A Matlab software for approximate solution of 2D elliptic problems by means of the meshless Monte Carlo random walk method

  • Sławomir Milewski
Open Access
Original Paper


This paper is devoted to the development of an innovative Matlab software, dedicated to the numerical analysis of two-dimensional elliptic problems, by means of the probabilistic approach. This approach combines features of the Monte Carlo random walk method with discretization and approximation techniques, typical for meshless methods. It allows for determination of an approximate solution of elliptic equations at the specified point (or group of points), without a necessity to generate large system of equations for the entire problem domain. While the procedure is simple and fast, the final solution may suffer from both stochastic and discretization errors. The attached Matlab software is based on several original author’s concepts. It permits the use of arbitrarily irregular clouds of nodes, non-homogeneous right-hand side functions, mixed type of boundary conditions as well as variable material coefficients (of anisotropic materials). The paper is illustrated with results of analysis of selected elliptic problems, obtained by means of this software.


Monte Carlo method Random walk technique Meshless methods Elliptic problems Finite difference method Implementation in Matlab 


Funding information

This research was supported by the National Science Centre, Poland, under the scientific project 2015/19/D/ST8/00816 (Computational coupled FEM / meshless FDM analysis dedicated to engineering non-stationary thermo-elastic and thermoplastic problems).


  1. 1.
    Metropolis, N., Ulam, S.: The Monte Carlo method. J. Am. Stat. Assoc. American Statistical Association 44(247), 335–341 (1949)CrossRefzbMATHGoogle Scholar
  2. 2.
    Forsythe, G.E., Leibler, R.A.: Matrix inversion by a Monte Carlo method. Math. Tab. Aids Comput. 4, 127–129 (1950)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Donsker, M.D., Kac, M.: A sampling method for determining the lowest eigenvalue and the principle eigenfunction of Schrödinger’s equation. J. Res. Natl. Bur. Stand. 44, 551–557 (1951)CrossRefGoogle Scholar
  4. 4.
    Curtiss, J.H.: Monte Carlo methods for the iteration of linear operators. J. Math. Phys. 32, 209–232 (1953)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Curtiss, J.H.: A theoretical comparison of the efficiencies of two classical methods and a Monte Carlo method for computing one component of the solution of a set of linear algebraic equations. In: Meyer, H.A. (ed.) Symp. Monte Carlo Methods, pp 191–233. Wiley, New York (1956)Google Scholar
  6. 6.
    Muller, M.E.: Some continuous Monte Carlo methods for the Dirichlet problem. Ann. Math. Stat 27, 569–589 (1956)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Reynolds, J.F.: A proof of the random-walk method for solving Laplace’s equation in 2-D. Math. Gaz. 49(370), 416–420 (1965). Mathematical AssociationCrossRefzbMATHGoogle Scholar
  8. 8.
    Haji-Sheikh, A., Sparrow, E.M.: The floating random walk and its application to Monte Carlo solutions of heat equations. J. SIAM, Appl. Math. 14(2), 370–389 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hoshino, S., Ichida, K.: Solution of partial differential equations by a modified random walk. Numer. Math 18, 61–72 (1971). SpringerMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Perrone, N., Kao, R.: A general finite difference method for arbitrary meshes. Comput. Struct. 5, 45–58 (1975)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Wyatt, M.J., Davies, G., Snell, C.: A new difference based finite element method. Instn. Eng. 59(2), 395–409 (1975)Google Scholar
  12. 12.
    Brandt, A.: Multi-level adaptive solutions to boundary value problems. Math. Comp. 31, 333–390 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Niederreiter, H.: Quasi-Monte Carlo methods and pseudo-random numbers. Bull. Amer. Math. Soc. 84, 957–1041 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Baclawski, K., Donsker, M.D., Kac, M.: Probability, Number Theory, and Statistical Physics, Selected Papers, pp 268–280. The MIT Press, Cambridge (1979)Google Scholar
  15. 15.
    Sipin, A.S.: Solving first boundary value problem for elliptic equations by Monte Carlo method. Monte Carlo Methods in Computational Mathematics and Mathematical Physics 2, 113–119 (1979)Google Scholar
  16. 16.
    Liszka, T., Orkisz, J.: The finite difference method at arbitrary irregular grids and its application in applied mechanics. Comput. Struct. 11, 83–95 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Booth, T.E.: Exact Monte Carlo solutions of elliptic partial differential equations. J. Comput. Phys. 39, 396–404 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Lancaster, P., Salkauskas, K.: Surfaces generated by moving least-squares method. Math. Comput. 155(37), 141–158 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Anderson, D.A., Tannenhill, J., Fletcher, R.H.: Computational Fluid Mechanics and Heat Transfer. McGraw-Hill, Washington (1984)Google Scholar
  20. 20.
    Ghoniem, A.F., Sherman, F.S.: Grid-free simulation of diffusion using random walk methods. J. Comp. Phys. 61, 1–37 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Eckhardt, R., Ulam, S., von Neumann, J.: The Monte Carlo method Los Alamos Science, Special Issue (15) , pp. 131–137 (1987)Google Scholar
  22. 22.
    Atkinson, K.E.: An introduction to numerical analysis. Wiley Ed., New York (1988)Google Scholar
  23. 23.
    Ermakov, S.M., Nekrutkin, V.V., Sipin, A.S.: Random Processes for Classical Equations of Mathematical Physics. Kluwer Academic Publishers, Dordrecht (1989)CrossRefzbMATHGoogle Scholar
  24. 24.
    Goldberg, D.: Genetic Algorithms in Search, Optimization and Machine Learning. Addison-Wesley Professional, Reading (1989)zbMATHGoogle Scholar
  25. 25.
    Lancaster, P., Salkauskas, K.: Curve and Surface Fitting. Academic Press Inc, Cambridge (1990)zbMATHGoogle Scholar
  26. 26.
    Mascagni, M.: High dimensional numerical integration and massively parallel computing. Contemp. Math. 1, 115:53–73 (1991)zbMATHGoogle Scholar
  27. 27.
    Sabelfeld, K.K.: Monte Carlo Methods in Boundary Value Problems. Springer, Berlin (1991)CrossRefGoogle Scholar
  28. 28.
    Dimov, I., Tonev, O.: Random walk on distant mesh points Monte Carlo methods. J. Stat. Phys. 70. Plenum Publishing Corporation (1993)Google Scholar
  29. 29.
    Liu, W.K., Jun, S., Zhang, Y.F.: Reproducing kernel particle methods. Int. J. Numer. Methods Eng. 20, 1081–1106 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Mikhailov, G.A.: New Monte Carlo Methods with Estimating Derivatives. V. S. P. Publishers, London (1995)zbMATHGoogle Scholar
  31. 31.
    Orkisz, J.: Finite difference method (part III), Handbook of Computational Solid Mechanics, pp 336–431. Springer, Berlin (1998)Google Scholar
  32. 32.
    Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes: the Art of Parallel Scientific Computing. Cambridge University Press, Cambridge (1999)zbMATHGoogle Scholar
  33. 33.
    Li, S., Liu, W.K.: Meshfree and particle methods and their applications. Appl. Mech. Rev. 55, 1–34 (2002)CrossRefGoogle Scholar
  34. 34.
    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(3-6), 347–355 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Liu, G.R.: Mesh Free Methods: Moving Beyond the Finite Element Method. CRC Press, Boca Raton (2003)zbMATHGoogle Scholar
  36. 36.
    Moller, B., Beer, M.: Fuzzy Randomness, Uncertainty in Civil Engineering and Computational Mechanics. Springer, Berlin (2004)CrossRefzbMATHGoogle Scholar
  37. 37.
    Zienkiewicz, O.C., Taylor, R.L.: Finite Element Method its Basis and Fundamentals. Elsevier, Amsterdam (2005)Google Scholar
  38. 38.
    Ramachandran, P., Ramakrishna, M., Rajan, S.C.: Efficient random walks in the presence of complex two-dimensional geometries. Computers and Mathematics with Applications 53(2), 329–344 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Ripley, B.D.: Pattern Recognition and Neural Networks. Cambridge University Press, Cambridge (2007)zbMATHGoogle Scholar
  40. 40.
    Orkisz, J., Milewski, S.: A’posteriori error estimation based on higher order approximation in the meshless finite difference method, Lecture Notes in Computational Science and Engineering: Meshfree Methods for Partial Differential Equations IV, pp. 189–213 (2008)Google Scholar
  41. 41.
    Sabelfeld, K., Mozartova, N.: Sparsified randomization algorithms for large systems of linear equations and a new version of the random walk on boundary method. Monte Carlo Methods Appl. 15(3), 257–284 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Sadiku, M.N.: Monte Carlo Methods for Electromagnetics. Taylor (2009)Google Scholar
  43. 43.
    Milstein, G., Tretyakov, M.V.: Stochastic Numerics for Mathematical Physics. Springer, Berlin (2010)zbMATHGoogle Scholar
  44. 44.
    Jurlewicz, A., Kern, P., Meerschaert, M.M., Scheffler, H.-P.: Fractional governing equations for coupled random walks. Computers and Mathematics with Applications 64(10), 3021–3036 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Milewski, S.: Meshless finite difference method with higher order approximation - applications in mechanics. Arch. Comput. Meth. Eng. 19(1), 1–49 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Milewski, S.: Selected computational aspects of the meshless finite difference method. Numer. Algorithms 63(1), 107–126 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Yu, W., Wang, X.: Advanced Field-Solver Techniques for RC Extraction of Integrated Circuits. Springer, Berlin (2014)CrossRefGoogle Scholar
  48. 48.
    Jaśkowiec, J., Milewski, S.: The effective interface approach for coupling of the FE and meshless FD methods and applying essential boundary conditions. Computers and Mathematics with Applications 70(5), 962–979 (2015)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Bernal, F., Acebron, J.A.: A comparison of higher-order weak numerical schemes for stopped stochastic differential equations. Communications in Computational Physics 20, 703–732 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Angstmann, C.N., Henry, B.I., Ortega-Piwonka, I.: Generalized master equations and fractional Fokker-Planck equations from continuous time random walks with arbitrary initial conditions. Computers and Mathematics with Applications 73 (6), 1315–1324 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Keady, K.P., Cleveland, M.A.: An improved random walk algorithm for the implicit Monte Carlo method. J. Comput. Phys. 328, 160–176 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Talebi, S., Gharehbash, K., Jalali, H.R.: Study on random walk and its application to solution of heat conduction equation by Monte Carlo method. Prog. Nucl. Energy 96, 18–35 (2017)CrossRefGoogle Scholar
  53. 53.
    Milewski, S.: Combination of the meshless finite difference approach with the Monte Carlo random walk technique for solution of elliptic problems. Computers & Mathematics with Applications 76(4), 854–876 (2018)MathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) 2019

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Civil Engineering DepartmentCracow University of TechnologyKrakówPoland

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