Metric‐based upscaling

Owhadi, Houman; Zhang, Lei

doi:10.1002/cpa.20163

Cited by 199 publications

(187 citation statements)

References 66 publications

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“…Offline stage Global information is important as it was demonstrated in [20,12] theoretically and numerically. This is particularly true when the problem is solved multiple times.…”

Section: Online Stagementioning

confidence: 99%

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

2013

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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

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“…Offline stage Global information is important as it was demonstrated in [20,12] theoretically and numerically. This is particularly true when the problem is solved multiple times.…”

Section: Online Stagementioning

confidence: 99%

“…In Figure 5 we used LSO. We choose the numbers of samples at each level as M = (70, 20,10) Table 2: Parameters and errors for the single-phase flow example.…”

Section: Single-phase Flowmentioning

confidence: 99%

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

2013

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“…Another approach advocated by Owhadi and Zhang in [53], whose origin is not the homogenization theory properly speaking, is based on the notion of harmonic coordinates. The starting point of their work is that although solutions u ∈ H 1 0 (D) to the equation…”

Section: −ˆQmentioning

confidence: 99%

Numerical homogenization: survey, new results, and perspectives

Gloria

2012

ESAIM: Proc.

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Abstract. These notes give a state of the art of numerical homogenization methods for linear elliptic equations. The guideline of these notes is analysis. Most of the numerical homogenization methods can be seen as (more or less different) discretizations of the same family of continuous approximate problems, which H-converges to the homogenized problem. Likewise numerical correctors may also be interpreted as approximations of Tartar's correctors. Hence the convergence analysis of these methods relies on the H-convergence theory. When one is interested in convergence rates, the story is different. In particular one first needs to make additional structure assumptions on the heterogeneities (say periodicity for instance). In that case, a crucial tool is the spectral interpretation of the corrector equation by Papanicolaou and Varadhan. Spectral analysis does not only allow to obtain convergence rates, but also to devise efficient new approximation methods. For both qualitative and quantitative properties, the development and the analysis of numerical homogenization methods rely on seminal concepts of the homogenization theory. These notes contain some new results.Résumé. Ces notes de cours dressent unétat de l'art des méthodes d'homogénéisation numérique pour leséquations elliptiques linéaires. Le fil conducteur choisi est l'analyse. La plupart des méthodes d'homogénéisation numérique s'interprète comme des discrétisations (plus ou moins différentes) d'une même famille de problèmes continus approchés qui H-converge vers le problème homogénéisé. De même, le concept de correcteur numérique s'interprète comme une approximation des correcteurs introduits par Tartar. Ainsi l'analyse de convergence repose essentiellement sur la théorie de la H-convergence. Si on s'intéresse aux estimations quantitatives d'erreur, il faut faire des hypothèses supplémentaires de structure sur les hétérogénéités (périodicité par exemple). Dans ce cas, un outil important est l'interprétation spectrale de l'équation du correcteur introduite par Papanicolaou et Varadhan, qui permet non seulement de démontrer des résultats quantitatifs, mais aussi de développer des méthodes numériques efficaces. Qu'il s'agisse de propriétés qualitatives ou quantitatives, le développement et l'analyse de méthodes d'homogénéisation numérique reposent sur des concepts fondateurs de la théorie de l'homogénéisation. Ces notes contiennent quelques résultats nouveaux. Article published online by EDP Sciences and available at http://www.esaim-proc.org or http://dx

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“…In this paper, we improve these results to the more practical situation in which we assume that the local problem is solved by some high-order finite elements as in [3]. New discussion of the harmonic coordinate in the multiscale method can be found in [19], where the globally defined harmonic coordinate is solved. Other high-order methods for generalized finite element methods are discussed in [17].…”

mentioning

confidence: 99%

“…These solutions are used to build the multiscale finite element basis to capture the small scale information of the leading-order differential operator. There are several alternatives to this approach for multiscale methods, for example, the multiscale variational method [15] and the heterogeneous multiscale method (HMM) [1], and additional methods are discussed in [4,19,7]. In this work, however, we focus on techniques based on multiscale finite element formulations.…”

mentioning

confidence: 99%

High-Order Multiscale Finite Element Method for Elliptic Problems

Hesthaven¹,

Zhang²,

Zhu³

2014

Multiscale Model. Simul.

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Abstract. In this paper, a new high-order multiscale finite element method (MsFEM) is developed for elliptic problems with highly oscillating coefficients. The method is inspired by the MsFEM developed in [G. Allaire and R. Brizzi, Multiscale Model. Simul., 4 (2005) 1. Introduction. The development of numerical methods for problems with highly oscillating coefficients is an increasingly active field of research. To overcome the computational cost of resolving the fine scale, multiscale finite element methods (MsFEMs) have been developed in [12,13,11,8,14,10]. Accuracy is achieved by solving a fine scale problem locally. These solutions are used to build the multiscale finite element basis to capture the small scale information of the leading-order differential operator. There are several alternatives to this approach for multiscale methods, for example, the multiscale variational method [15] and the heterogeneous multiscale method (HMM) [1], and additional methods are discussed in [4,19,7]. In this work, however, we focus on techniques based on multiscale finite element formulations.Originally, MsFEMs were proposed for linear finite elements; see [12,13]. For many applications, e.g., elliptic problems with singular forcing or nonconvex domains, and wave equations, high-order finite elements are known to be advantageous in terms of accuracy and efficiency, in particular for large problems. Allaire and Brizzi [3] generalized the original approach to enable the use of high-order (order higher than one) elements by local harmonic coordinates. This method uses a composite rule to change the local coordinates. Inspired by that work, we propose in this work a new high-order accurate MsFEM. However, in contrast to the previous work, we do not use a composite rule but approach the development in a more explicit way. Additionally,

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Metric‐based upscaling

Cited by 199 publications

References 66 publications

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

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

Numerical homogenization: survey, new results, and perspectives

High-Order Multiscale Finite Element Method for Elliptic Problems

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