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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, 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
Parameter estimation for multiscale diffusions
 J. Statist. Phys
, 2007
"... We study the problem of parameter estimation for timeseries possessing two, widely separated, characteristic time scales. The aim is to understand situations where it is desirable to fit a homogenized singlescale model to such multiscale data. We demonstrate, numerically and analytically, that if ..."
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Cited by 30 (10 self)
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We study the problem of parameter estimation for timeseries possessing two, widely separated, characteristic time scales. The aim is to understand situations where it is desirable to fit a homogenized singlescale model to such multiscale data. We demonstrate, numerically and analytically, that if the data is sampled too finely then the parameter fit will fail, in that the correct parameters in the homogenized model are not identified. We also show, numerically and analytically, that if the data is subsampled at an appropriate rate then it is possible to estimate the coefficients of the homogenized model correctly. The ideas are studied in the context of thermally activated motion in a twoscale potential. However the ideas may be expected to transfer to other situations where it is desirable to fit an averaged or homogenized equation to multiscale data.
Consistency and stability of tau leaping schemes for chemical reaction systems
 SIAM Multiscale Modeling
, 2005
"... Abstract. We develop a theory of local errors for the explicit and implicit tauleaping methods for simulating stochastic chemical systems, and we prove that these methods are firstorder consistent. Our theory provides local error formulae that could serve as the basis for future stepsize control t ..."
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Cited by 29 (7 self)
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Abstract. We develop a theory of local errors for the explicit and implicit tauleaping methods for simulating stochastic chemical systems, and we prove that these methods are firstorder consistent. Our theory provides local error formulae that could serve as the basis for future stepsize control techniques. We prove that, for the special case of systems with linear propensity functions, both tauleaping methods are firstorder convergent in all moments. We provide a stiff stability analysis of the mean of both leaping methods, and we confirm that the implicit method is unconditionally stable in the mean for stable systems. Finally, we give some theoretical and numerical examples to illustrate these results.
A MULTISCALE METHOD FOR HIGHLY OSCILLATORY ORDINARY DIFFERENTIAL EQUATIONS WITH RESONANCE IN MEMORY OF GERMUND DAHLQUIST
"... ABSTRACT. A multiscale method for computing the effective behavior of a class of stiff and highly oscillatory ordinary differential equations (ODEs) is presented. The oscillations may be in resonance with one another and thereby generate hidden slow dynamics. The proposed method relies on correctly ..."
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Cited by 16 (2 self)
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ABSTRACT. A multiscale method for computing the effective behavior of a class of stiff and highly oscillatory ordinary differential equations (ODEs) is presented. The oscillations may be in resonance with one another and thereby generate hidden slow dynamics. The proposed method relies on correctly tracking a set of slow variables whose dynamics is closed up to ɛ perturbation, and is sufficient to approximate any variable and functional that are slow under the dynamics of the ODE. This set of variables is detected numerically as a preprocessing step in the numerical methods. Error and complexity estimates are obtained. The advantages of the method is demonstrated with a few examples, including a commonly studied problem of Fermi, Pasta, and Ulam. 1.
A general strategy for designing seamless multiscale methods
 J. Comput. Phys
, 2009
"... Abstract We present a new general framework for designing multiscale methods. Compared with previous work such as Brandt's systematic upscaling, the heterogeneous multiscale method and the "equationfree" approach, this new framework has the distinct feature that it does not require ..."
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Cited by 16 (1 self)
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Abstract We present a new general framework for designing multiscale methods. Compared with previous work such as Brandt's systematic upscaling, the heterogeneous multiscale method and the "equationfree" approach, this new framework has the distinct feature that it does not require reinitializing the microscale model at each macro time step or each macro iteration step. In the new strategy, the macroand micromodels evolve simultaneously using different time steps (and therefore different clocks), and they exchange data at every step. The micromodel uses its own appropriate time step. The macromodel runs at a slower pace than required by accuracy and stability considerations for the macroscale dynamics, in order for the micromodel to relax. Examples are discussed and application to modeling complex fluids is presented.
Maximum Likelihood Drift Estimation for Multiscale Diffusions
 Stochastic Processes Applications
"... We study the problem of parameter estimation using maximum likelihood for fast/slow systems of stochastic differential equations. Our aim is to shed light on the problem of model/data mismatch at small scales. We consider two classes of fast/slow problems for which a closed coarsegrained equation f ..."
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Cited by 14 (5 self)
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We study the problem of parameter estimation using maximum likelihood for fast/slow systems of stochastic differential equations. Our aim is to shed light on the problem of model/data mismatch at small scales. We consider two classes of fast/slow problems for which a closed coarsegrained equation for the slow variables can be rigorously derived, which we refer to as averaging and homogenization problems. We ask whether, given data from the slow variable in the fast/slow system, we can correctly estimate parameters in the drift of the coarsegrained equation for the slow variable, using maximum likelihood. We show that, whereas the maximum likelihood estimator is asymptotically unbiased for the averaging problem, for the homogenization problem maximum likelihood fails unless we subsample the data at an appropriate rate. An explicit formula for the asymptotic error in the log likelihood function is presented. Our theory is applied to two simple examples from molecular dynamics.
Effectiveness of implicit methods for stiff stochastic differential equations
 Commun. Comput. Phys
, 2000
"... differential equations ..."
The moment map: Nonlinear dynamics of density evolution via a few moments
 SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS
, 2005
"... We explore situations in which certain stochastic and highdimensional deterministic systems behave effectively as lowdimensional dynamical systems. We define and study moment maps, maps on spaces of loworder moments of evolving distributions, as a means of understanding equationsfree multiscale ..."
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Cited by 11 (0 self)
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We explore situations in which certain stochastic and highdimensional deterministic systems behave effectively as lowdimensional dynamical systems. We define and study moment maps, maps on spaces of loworder moments of evolving distributions, as a means of understanding equationsfree multiscale algorithms for these systems. We demonstrate how nonlinearity arises in these maps and how this results in the stabilization of metastable states. Examples are shown for a hierarchy of models, ranging from simple stochastic differential equations to molecular dynamics simulations of a particle in contact with a heat bath.
SROCK methods for stiff Ito SDEs
 Commun. Math. Sci
, 2008
"... Abstract. In this paper, we present a class of explicit numerical methods for stiff Itô stochastic differential equations (SDEs). These methods are as simple to program and to use as the wellknown EulerMaruyama method, but much more efficient for stiff SDEs. For such problems, it is well known tha ..."
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Cited by 10 (9 self)
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Abstract. In this paper, we present a class of explicit numerical methods for stiff Itô stochastic differential equations (SDEs). These methods are as simple to program and to use as the wellknown EulerMaruyama method, but much more efficient for stiff SDEs. For such problems, it is well known that standard explicit methods face stepsize reduction. While semiimplicit methods can avoid these problems at the cost of solving (possibly large) nonlinear systems, we show that the stepsize reduction phenomena can be reduced significantly for explicit methods by using stabilization techniques. Stabilized explicit numerical methods called SROCK (for stochastic orthogonal RungeKutta Chebyshev) have been introduced in [C. R. Acad. Sci. Paris, 345(10), 2007] as an alternative to (semi) implicit methods for the solution of stiff stochastic systems. In this paper we discuss a genuine Itô version of the SROCK methods which avoid the use of transformation formulas from Stratonovich to Itô calculus. This is important for many applications. We present two families of methods for onedimensional and multidimensional Wiener processes. We show that for stiff problems, significant improvement over classical explicit methods can be obtained. Convergence and stability properties of the methods are discussed and numerical examples as well as applications to the simulation of stiff chemical Langevin equations are presented.