Analysis of multiscale methods for stochastic differential equations

Communications on Pure and Applied Mathematics

Volumn 58, Issue 11, 2005, Pages 1544-1585

  E, Weinan ^a   Liu, Di ^b   Vanden Eijnden, Eric ^b  

^a PRINCETON UNIVERSITY   (United States)

^b NEW YORK UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 27744467603     PISSN: 00103640     EISSN: None     Source Type: Journal    
DOI: 10.1002/cpa.20088     Document Type: Article

References (28)

1. Car, R.; Parrinello, M. Unified approach for molecular dynamics and density-functional theory. Phys. Rev. Lett. 55 (1985), no. 22, 2471-2474.
2. Chorin, A. J. Computational fluid mechanics. Selected papers. Academic Press, Boston, 1989.
3. Chorin, A. J.; Kast, A. P.; Kupferman, R. Optimal prediction of underresolved dynamics. Proc. Natl. Acad. Sci. USA 95 (1998), no. 8, 4094-4098.
4. Chorin, A. J.; Kast, A. P.; Kupferman, R. Unresolved computation and optimal predictions. Comm. Pure Appl. Math. 52 (1999), no. 10, 1231-1254.
5. Da Prato, G.; Zabczyk, J. Stochastic equations in infinite dimensions. Encyclopedia of Mathematics and Its Applications, 44. Cambridge University Press, Cambridge, 1992.
6. E, W. Analysis of the heterogeneous multiscale method for ordinary differential equations. Commun. Math. Sci. 1 (2003), no. 3, 423-436.
7. E, W.; Engquist, B. The heterogeneous multiscale methods. Commun. Math. Sci. 1 (2003), no. 1, 87-132.
8. E, W.; Engquist, B. Multiscale modeling and computation. Notices Amer. Math. Soc. 50 (2003), no. 9, 1062-1070.
9. E, W.; Ming, P. B.; Zhang, P. W. Analysis of the heterogeneous multiscale method for elliptic homogenization problems. J. Amer. Math. Soc. 18, no. 1, 121-156.
10. Engquist, B.; Tsai, R. The heterogeneous multiscale method for stiff ODEs. Math. Comp., in press.
11. Fatkullin, I.; Vanden-Eijnden, E. A computational strategy for multiscale systems with applications to Lorenz 96 model. J. Comput. Phys. 200 (2004), no. 2, 605-638.
12. Freidlin, M. I.; Wentzell, A. D. Random perturbations of dynamical systems. Second edition. Grundlehren der Mathematischen Wissenschaften, 260. Springer-Verlag, New York, 1998.
13. Gear, C. W.; Kevrekidis, I. G. Projective methods for stiff differential equations: problems with gaps in their eigenvalue spectrum. SIAM J. Sci. Comput. 24 (2003), no. 4, 1091-1106.
14. Hasḿinskiǐ, R. Z. Stochastic stability of differential equations. Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics and Analysis, 7. Sijthoff & Noordhoff, Alphen aan den Rijn-Germantown, Md., 1980.
15. Kloeden, P. E.; Platen, E. Numerical solution of stochastic differential equations. Applications of Mathematics (New York), 23. Springer-Verlag, Berlin, 1992.
16. Kolesik, M.; Novotny, M. A.; Rikvold, P. A. Projection method for statics and dynamics of lattice spin systems. Phys. Rev. Lett. 80 (1998), no. 15, 3384-3387.
17. Kurtz, T. G. A limit theorem for perturbed operator semigroups with applications to random evolutions. J. Functional Analysis 12 (1973), 55-67.
18. Lamba, H.; Mattingly, J. C.: Stuart, A. M. An adaptive Euler-Maruyama scheme for SDEs: Part I, convergence, Part II, stability. Preprint, 2004.
19. Lebedev, V. I.; Finogenov, S. I. Explicit methods of second order for the solution of stiff systems of ordinary differential equations. Zh. Vychisl. Mat. Mat. Fiz. 16 (1976), 895-910.
20. Majda, A. J.; Timofeyev, I.; Vanden-Eijnden, E. A mathematical framework for stochastic climate models. Comm. Pure Appl. Math. 54 (2001), no. 8, 891-974.
21. Meyn, S. P.; Tweedie, R. L. Stability of Markovian processes. I. Criteria for discrete-time chains. Adv. in Appl. Probab. 24 (1992), no. 3, 542-574.
22. Continuous-time processes and sampled chains. Adv. in Appl. Probab. 25 (1993), no. 3, 487-517.
23. Foster-Lyapunov criteria for continuous-time processes. Adv. in Appl. Probab. 25 (1993), no. 3, 518-548.
24. Mikulyavichyus, R. Some properties of solutions of stochastic differential equations. Litovsk. Mat. Sb. 23 (1983), no. 4, 18-31.
25. Ming, P. B.; Zhang, P. W. Analysis of the heterogeneous multiscale method for parabolic homogenization problems. Submitted to Math. Comp.
26. Papanicolaou, G. C. Introduction to the asymptotic analysis of stochastic equations. Modern modeling of continuum phenomena (Ninth Summer Sem. Appl. Math., Rensselaer Polytech. Inst., Troy, N.Y., 1975), 109-147. Lectures in Applied Mathematics, Vol. 16. American Mathematical Society, Providence, R.I., 1977.
27. Talay, D. Second order discretization schemes of stochastic differential systems for the computation of the invariant law. Stock. Stock. Rep. 29 (1990), 13-36.
28. Vanden-Eijnden, E. Numerical techniques for multi-scale dynamical systems with stochastic effects. Commun. Math. Sci. 1 (2003), no. 2, 385-391.