Communications in Mathematical Sciences

ON HMM-like integrators and projective integration methods for systems with multiple time scales

Eric Vanden-Eijnden

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HMM-like multiscale integrators and projective integration methods are two different types of multiscale integrators which have been introduced to simulate efficiently systems with widely disparate time scales. The original philosophies of these methods, reviewed here, were quite different. Recently, however, projective integration methods seem to have evolved in a way that make them increasingly similar to HMM-integrators and quite different from what they were originally. Nevertheless, the strategy of extrapolation which was at the core of the original projective integration methods has its value and should be extended rather than abandoned. An attempt in this direction is made here and it is shown how the strategy of extrapolation can be generalized to stochastic dynamical systems with multiple time scales, in a way reminiscent of Chorin’s artificial compressibility method and the Car-Parrinello method used in molecular dynamics. The result is a seamless integration scheme, i.e. one that does not require knowing explicitly what the slow and fast variables are.

Article information

Commun. Math. Sci., Volume 5, Issue 2 (2007), 495-505.

First available in Project Euclid: 9 July 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34A50 65C30: Stochastic differential and integral equations 65L20: Stability and convergence of numerical methods 60H35: Computational methods for stochastic equations [See also 65C30]

multiscale integrators HMM projective integration methods stiff ODEs averaging theorems


Vanden-Eijnden, Eric. ON HMM-like integrators and projective integration methods for systems with multiple time scales. Commun. Math. Sci. 5 (2007), no. 2, 495--505.

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