online ensemble level method, one uses the precomputed functions to construct a multiscale basis for a particular realization. With this basis, the solution corresponding to this particular realization is approximated in LSO mixed multiscale finite element method (MsFEM). In both approaches, the accuracy of the method is related to the number of snapshots computed based on different realizations that one uses to precompute a multiscale basis. In this paper, ensemble level multiscale methods are used in multilevel Monte Carlo methods (Giles 2008a, Oper.Res. 56(3):607-617, b). In multilevel Monte Carlo methods, more accurate (and expensive) forward simulations are run with fewer samples, while less accurate (and inexpensive) forward simulations are run with a larger number of samples. Selecting the number of expensive and inexpensive simulations based on the number of coarse degrees of freedom, one can show that MLMC methods can provide better accuracy at the same cost as Monte Carlo (MC) methods. The main objective of the paper is twofold. First, we would like to compare NLSO and LSO mixed MsFEMs. Further, we use both approaches in the context of MLMC to speedup MC calculations.

[1]  J. van Leeuwen.  Large-Scale Scientific Computing: Third International Conference, LSSC 2001 Sozopol, Bulgaria, June 6–10, 2001 Revised Papers , 2001, LSSC.

[2]  Louis J. Durlofsky,et al.  Ensemble-Level Upscaling for Efficient Estimation of Fine-Scale Production Statistics , 2008 .

[3]  H. Owhadi,et al.  Metric‐based upscaling , 2007 .

[4]  Thomas Y. Hou,et al.  A Multiscale Finite Element Method for Elliptic Problems in Composite Materials and Porous Media , 1997 .

[5]  Stefan Heinrich,et al.  Multilevel Monte Carlo Methods , 2001, LSSC.

[6]  Stein Krogstad,et al.  A Hierarchical Multiscale Method for Two-Phase Flow Based upon Mixed Finite Elements and Nonuniform Coarse Grids , 2006, Multiscale Model. Simul..

[7]  K. A. Cliffe,et al.  Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients , 2011, Comput. Vis. Sci..

[8]  Andrea Barth,et al.  Multi-level Monte Carlo Finite Element method for elliptic PDEs with stochastic coefficients , 2011, Numerische Mathematik.

[9]  Yalchin Efendiev,et al.  Mixed Multiscale Finite Element Methods for Stochastic Porous Media Flows , 2008, SIAM J. Sci. Comput..

[10]  Andrea Barth,et al.  Multi-level Monte Carlo finite element method for elliptic PDE's with stochastic coefficients , 2010 .

[11]  P. Deuflhard,et al.  Large Scale Scientific Computing , 1987 .

[12]  Yalchin Efendiev,et al.  Mixed Multiscale Finite Element Methods Using Limited Global Information , 2008, Multiscale Model. Simul..

[13]  E. Rank,et al.  A multiscale finite-element method , 1997 .

[14]  Yalchin Efendiev,et al.  A Systematic Coarse-Scale Model Reduction Technique for Parameter-Dependent Flows in Highly Heterogeneous Media and Its Applications , 2012, Multiscale Model. Simul..

[15]  Stein Krogstad,et al.  Open-source MATLAB implementation of consistent discretisations on complex grids , 2012, Computational Geosciences.

[16]  M. Giles.  Improved Multilevel Monte Carlo Convergence using the Milstein Scheme , 2008 .

[17]  Peter Orbanz.  Probability Theory II , 2011 .

[18]  Dimitrios V. Rovas,et al.  Reduced-basis output bound methods for parabolic problems , 2006 .

[19]  J. Hesthaven,et al.  Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations , 2007 .

[20]  H. Niederreiter,et al.  Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing , 1995 .

[21]  L. Durlofsky,et al.  A coupled local-global upscaling approach for simulating flow in highly heterogeneous formations , 2003 .

[22]  Yalchin Efendiev,et al.  Multiscale Finite Element Methods: Theory and Applications , 2009 .

[23]  Harald Niederreiter,et al.  Monte Carlo and Quasi-Monte Carlo Methods 2006 , 2007 .

[24]  Zhiming Chen,et al.  A mixed multiscale finite element method for elliptic problems with oscillating coefficients , 2003, Math. Comput..

[25]  Cornelia Kronsbein,et al.  On selected efficient numerical methods for multiscale problems with stochastic coefficients , 2013 .

[26]  Michael B. Giles,et al.  Multilevel Monte Carlo Path Simulation , 2008, Oper. Res..

[27]  Jørg E. Aarnes,et al.  On the Use of a Mixed Multiscale Finite Element Method for GreaterFlexibility and Increased Speed or Improved Accuracy in Reservoir Simulation , 2004, Multiscale Model. Simul..

[28]  Knut-Andreas Lie,et al.  An Introduction to the Numerics of Flow in Porous Media using Matlab , 2007, Geometric Modelling, Numerical Simulation, and Optimization.