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Numerical Solution of the Robin Problem of Laplace Equations with a Feynman–Kac Formula and Reflecting Brownian Motions

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Abstract

In this paper, we present numerical methods to implement the probabilistic representation of third kind (Robin) boundary problem for the Laplace equations. The solution is based on a Feynman–Kac formula for the Robin problem which employs the standard reflecting Brownian motion (SRBM) and its boundary local time arising from the Skorokhod problem. By simulating SRBM paths through Brownian motion using Walk on Spheres method, approximation of the boundary local time is obtained and the Feynman–Kac formula is then calculated by evaluating the average of all path integrals over the boundary under a measure defined through the local time. Numerical results demonstrate the accuracy and efficiency of the proposed method for finding a local solution of the Laplace equations with Robin boundary conditions.

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References

  1. Audus, D.J., Hassan, A.M., Garboczi, E.J., Douglas, J.F.: Interplay of particle shape and suspension properties: a study of cube-like particles. Soft Matter 11(17), 3360–3366 (2015)

    Article  Google Scholar 

  2. Binder, I., Braverman, M.: The rate of convergence of the walk of sphere algorithm. Geom. Funct. Anal. 22, 558–587 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Burdzy, K., Chen, Z., Sylvester, J.: The heat equation and reflected Brownian motion in time-dependent domains. Annu. Probab. 32(1B), 775–804 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chung, K.L.: Green, Brown, and Probability and Brownian Motion on the Line. World Scientific Pub Co Inc, Singapore (2002)

    Book  MATH  Google Scholar 

  5. Douglas, J.F.: Integral equation approach to condensed matter relaxation. J. Phys. Condens. Matter 11(10A), A329 (1999)

    Article  Google Scholar 

  6. Feynman, R.P.: Space-time approach to nonrelativistic quantum mechanics. Rev. Mod. Phys. 20, 367–387 (1948)

    Article  MathSciNet  Google Scholar 

  7. Freidlin, M.: Functional Integration and Partial Differential Equations. Princeton University Press, Princeton (1985)

    Book  MATH  Google Scholar 

  8. Hsu, (Elton) P.: Reflecting Brownian motion, boundary local time and the Neumann problem, Dissertation Abstracts International Part B: Science and Engineering [DISS. ABST. INT. PT. B- SCI. ENG.], Vol. 45, No. 6 (1984)

  9. Hwang, C.O., Mascagni, M., Given, J.A.: A Feynman–Kac path-integral implementation for Poisson’s equation using an h-conditioned Green’s function. Math. Comput. Simul. 62(3), 347–355 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kac, M.: On distributions of certain Wiener functionals. Trans. Am. Math. Soc. 65, 1–13 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kac, M.: On some connections between probability theory and differential and integral equations. In: Proceedings of 2nd Berkeley Symposium Math. Stat. and Probability, vol. 65, pp. 189–215 (1951)

  12. Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus. Springer, New York (1988)

    Book  MATH  Google Scholar 

  13. Lejay, A., Maire, S.: New Monte Carlo schemes for simulating diffusions in discontinuous media. J. Comput. Appl. Math. 245, 97–116 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lions, P.L., Sznitman, A.S.: Stochastic differential equations with reflecting boundary conditions. Commun. Pure Appl. Math. 37(4), 511–537 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  15. Maire, S., Tanré, E.: Monte Carlo approximations of the Neumann problem. Monte Carlo Methods Appl. 19(3), 201–236 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Morillon, J.-P.: Numerical solutions of linear mixed boundary value problems using stochastic representations. Int. J. Numer. Methods Eng. 40, 387–405 (1997)

    Article  MathSciNet  Google Scholar 

  17. Müller, M.E.: Some continuous Monte Carlo methods for the Dirichlet problem. Ann. Math. Stat. 27(3), 569–589 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  18. Øksendal, B.: Stochastic Differential Equations. Springer, Berlin (2003)

    Book  MATH  Google Scholar 

  19. Papanicolaou, V.G.: The probabilistic solution of the third boundary value problem for second order elliptic equations. Probab. Theory Relat. Fields 87, 27–77 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  20. Sabelfeld, K.K., Simonov, N.A.: Random walks on boundary for solving PDEs, Walter de Gruyter (1994)

  21. Skorokhod, A.V.: Stochastic equations for diffusion processes in a bounded region. Theory Probab. Appl. 6(3), 264–274 (1961)

    Article  MATH  Google Scholar 

  22. Souza de Cursi, J.E.: Numerical methods for linear boundary value problems based on Feynman–Kac representations. Math. Comput. Simul. 36(1), 1–16 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  23. Tanaka, H.: Stochastic differential equations with reflecting boundary condition in convex regions. Hiroshima Math. J. 9(1), 163–177 (1979)

    MathSciNet  MATH  Google Scholar 

  24. Yan, C., Cai, W., Zeng, X.: A parallel method for solving Laplace equations with Dirichlet data using local boundary integral equations and random walks. SIAM J. Sci. Comput. 35(4), B868–B889 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Zhou, Y., Cai, W., Hsu, (Elton) P.: Local Time of Reflecting Brownian Motion and Probabilistic Representation of the Neumann Problem, Preprint (2015)

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Acknowledgments

The authors Y.J.Z. and W.C. acknowledge the support of the National Science Foundation (DMS-1315128) and the National Natural Science Foundation of China (No. 91330110) for the work in this paper.

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  1. Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC, 28223-0001, USA

    Yijing Zhou & Wei Cai

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  1. Yijing Zhou
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  2. Wei Cai
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Correspondence to Wei Cai.

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Zhou, Y., Cai, W. Numerical Solution of the Robin Problem of Laplace Equations with a Feynman–Kac Formula and Reflecting Brownian Motions. J Sci Comput 69, 107–121 (2016). https://doi.org/10.1007/s10915-016-0184-y

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  • DOI: https://doi.org/10.1007/s10915-016-0184-y

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