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102
The heterogeneous multiscale method: A review
 COMMUN. COMPUT. PHYS
, 2007
"... This paper gives a systematic introduction to HMM, the heterogeneous multiscale method, including the fundamental design principles behind the HMM philosophy and the main obstacles that have to be overcome when using HMM for a particular problem. This is illustrated by examples from several applic ..."
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Cited by 100 (5 self)
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This paper gives a systematic introduction to HMM, the heterogeneous multiscale method, including the fundamental design principles behind the HMM philosophy and the main obstacles that have to be overcome when using HMM for a particular problem. This is illustrated by examples from several application areas, including complex fluids, microfluidics, solids, interface problems, stochastic problems, and statistically selfsimilar problems. Emphasis is given to the technical tools, such as the various constrained molecular dynamics, that have been developed, in order to apply HMM to these problems. Examples of mathematical results on the error analysis of HMM are presented. The paper ends with a discussion on some of
An optimal variance estimate in stochastic homogenization of discrete elliptic equations
 Ann. Probab
, 2011
"... We consider a discrete elliptic equation on the ddimensional lattice Zd with random coefficients A of the simplest type: they are identically distributed and independent from edge to edge. On scales large w.r.t. the lattice spacing (i.e., unity), the solution operator is known to behave like the so ..."
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Cited by 68 (26 self)
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We consider a discrete elliptic equation on the ddimensional lattice Zd with random coefficients A of the simplest type: they are identically distributed and independent from edge to edge. On scales large w.r.t. the lattice spacing (i.e., unity), the solution operator is known to behave like the solution operator of a (continuous) elliptic equation with constant deterministic coefficients. This symmetric “homogenized ” matrix Ahom = ahom Id is characterized by ξ · Ahomξ =〈(ξ +∇φ) · A(ξ +∇φ) 〉 for any direction ξ ∈ Rd, where the random field φ (the “corrector”) is the unique solution of −∇ ∗ · A(ξ +∇φ) = 0 such that φ(0) = 0, ∇φ is stationary and 〈∇φ〉=0, 〈· 〉 denoting the ensemble average (or expectation). It is known (“by ergodicity”) that the above ensemble average of the energy density E = (ξ +∇φ) · A(ξ +∇φ), which is a stationary random field, can be recovered by a system average. We quantify this by proving that the variance of a spatial average of E on length scales L satisfies the optimal estimate, that is, var [ ∑ EηL] � L−d, where the averaging function [i.e., ∑ ηL = 1, supp(ηL) ⊂{x≤L}] has to be smooth in the sense that
Finite element heterogeneous multiscale methods with near optimal . . .
"... This paper is concerned with a numerical method for multiscale elliptic problems. Using the framework of the Heterogeneous Multiscale Methods (HMM), we propose a micromacro approache which combines finite element method (FEM) for the macroscopic solver and the pseudospectral method for the micro s ..."
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Cited by 58 (24 self)
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This paper is concerned with a numerical method for multiscale elliptic problems. Using the framework of the Heterogeneous Multiscale Methods (HMM), we propose a micromacro approache which combines finite element method (FEM) for the macroscopic solver and the pseudospectral method for the micro solver. Unlike the micromacro methods based on standard FEM proposed so far in HMM we obtain, for periodic homogenization problems, a method that has almostlinear complexity in the number of degrees of freedom of the discretization of the macro (slow) variable.
Domain decomposition for multiscale PDEs
 Numer. Math
"... We consider additive Schwarz domain decomposition preconditioners for piecewise linear finite element approximations of elliptic PDEs with highly variable coefficients. In contrast to standard analyses, we do not assume that the coefficients can be resolved by a coarse mesh. This situation arises of ..."
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Cited by 50 (19 self)
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We consider additive Schwarz domain decomposition preconditioners for piecewise linear finite element approximations of elliptic PDEs with highly variable coefficients. In contrast to standard analyses, we do not assume that the coefficients can be resolved by a coarse mesh. This situation arises often in practice, for example in the computation of flows in heterogeneous porous media, in both the deterministic and (MonteCarlo simulated) stochastic cases. We consider preconditioners which combine local solves on general overlapping subdomains together with a global solve on a general coarse space of functions on a coarse grid. We perform a new analysis of the preconditioned matrix, which shows rather explicitly how its condition number depends on the variable coefficient in the PDE as well as on the coarse mesh and overlap parameters. The classical estimates for this preconditioner with linear coarsening guarantee good conditioning only when the coefficient varies mildly inside the coarse grid elements. By contrast, our new results show that, with a good choice of subdomains and coarse space basis functions, the preconditioner can still be robust even for large coefficient variation inside domains,
On apriori error analysis of fully discrete heterogeneous multiscale FEM
, 2004
"... Abstract. Heterogeneous multiscale methods have been introduced by E and Engquist [Commun. Math. Sci., 1 (2003), pp. 87–132] as a methodology for the numerical computation of problems with multiple scales. Analyses of the methods for various homogenization problems have been done by several authors ..."
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Cited by 45 (23 self)
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Abstract. Heterogeneous multiscale methods have been introduced by E and Engquist [Commun. Math. Sci., 1 (2003), pp. 87–132] as a methodology for the numerical computation of problems with multiple scales. Analyses of the methods for various homogenization problems have been done by several authors. These results were obtained under the assumption that the microscopic models (the cell problems in the homogenization context) are analytically given. For numerical computations, these microscopic models have to be solved numerically. Therefore, it is important to analyze the error transmitted on the macroscale by discretizing the fine scale. We give in this paper H1 and L2 a priori estimates of the fully discrete heterogeneous multiscale finite element method. Numerical experiments confirm that the obtained a priori estimates are sharp.
Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on Glauber dynamics
 In preparation
"... We study the effective largescale behavior of discrete elliptic equations on the lattice Z d with random coefficients. The theory of stochastic homogenization relates the random but stationary field of coefficients with a deterministic matrix of effective coefficients. This is done via the correcto ..."
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Cited by 31 (17 self)
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We study the effective largescale behavior of discrete elliptic equations on the lattice Z d with random coefficients. The theory of stochastic homogenization relates the random but stationary field of coefficients with a deterministic matrix of effective coefficients. This is done via the corrector problem, which can be viewed as a highly degenerate elliptic equation on the infinitedimensional space of admissible coefficient fields. In this contribution we develop quantitative methods for the corrector problem assuming that the ensemble of coefficient fields satisfies a spectral gap estimate w. r. t. a Glauber dynamics. As a main result we prove an optimal estimate for the decay in time of the parabolic equation associated to the corrector problem (i. e. for the “random environment as seen from a random walker”). As a corollary we obtain existence and moment bounds for stationary correctors (in dimension d> 2) and optimal estimates for regularized versions of the corrector (in dimensions d ≥ 2). We also give a selfcontained proof for a new estimate on the gradient of the parabolic, variablecoefficient Green’s function, which is a crucial analytic ingredient in our method. As an application, we study the approximation of the homogenized coefficients via a representative volume element. The approximation introduces two types of errors. Based on our quantitative
A computational strategy for multiscale systems with applications to Lorenz 96 model
, 2004
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Multiscale modeling of physical phenomena: Adaptive control of models
 SIAM J. Sci. Comput
, 2005
"... Abstract. It is common knowledge that the accuracy with which computer simulations can depict physical events depends strongly on the choice of the mathematical model of the events. Perhaps less appreciated is the notion that the error due to modeling can be defined, estimated, and used adaptively t ..."
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Cited by 27 (6 self)
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Abstract. It is common knowledge that the accuracy with which computer simulations can depict physical events depends strongly on the choice of the mathematical model of the events. Perhaps less appreciated is the notion that the error due to modeling can be defined, estimated, and used adaptively to control modeling error, provided one accepts the existence of a base model that can serve as a datum with respect to which other models can be compared. In this work, it is shown that the idea of comparing models and controlling model error can be used to develop a general approach for multiscale modeling, a subject of growing importance in computational science. A posteriori estimates of modeling error in socalled quantities of interest are derived and a class of adaptive modeling algorithms is presented. Several applications of the theory and methodology are presented. These include preliminary work on random multiphase composite materials, molecular statics simulations with applications to problems in nanoindentation, and analysis of molecular dynamics models using various techniques for scale bridging. Key words. multiscale modeling, goaloriented adaptive modeling, a posteriori error estimation
The heterogeneous multiscale finite element method for elliptic homogenization problems in perforated domains
 Numer. Math
"... (Communicated by Alfio Quarteroni) Abstract. This contribution is concerned with the formulation of a heterogeneous multiscale finite elements method (HMM) for solving linear advectiondiffusion problems with rapidly oscillating coefficient functions and a large expected drift. We show that, in the c ..."
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Cited by 27 (16 self)
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(Communicated by Alfio Quarteroni) Abstract. This contribution is concerned with the formulation of a heterogeneous multiscale finite elements method (HMM) for solving linear advectiondiffusion problems with rapidly oscillating coefficient functions and a large expected drift. We show that, in the case of periodic coefficient functions, this approach is equivalent to a discretization of the twoscale homogenized equation by means of a Discontinuous Galerkin Time Stepping Method with quadrature. We then derive an optimal order apriori error estimate for this version of the HMM and finally provide numerical experiments to validate the method. 1. Introduction. In
A framework for adaptive multiscale methods for elliptic problems. Multiscale Modeling
 Simulation
"... Abstract. We describe a projection framework for developing adaptive multiscale methods for computing approximate solutions to elliptic boundary value problems. The framework is consistent with homogenization when there is scale separation. We introduce an adaptive form of the finite element algorit ..."
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Cited by 25 (0 self)
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Abstract. We describe a projection framework for developing adaptive multiscale methods for computing approximate solutions to elliptic boundary value problems. The framework is consistent with homogenization when there is scale separation. We introduce an adaptive form of the finite element algorithms for solving problems with no clear scale separation. We present numerical simulations demonstrating the effectiveness and adaptivity of the multiscale method, assess its computational complexity, and discuss the relationship between this framework and other multiscale methods, such as wavelets, multiscale finite element methods, and the use of harmonic coordinates. We prove in detail that the projection based method captures homogenization when there is strong scale separation. Key words. Multiscale finite elements, numerical homogenization, variational multiscale method AMS subject classifications. 65N30, 35J20, 35B27 1. Introduction. In