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Efficient Parallel Solution of Nonlinear Parabolic Partial Differential Equations by a Probabilistic Domain Decomposition

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Abstract

Initial- and initial-boundary value problems for nonlinear one-dimensional parabolic partial differential equations are solved numerically by a probabilistic domain decomposition method. This is based on a probabilistic representation of solutions by means of branching stochastic processes. Only few values of the solution inside the space-time domain are generated by a Monte Carlo method, and an interpolation is then made so to approximate suitable interfacial values of the solution inside the domain. In this way, a fully decoupled set of sub-problems is obtained. This method allows for an efficient massively parallel implementation, is scalable and fault tolerant. Numerical examples, including some for the KPP equation and beyond are given to show the performance of the algorithm.

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References

  1. Acebrón, J.A., Busico, M.P., Lanucara, P., Spigler, R.: Domain decomposition solution of elliptic boundary-value problems. SIAM J. Sci. Comput. 27(2), 440–457 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  2. Acebrón, J.A., Busico, M.P., Lanucara, P., Spigler, R.: Probabilistically induced domain decomposition methods for elliptic boundary-value problems. J. Comput. Phys. 210(2), 421–438 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  3. Acebrón, J.A., Spigler, R.: Supercomputing applications to the numerical modeling of industrial and applied mathematics problems. J. Supercomput. 40, 67–80 (2007)

    Article  Google Scholar 

  4. Acebrón, J.A., Spigler, R.: A fully scalable parallel algorithm for solving elliptic partial differential equations. In: Lect. Notes in Comput. Sci., vol. 4641, pp. 727–736. Springer, Berlin (2007)

    Google Scholar 

  5. Acebrón, J.A., Rodríguez-Rozas, A., Spigler, R.: Domain decomposition solution of nonlinear two-dimensional parabolic problems by random trees. J. Comput. Phys. 228(15), 5574–5591 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  6. Arnold, L.: Stochastic Differential Equations: Theory and Applications. Wiley, New York (1974)

    MATH  Google Scholar 

  7. Brazhnik, P.K., Tyson, J.J.: On travelling wave solutions of Fishers equation in two spatial dimensions. SIAM J. Appl. Math. 60, 371–391 (1999)

    MathSciNet  Google Scholar 

  8. Buchmann, F.M.: Simulation of stopped diffusions. J. Comput. Phys. 202, 446–462 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  9. Chan, T.F., Mathew, T.P.: Domain decomposition algorithms. Acta Numer. 61–143 (1994) [Cambridge University Press, Cambridge, 1994]

  10. Chauvin, B., Rouault, A.: A stochastic simulation for solving scalar reaction-diffusion equations. Adv. Appl. Probab. 22, 88–100 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  11. Chorin, A.J.: Numerical study of slightly viscous flow. J. Fluid Mech. 57, 785–796 (1973)

    Article  MathSciNet  Google Scholar 

  12. Dongarra, J., Fox, G., Kennedy, K., Torczon, L., Gropp, W., Foster, I., White, A.: The Sourcebook of Parallel Computing. Morgan Kaufmann, San Mateo (2002)

    Google Scholar 

  13. DuChateau, P., Zachmann, D.: Applied Partial Differential Equations. Dover, New York (2002)

    Google Scholar 

  14. Freidlin, M.: Functional Integration and Partial Differential Equations. Annals of Mathematics Studies, vol. 109. Princeton Univ. Press, Princeton (1985)

    MATH  Google Scholar 

  15. Floriani, E., Lima, R., Vilela Mendes, R.: Poisson-Vlasov: Stochastic representation and numerical codes. Eur. Phys. J. D 46, 295–302 (2008)

    Article  Google Scholar 

  16. Jansons, K.M., Lythe, G.D.: Efficient numerical solution of stochastic differential equations using exponential timestepping. J. Stat. Phys. 100(5/6), 1097–1109 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  17. Jansons, K.M., Lythe, G.D.: Exponential timestepping with boundary test for stochastic differential equations. SIAM J. Sci. Comput. 24, 1809–1822 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  18. Kalos, M.H., Withlock, P.A.: Monte Carlo Methods, Vol. I: Basics. Wiley, New York (1986)

    Book  Google Scholar 

  19. Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus, 2nd edn. Springer, Berlin (1991)

    MATH  Google Scholar 

  20. Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations. Springer, Berlin (1992)

    MATH  Google Scholar 

  21. Lapeyre, B., Pardoux, E., Sentis, R.: Introduction to Monte-Carlo Methods for Transport and Diffusion Equations. Oxford Univ. Press, London (2003)

    MATH  Google Scholar 

  22. Le Jan, Y., Sznitman, A.S.: Stochastic cascades and 3-dimensional Navier-Stokes equations. Probab. Theory Relat. Fields 109, 343–336 (1997)

    Article  MATH  Google Scholar 

  23. McKean, H.P.: Application of Brownian motion to the equation of Kolmogorov-Petrovskii-Piskunov. Commun. Pure Appl. Math. 28, 323–331 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  24. Milstein, G.N., Tretyakov, M.V.: Stochastic Numerics for Mathematical Physics. Springer, Berlin (2004)

    MATH  Google Scholar 

  25. Petersen, W., Arbenz, P.: Introduction to Parallel Computing. A Practical Guide with Examples in C. Oxford Univ. Press, London (2004)

    MATH  Google Scholar 

  26. Peirano, E., Talay, D.: Domain decomposition by stochastic methods. In: Herrera, I., Keyes, D.E., Widlund, O.B., Yates, R. (eds.) Domain Decomposition Methods in Science and Engineering, Proc. of the 14th Int. Conf. on Domain Decomposition Methods, Natl. Auton. Univ. Mex., Cocoyoc, México, 2002, pp. 131–147 (2003) (electronic)

  27. Quarteroni, A., Valli, A.: Domain Decomposition Methods for Partial Differential Equations. Oxford Science Publications. Clarendon Press, Oxford (1999)

    MATH  Google Scholar 

  28. Ramirez, J.M.: Multiplicative cascades applied to PDEs (two numerical examples). J. Comput. Phys. 214, 122–136 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  29. Regnier, H., Talay, D.: Vitesse de convergence d’une méthode particulaire stochastique avec branchements. C. R. Acad. Sci. Paris, Sér. I, Math. 332, 933–938 (2001)

    MATH  MathSciNet  Google Scholar 

  30. Regnier, H., Talay, D.: Convergence rate of the Sherman and Peskin branching stochastic particle method. Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 460, 199–220 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  31. Sherman, A.S., Peskin, C.S.: A Monte Carlo method for scalar reaction diffusion equations. SIAM J. Sci. Stat. Comput. 7, 1360–1372 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  32. Sherman, A., Mascagni, M.: A gradient random walk method fr two-dimensional reaction-diffusion equations. SIAM J. Sci. Comput. 15, 1280–1293 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  33. Talay, D.: 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, pp. 48–196. Springer, Berlin (1996)

    Chapter  Google Scholar 

  34. Waymire, E.: Probability and incompressible Navier-Stokes equations: An overview of some recent developments. Probab. Surv. 2, 1–32 (2005)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Center for Mathematics and its Applications, Department of Mathematics, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001, Lisbon, Portugal

    Juan A. Acebrón & Ángel Rodríguez-Rozas

  2. Dipartimento di Matematica, Università “Roma Tre”, Largo S.L. Murialdo 1, 00146, Rome, Italy

    Renato Spigler

Authors
  1. Juan A. Acebrón
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  2. Ángel Rodríguez-Rozas
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  3. Renato Spigler
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Correspondence to Juan A. Acebrón.

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Acebrón, J.A., Rodríguez-Rozas, Á. & Spigler, R. Efficient Parallel Solution of Nonlinear Parabolic Partial Differential Equations by a Probabilistic Domain Decomposition. J Sci Comput 43, 135–157 (2010). https://doi.org/10.1007/s10915-010-9349-2

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  • DOI: https://doi.org/10.1007/s10915-010-9349-2

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