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277
Extracting macroscopic dynamics: model problems and algorithms
- NONLINEARITY
, 2004
"... In many applications, the primary objective of numerical simulation of time-evolving systems is the prediction of macroscopic, or coarse-grained, quantities. A representative example is the prediction of biomolecular conformations from molecular dynamics. In recent years a number of new algorithmic ..."
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Cited by 111 (8 self)
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In many applications, the primary objective of numerical simulation of time-evolving systems is the prediction of macroscopic, or coarse-grained, quantities. A representative example is the prediction of biomolecular conformations from molecular dynamics. In recent years a number of new algorithmic approaches have been introduced to extract effective, lower-dimensional, models for the macroscopic dynamics; the starting point is the full, detailed, evolution equations. In many cases the effective low-dimensional dynamics may be stochastic, even when the original starting point is deterministic. This review surveys a number of these new approaches to the problem of extracting effective dynamics, highlighting similarities and differences between them. The importance of model problems for the evaluation of these new approaches is stressed, and a number of model problems are described. When the macroscopic dynamics is stochastic, these model problems are either obtained through a clear separation of time-scales, leading to a stochastic effect of the fast dynamics on the slow dynamics, or by considering high dimensional ordinary differential equations which, when projected onto a low dimensional subspace, exhibit stochastic behaviour through the presence of a broad frequency spectrum. Models whose stochastic microscopic behaviour leads to deterministic macroscopic dynamics are also introduced. The algorithms we overview include SVD-based methods for nonlinear problems, model reduction for linear control systems, optimal prediction techniques, asymptotics-based mode elimination, coarse timestepping methods and transfer-operator based methodologies.
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 104 (7 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, micro-fluidics, solids, interface problems, stochastic problems, and statistically self-similar 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
Analysis of the heterogeneous multiscale method for ordinary differential equations
- Commun. Math. Sci
"... Abstract. The heterogeneous multiscale method (HMM) is applied to various parabolic problems with multiscale coefficients. These problems can be either linear or nonlinear. Optimal estimates are proved for the error between the HMM solution and the homogenized solution. ..."
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Cited by 103 (11 self)
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Abstract. The heterogeneous multiscale method (HMM) is applied to various parabolic problems with multiscale coefficients. These problems can be either linear or nonlinear. Optimal estimates are proved for the error between the HMM solution and the homogenized solution.
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 micro-macro 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 56 (21 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 micro-macro approache which combines finite element method (FEM) for the macroscopic solver and the pseudospectral method for the micro solver. Unlike the micro-macro 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.
Analysis of a force-based quasicontinuum approximation
- M2AN Math. Model. Numer. Anal
"... Abstract. We analyze a force-based quasicontinuum approximation to a one-dimensional system of atoms that interact by a classical atomistic potential. This force-based quasicontinuum approximation is derived as the modification of an energy-based quasicontinuum approximation by the addition of nonco ..."
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Cited by 56 (25 self)
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Abstract. We analyze a force-based quasicontinuum approximation to a one-dimensional system of atoms that interact by a classical atomistic potential. This force-based quasicontinuum approximation is derived as the modification of an energy-based quasicontinuum approximation by the addition of nonconservative forces to correct nonphysical “ghost ” forces that occur in the atomistic to continuum interface. We prove that the force-based quasicontinuum equations have a unique solution under suitable restrictions on the loads. For Lennard-Jones next-nearest-neighbor interactions, we show that unique solutions exist for loads in a symmetric region extending nearly to the tensile limit. We give an analysis of the convergence of the ghost force iteration method to solve the equilibrium equations for the force-based quasicontinuum approximation. We show that the ghost force iteration is a contraction and give an analysis for its convergence rate. 1.
Analysis of a one-dimensional nonlocal quasi-continuum method
- SIMULATION 7(4), 1838–1875 (2009). DOI 10.1137/080725842. URL HTTP://LINK.AIP.ORG/LINK/?MMS/7/1838/1
"... The accuracy of the quasicontinuum method is analyzed using a series of models with increasing complexity. It is demonstrated that the existence of the ghost force may lead to large errors. It is also shown that the ghost force removal strategy proposed by E, Lu and Yang leads to a version of the qu ..."
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Cited by 55 (4 self)
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The accuracy of the quasicontinuum method is analyzed using a series of models with increasing complexity. It is demonstrated that the existence of the ghost force may lead to large errors. It is also shown that the ghost force removal strategy proposed by E, Lu and Yang leads to a version of the quasicontinuum method with uniform accuracy.
Accurate multiscale finite element methods for two-phase flow simulations
, 2006
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Heterogeneous multiscale methods for stiff ordinary differential equations. 2003. Under review
"... Abstract. The heterogeneous multiscale methods (HMM) is a general framework for the numerical approximation of multiscale problems. It is here developed for ordinary differential equations containing different time scales. Stability and convergence results for the proposed HMM methods are presented ..."
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Cited by 46 (9 self)
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Abstract. The heterogeneous multiscale methods (HMM) is a general framework for the numerical approximation of multiscale problems. It is here developed for ordinary differential equations containing different time scales. Stability and convergence results for the proposed HMM methods are presented together with numerical tests. The analysis covers some existing methods and the new algorithms that are based on higher-order estimates of the effective force by kernels satisfying certain moment conditions and regularity properties. These new methods have superior computational complexity compared to traditional methods for stiff problems with oscillatory solutions.
Multiscale finite element methods for nonlinear problems and their applications
- Comm. Math. Sci
, 2004
"... In this paper we propose a generalization of multiscale finite element methods (MsFEM) to nonlinear problems. We study the convergence of the proposed method for nonlinear elliptic equations and propose an oversampling technique. Numerical examples demonstrate that the oversampling technique greatly ..."
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Cited by 44 (18 self)
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In this paper we propose a generalization of multiscale finite element methods (MsFEM) to nonlinear problems. We study the convergence of the proposed method for nonlinear elliptic equations and propose an oversampling technique. Numerical examples demonstrate that the oversampling technique greatly reduces the error. The application of MsFEM to porous media flows is considered. Finally, we describe further generalizations of MsFEM to nonlinear time-dependent equations and discuss the convergence of the method for various kinds of heterogeneities.
A Multiscale Finite Element Method For Numerical Homogenization
, 2004
"... This paper is concerned with a multiscale finite element method for numerically solving second order scalar elliptic boundary value problems with highly oscillating coefficients. In the spirit of previous other works, our method is based on the coupling of a coarse global mesh and of a fine local ..."
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Cited by 43 (3 self)
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This paper is concerned with a multiscale finite element method for numerically solving second order scalar elliptic boundary value problems with highly oscillating coefficients. In the spirit of previous other works, our method is based on the coupling of a coarse global mesh and of a fine local mesh, the latter one being used for computing independently an adapted finite element basis for the coarse mesh. The main new idea is the introduction of a composition rule, or change of variables, for the construction of this finite element basis. In particular, this allows for a simple treatment of high order finite element methods. We provide
