A stochastic particle method for the Mckean-Vlasov and the burgers equation

Mathematics of Computation

Volumn 66, Issue 217, 1997, Pages 157-192

  Bossy, Mireille ^a   Talay, Denis ^a  

^a INRIA   (France)

Author keywords

Burgers equation; Interacting particle systems; Stochastic particle methods

Indexed keywords

EID: 0031539944     PISSN: 00255718     EISSN: None     Source Type: Journal    
DOI: 10.1090/s0025-5718-97-00776-x     Document Type: Article

References (28)

1. P. Bernard, D. Talay, and L. Tubaro. Rate of convergence of a stochastic particle method for the Kolmogorov equation with variable coefficients. Math. Comp., 63(208):555-587, 1994. MR 95c:65139
2. M. Bossy. Vitesse de Convergence d'Algorithmes Particulaires Stochastiques et Application à l'Equation de Burgers. PhD thesis, Université de Provence, 1995.
3. M. Bossy and D. Talay. Convergence rate for the approximation of the limit law of weakly interacting particles: application to the Burgers equation. Ann. Appl. Probab., To appear.
4. P. Cannarsa and V. Vespri. Generation of analytic semigroups by elliptic operators with unbounded coefficients. SIAM J. Math. Anal., 18:857-872, 1987. MR 88d:47053
5. A.J. Chorin and J.E. Marsden. A Mathematical Introduction to Fluid Mechanics. Springer Verlag, New York, 1993. MR 94c:76002
6. A.L. Chorin. Vortex methods and Vortex Statistics - Lectures for Les Houches Summer School of Theoretical Physics. Lawrence Berkeley Laboratory Prepublications, 1993.
7. R.L. Dobrushin. Prescribing a system of random variables by conditional distributions. Theory Probab. Appl., 3:469, 1970.
8. A. Friedman. Stochastic Differential Equations and Applications, vol. 1. Academic Press, New York, 1975. MR 58:13350a
9. J. Gärtner. On the Mckean-Vlasov limit for interacting diffusions. Math. Nachr., 137:197-248, 1988. MR 90a:60184
10. J. Goodman. Convergence of the random vortex method. Comm. Pure Appl. Math., 40:189-220, 1987. MR 88d:35159
11. K.E. Gustafson and J.A. Sethian (eds.). Vortex Methods and Vortex Motion. SIAM, 1991. MR 93f:76078
12. O.H. Hald. Convergence of random methods for a reaction diffusion equation. SIAM J. Sci. Statist. Comput., 2:85-94, 1981. MR 83c:65210
13. _, Convergence of a random method with creation of vorticity. SIAM J. Sci. Statist. Comput., 7:1373-1386, 1986.
14. xx. Comm. Pure Appl. Math., 3:201-230, 1950. MR 13:846c
15. I. Karatzas and S.E. Shreve. Brownian Motion and Stochastic Calculus. Springer-Verlag, New York, 1988. MR 89c:60096
16. S. N. Kružkov. First order quasilinear equation in several independent variables. Math. USSR Sbornik, 10:217-243, 1970. MR 42:2159
17. C. Léonard. Une loi des grands nombres pour des systèmes de diffusions avec interaction et à coefficients non bornés. Ann. Inst. Henri Poincaré, 22(2):237-262, 1986. MR 88f:60136
18. D.G. Long. Convergence of the random vortex method in two dimensions. J. Amer. Math. Soc., 1(4), 1988. MR 90a:65202
19. C. Marchioro and M. Pulvirenti. Hydrodynamics in two dimensions and vortex theory. Comm. Math. Phys., 84:483-503, 1982. MR 84e:35126
20. S. Méléard and S. Roelly-Coppoletta. A propagation of chaos result for a system of particles with moderate interaction. Stochastic Process. Appl., 26:317-332, 1987. MR 89e:60201
21. M. Métivier. Quelques problèmes liés aux systèmes infinis de particules et leurs limites. Lectures Notes in Math., 1204:426-446, 1984. MR 90h:60100
22. G.N. Milshtein. Approximate integration of stochastic differential equations. Theory Probab. Appl., 19:557-562, 1974.
23. K. Oelschläger. A martingale approach to the law of large numbers for weakly interacting stochastic processes. Ann. Probab., 12:458-479, 1984. MR 85j:60191
24. E.G. Puckett. A study of the vortex sheet method and its rate of convergence. SIAM J. Sci. Statist. Comput., 10(2):298-327, 1989. MR 90c:76016
25. _, Convergence of a random particle method to solutions of the Kolmogorov equation. Math. Comp., 52(186):615-645, 1989. MR 90h:65008
26. P.A. Raviart. An analysis of particle methods. In F. Brezzi, ed., Numerical Methods in Fluid Dynamics, vol. 1127 of Lecture Notes in Math., pp. 243-324. Springer-Verlag, Berlin, Heidelberg, New York, 1985. MR 87h:76010
27. S. Roberts. Convergence of a random walk method for the Burgers equation. Math. Comp., 52(186):647-673, 1989. MR 89i:65090
28. A.-S. Sznitman. Topics in propagation of chaos. In P.L. Hennequin, ed., Ecole d'Eté de Probabilités de Saint-Flour XIX - 1989, vol. 1464 of Lecture Notes in Math., pp. 165-251. Springer-Verlag, Berlin, Heidelberg, New York, 1991. MR 93b:60179