Skip to main content
SpringerLink
Log in

Probabilistic numerical methods for partial differential equations: Elements of analysis

  • Chapter
  • First Online:
Probabilistic Models for Nonlinear Partial Differential Equations

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 1627))

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 54.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 69.95
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bally, V., Talay, D., “The law of the Euler scheme for stochastic differential equations (I): convergence rate of the distribution function”, Probability Theory and Related Fields, 104, 43–60 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  2. Bally, V., Talay, D., “The law of the Euler scheme for stochastic differential equations (II): convergence rate of the density”, Rapport de Recherche 2675, INRIA (1995). To appear in Monte Carlo Methods and Applications.

    Google Scholar 

  3. Bensoussan A., Lions, J.L., Applications des Inéquations Variationnelles en Contrôle Stochastique, Dunod (1978).

    Google Scholar 

  4. Bernard, P., Talay, D., Tubaro, L., “Rate of convergence of a stochastic particle method for the Kolmogorov equation with variable coefficients”, Math. Comp., 63(208), 557–587 (1994).

    MathSciNet  MATH  Google Scholar 

  5. Bossy, M., Vitesse de Convergence d'Algorithmes Particulaires Stochastiques et Application à l'Equation de Burgers, PhD thesis, Université de Provence (1995).

    Google Scholar 

  6. Bossy, M., Talay, D., “Convergence rate for the approximation of the limit law of weakly interacting particles: application to the Burgers equation” (1995). To appear in Ann. Appl. Probab..

    Google Scholar 

  7. Bossy, M., Talay, D., “A stochastic particle method for the McKean-Vlasov and the Burgers equation” (1995). To appear in Math. Comp.

    Google Scholar 

  8. Calzolari, A., Costantini, C., Marchetti F., “A confidence interval for Monte Carlo methods with an application to simulation of obliquely reflecting Brownian motion”, Stochastic Processes and their Applications, 29, 209–222 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  9. Cannarsa, P., Vespri, C., “Generation of analytic semigroups by elliptic operators with unbounded coefficients”, SIAM J. Math. Anal., 18(3), 857–872 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  10. Chorin, A.J., Marsden, J.E., A Mathematical Introduction to Fluid Mechanics, Springer Verlag (1993).

    Google Scholar 

  11. Chorin, A.J., “Vortex methods and Vortex Statistics—Lectures for Les Houches Summer School of Theoretical Physics”, Lawrence Berkeley Laboratory Prepublications (1993).

    Google Scholar 

  12. Costantini, C., Pacchiarotti, B., SARTORETTO, F., “Numerical approximation of functionals of diffusion processes”, (1995).

    Google Scholar 

  13. Faure, O., Simulation du Mouvement Brownien et des Diffusions, PhD thesis, Ecole Nationale des Ponts et Chaussées (1992).

    Google Scholar 

  14. Freidlin, M., Functional Integration and Partial Differential Equations. Annals of Mathematics Studies, Princeton University (1985).

    Google Scholar 

  15. Friedman, A., Stochastic Differential Equations and Applications, volume 1, Academic Press, New-York (1975).

    MATH  Google Scholar 

  16. Gaines, J.G., Lyons, T.J., “Random generation of stochastic area integrals”, SIAM Journal of Applied Mathematics, 54(4), 1132–1146 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  17. Goodman, J., “Convergence of the random vortex method”, Comm. Pure Appl. Math., 40, 189–220 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  18. Gustafson, K.E., Sethian, J.A., (eds.), Vortex Methods and Vortex Motions, SIAM (1991).

    Google Scholar 

  19. Hald, O.H., “Convergence of random methods for a reaction diffusion equation”, SIAM J. Sci. Stat. Comput., 2, 85–94 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  20. Hald, O.H., “Convergence of a random method with creation of vorticity”, SIAM J. Sci. Stat. Comput., 7, 1373–1386 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  21. Has'minskii, R.Z., Stochastic Stability of Differential Equations, Sijthoff & Noordhoff (1980).

    Google Scholar 

  22. Ikeda, N., Watanabe, S., Stochastic Differential Equations and Diffusion Processes, North Holland (1981).

    Google Scholar 

  23. Kanagawa, S., “On the rate of convergence for Maruyama's approximate solutions of stochastic differential equations”, Yokohama Math. J., 36, 79–85 (1988).

    MathSciNet  MATH  Google Scholar 

  24. Kunita, H., “Stochastic differential equations and stochastic flows of diffeomorphisms”, Ecole d'Été de Saint-Flour XII, LNM 1097, Springer (1984).

    Google Scholar 

  25. Kurtz, T.G., Protter, P., “Wong-Zakai corrections, random evolutions and numerical schemes for S.D.E.'s”, in Stochastic Analysis: Liber Amicorum for Moshe Zakai, 331–346 (1991).

    Google Scholar 

  26. Kusuoka, S., Stroock, D., “Applications of the Malliavin Calculus, part II”, J. Fac. Sci. Univ. Tokyo, 32, 1–76 (1985).

    MathSciNet  MATH  Google Scholar 

  27. Lépingle, D., “Un schéma d'Euler pour équations différentielles réfléchies”, Note aux Comptes-Rendus de l'Académie des Sciences, 316(I), 601–605 (1993).

    Google Scholar 

  28. Lépingle, D., “Euler scheme for reflected stochastic differential equations”, Mathematics and Computers in Simulation, 38 (1995).

    Google Scholar 

  29. Liu, Y., “Numerical approaches to reflected diffusion processes”, (submitted for publication).

    Google Scholar 

  30. Long, D.G., Convergence of the random vortex method in two dimensions, J. Amer. Math. Soc., 1(4) (1988).

    Google Scholar 

    Google Scholar 

  31. Marchioro, C., Pulvirenti, M., “Hydrodynamics in two dimensions and vortex theory”, Comm. Math. Phys., 84, 483–503 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  32. Milshtein, G.N., “The solving of the boundary value problem for parabolic equation by the numerical integration of stochastic equations” (to appear).

    Google Scholar 

  33. Newton, N.J., “Variance reduction for simulated diffusions”, SIAM J. Appl. Math., 54(6), 1780–1805 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  34. Nualart, D., Malliavin Calculus and Related Topics. Probability and its Applications, Springer-Verlag (1995).

    Google Scholar 

  35. Osada, H., “Propagation of chaos for the two dimensional Navier-Stokes equation”, in K.Itô and N. Ikeda, editors, Probabilistic Methods in Mathematical Physics, pages 303–334, Academic Press (1987).

    Google Scholar 

  36. Pardoux, E., “Filtrage non linéaire et équations aux dérivées partielles stochastiques associées”, Cours à l'Ecole d'Eté de Probabilités de Saint-Flour XIX, LNM 1464, Springer-Verlag (1991).

    Google Scholar 

  37. Protter, P., Stochastic Integration and Differential Equations, Springer-Verlag, Berlin (1990).

    Book  MATH  Google Scholar 

  38. Protter, P., Talay, D., “The Euler scheme for Lévy driven stochastic differential equations”, Rapport de Recherche 2621, INRIA, (1995). To appear in Annals Prob.

    Google Scholar 

  39. Puckett, E.G., A study of the vortex sheet method and its rate of convergence. SIAM J. Sci. Stat. Comput., 10(2), 298–327 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  40. Puckett, E.G., “Convergence of a random particle method to solutions of the Kolmogorov equation”, Math. Comp., 52(186), 615–645 (1989).

    MathSciNet  MATH  Google Scholar 

  41. Raviart, P.A., “An analysis of particle methods”, in Numerical Methods in Fluid Dynamics, F. Brezzi (ed.), vol 1127 of Lecture Notes in Math., 243–324. Springer-Verlag (1985).

    Google Scholar 

  42. Roberts, S., “Convergence of a random walk method for the Burgers equation”, Math. Comp., 52(186), 647–673 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  43. Roynette, B., “Approximation en norme Besov de la solution d'une équation différentielle stochastique”, Stochastics and Stochastic Reports, 49, 191–209 (1994).

    Article  MathSciNet  Google Scholar 

  44. Slominski, L., “On existence, uniqueness and stability of solutions of multidimensional SDE's with reflecting boundary conditions”, Ann. Inst. H. Poincaré, 29 (1993).

    Google Scholar 

  45. Slominski, L., “On approximation of solutions of multidimensional S.D.E.'s with reflecting boundary conditions”, Stochastic Processes and their Applications, 50(2), 197–219 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  46. Sznitman, A.S., “Topics in propagation of chaos”, Ecole d'Eté de Probabilités de Saint Flour XIX, (P.L. Hennequin, ed.), LNM 1464, Springer, Berlin, Heidelberg, New York (1989).

    Google Scholar 

  47. Talay, D., “Second order discretization schemes of stochastic differential systems for the computation of the invariant law”, Stochastics and Stochastic Reports, 29(1), 13–36 (1990).

    Article  MATH  Google Scholar 

  48. Talay, D., “Simulation and numerical analysis of stochastic differential systems: a review”, Probabilistic Methods in Applied Physics, (P. Krée and W. Wedig, eds.), Lecture Notes in Physics 451, 54–96, Springer-Verlag (1995).

    Google Scholar 

  49. Talay, D., Tubaro, L., Probabilistic Numerical Methods for Partial Differential Equations, (book in preparation).

    Google Scholar 

  50. Talay, D., Tubaro, L., “Expansion of the global error for numerical schemes solving stochastic differential equations”, Stochastic Analysis and Applications, 8(4), 94–120 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  51. Tweedie, R.L., “Sufficient conditions for ergodicity and recurrence of markov chains of a general state space”, Stochastic Processes and Applications, 3 (1975). *** DIRECT SUPPORT *** A00I6B81 00004

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. INRIA, 2004 Route des Lucioles, BP 93, F-06902, Sophia-Antipolis Cedex, France

    Denis Talay

Authors
  1. Denis Talay
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Denis Talay Luciano Tubaro

Rights and permissions

Reprints and permissions

Copyright information

© 1996 Springer-Verlag

About this chapter

Cite this chapter

Talay, D. (1996). Probabilistic numerical methods for partial differential equations: Elements of analysis. In: Talay, D., Tubaro, L. (eds) Probabilistic Models for Nonlinear Partial Differential Equations. Lecture Notes in Mathematics, vol 1627. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0093180

Download citation

  • DOI: https://doi.org/10.1007/BFb0093180

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-61397-8

  • Online ISBN: 978-3-540-68513-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation