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Numerical schemes are presented for dynamical systems with multiple timescales. Two classes of methods are discussed, depending on the time interval on which the evolution of the slow variables in the system is sought. On rather short time intervals, the slow variables satisfy ordinary differential equations. On longer time intervals, however, fluctuations… (More)

We study the action minimization problem that is formally associated to phase transformation in the stochastically perturbed Allen-Cahn equation. The sharp-interface limit is related to (but different from) the sharp-interface limits of the related energy functional and deterministic gradient flows. In the sharp-interface limit of the action minimization… (More)

We analyze a class of numerical schemes proposed in [26] for stochastic differential equations with multiple time scales. Both advective and diffusive time scales are considered. Weak as well as strong convergence theorems are proven. Most of our results are optimal. They in turn allow us to provide a thorough discussion on the efficiency as well as optimal… (More)

We present a systematic introduction to the basic concepts and techniques for determining transition pathways and transition rates in systems with multiple metastable states. After discussing the classical transition state theory and its limitations, we derive a new set of equations for the optimal dividing surfaces. We then discuss transition path… (More)

A new approach is proposed for stochastic parameterization of subgrid scale processes in models of atmospheric or oceanic circulation. The new approach relies on two key ingredients. First, the unresolved processes are represented by a Markov chain whose properties depend on the state of the resolved model variables. Second, the properties of this… (More)

- Giovanni Ciccotti, Raymond Kapral, Eric Vanden-Eijnden
- Chemphyschem : a European journal of chemical…
- 2005

We give a new formula expressing the components of the mean force in terms of a conditional expectation which can be computed by Blue Moon sampling. This generalizes to the vectorial case a formula first derived by Ruiz-Montero et al. for a scalar reaction coordinate. We also discuss how to compute this conditional average by means of constrained stochastic… (More)

- Weinan E, Bjorn Engquist, Xiantao Li, Weiqing Ren, Eric Vanden-Eijnden

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,… (More)

- Weinan E, Di Liu, Eric Vanden-Eijnden
- The Journal of chemical physics
- 2005

An efficient simulation algorithm for chemical kinetic systems with disparate rates is proposed. This new algorithm is quite general, and it amounts to a simple and seamless modification of the classical stochastic simulation algorithm (SSA), also known as the Gillespie [J. Comput. Phys. 22, 403 (1976); J. Phys. Chem. 81, 2340 (1977)] algorithm. The basic… (More)

- Weinan E, Di Liu, Eric Vanden-Eijnden
- J. Comput. Physics
- 2007

We present an efficient numerical algorithm for simulating chemical kinetic systems with multiple time scales. This algorithm is an improvement of the traditional stochastic simulation algorithm (SSA), also known as Gillespie's algorithm. It is in the form of a nested SSA and uses an outer SSA to simulate the slow reactions with rates computed from… (More)

Numerical schemes for systems with multiple spatio-temporal scales are investigated. The multiscale schemes use asymptotic results for this type of systems which guarantee the existence of an effective dynamics for some suitably defined modes varying slowly on the largest scales. The multiscale schemes are analyzed in general, then illustrated on a specific… (More)