×

Analysis of multiscale methods for stochastic differential equations. (English) Zbl 1080.60060

The paper studies numerical methods for multi-scale dynamical systems modelled by stochastic differential equations. Strong as well as weak numerical approximations are investigated and their convergence is proven. Some of the results are optimal in a certain sense. Both advective and diffusive time scales are covered.

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
65C20 Probabilistic models, generic numerical methods in probability and statistics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Car, Phys Rev Lett 55 pp 2471– (1985)
[2] Computational fluid mechanics. Selected papers. Academic Press, Boston, 1989. · Zbl 0699.76003
[3] Chorin, Proc Natl Acad Sci USA 95 pp 4094– (1998)
[4] Chorin, Comm Pure Appl Math 52 pp 1231– (1999)
[5] ; Stochastic equations in infinite dimensions. Encyclopedia of Mathematics and Its Applications, 44. Cambridge University Press, Cambridge, 1992. · doi:10.1017/CBO9780511666223
[6] E, Commun Math Sci 1 pp 423– (2003) · Zbl 1088.65552 · doi:10.4310/CMS.2003.v1.n3.a3
[7] E, Commun Math Sci 1 pp 87– (2003) · Zbl 1093.35012 · doi:10.4310/CMS.2003.v1.n1.a8
[8] E, Notices Amer Math Soc 50 pp 1062– (2003)
[9] E, J Amer Math Soc 18 pp 121–
[10] Engquist, Math Comp
[11] Fatkullin, J Comput Phys 200 pp 605– (2004)
[12] ; Random perturbations of dynamical systems. Second edition. Grundlehren der Mathematischen Wissenschaften, 260. Springer-Verlag, New York, 1998. · doi:10.1007/978-1-4612-0611-8
[13] Gear, SIAM J Sci Comput 24 pp 1091– (2003)
[14] 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. · doi:10.1007/978-94-009-9121-7
[15] ; Numerical solution of stochastic differential equations. Applications of Mathematics (New York), 23. Springer-Verlag, Berlin, 1992. · doi:10.1007/978-3-662-12616-5
[16] Kolesik, Phys Rev Lett 80 pp 3384– (1998)
[17] Kurtz, J Functional Analysis 12 pp 55– (1973)
[18] ; : An adaptive Euler-Maruyama scheme for SDEs: Part I, convergence, Part II, stability. Preprint, 2004.
[19] Lebedev, Zh Vychisl Mat Mat Fiz 16 pp 895– (1976)
[20] Majda, Comm Pure Appl Math 54 pp 891– (2001)
[21] Meyn, Adv in Appl Probab 24 pp 542– (1992)
[22] 25 (1993), 487-548.
[23] Mikulyavichyus, Litovsk Mat Sb 23 pp 18– (1983)
[24] Ming, Math Comp
[25] 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.
[26] Talay, Stoch Stoch Rep 29 pp 13– (1990) · Zbl 0697.60066 · doi:10.1080/17442509008833606
[27] Vanden-Eijnden, Commun Math Sci 1 pp 385– (2003) · Zbl 1088.60060 · doi:10.4310/CMS.2003.v1.n2.a11
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.