TY - JOUR

T1 - NUMERICAL TECHNIQUES FOR MULTI-SCALE DYNAMICAL SYSTEMS WITH STOCHASTIC EFFECTS

AU - Vanden-Eijnden, Eric

N1 - Funding Information:
The efficiency of the method can be analyzed as in the deterministic case, by comparing the number of evaluations of the functions g and h in (3.1), and of g, G, and h in (3.9) – to that should be added the cost of computing (3.17), which requires mM + mM2 + 21m2M2 function evaluations, and taking the square root of a(x, ε), which requires 61n3 operations; we assume that the cost associated with both is negligible. (3.1) must be solved using a micro-time-step δs = ε2∆τ, and R = ⌊∆s/ε2∆τ⌋ = O(ε−2) micro-time-steps are required to advance by one macro-time-step ∆s; hence, the number of evaluations of g and h is 2mR. On the other hand, the number of evaluations of g, G, and h simply is 3mM using the scheme based on (3.9). Therefore, as soon as M < 23R = O(ε−2), it becomes advantageous to use the new scheme instead of integrating (3.1) directly. (Note also that this is the worst case scenario, and it will usually be possible to improve it greatly by taking advantage of the special symmetries of the problem under consideration – for instance, if divxf(x,y,ε) = 0, then b2(x,ε) = 0, or if f(x,y,ε) = f(y,ε), then b2(x,ε) = 0 and a(x, ε) = a(ε) must be computed only once, etc.) Acknowledgments. The research reported here originated from discussions with Weinan E at a meeting at Oberwolfach in 1999; I thank Weinan for sharing many thoughts about these approaches since that time, and encouraging me to eventually write the present paper. I also thank Andy Majda for many interesting discussions. This work was supported in part by the NSF via grant DMS02-09959, and by the AMIAS (Association of Members of the Institute for Advanced Study).
Publisher Copyright:
© 2003 International Press

PY - 2003

Y1 - 2003

N2 - Numerical schemes are presented for dynamical systems with multiple time-scales. 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 become important, and stochastic differential equations are obtained. In both cases, the numerical methods compute the evolution of the slow variables without having to derive explicitly the effective equations beforehand; rather, the coefficients entering these equations are obtained on the fly using short simulations of appropriate auxiliary systems

AB - Numerical schemes are presented for dynamical systems with multiple time-scales. 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 become important, and stochastic differential equations are obtained. In both cases, the numerical methods compute the evolution of the slow variables without having to derive explicitly the effective equations beforehand; rather, the coefficients entering these equations are obtained on the fly using short simulations of appropriate auxiliary systems

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U2 - 10.4310/CMS.2003.v1.n2.a11

DO - 10.4310/CMS.2003.v1.n2.a11

M3 - Article

AN - SCOPUS:85128824735

SN - 1539-6746

VL - 1

SP - 385

EP - 391

JO - Communications in Mathematical Sciences

JF - Communications in Mathematical Sciences

IS - 2

ER -