Abstract
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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Metropolis, N., Ulam, S.: The Monte Carlo method. J. Am. Stat. Assoc. American Statistical Association 44(247), 335–341 (1949)
Forsythe, G.E., Leibler, R.A.: Matrix inversion by a Monte Carlo method. Math. Tab. Aids Comput. 4, 127–129 (1950)
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)
Curtiss, J.H.: Monte Carlo methods for the iteration of linear operators. J. Math. Phys. 32, 209–232 (1953)
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)
Muller, M.E.: Some continuous Monte Carlo methods for the Dirichlet problem. Ann. Math. Stat 27, 569–589 (1956)
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 Association
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)
Hoshino, S., Ichida, K.: Solution of partial differential equations by a modified random walk. Numer. Math 18, 61–72 (1971). Springer
Perrone, N., Kao, R.: A general finite difference method for arbitrary meshes. Comput. Struct. 5, 45–58 (1975)
Wyatt, M.J., Davies, G., Snell, C.: A new difference based finite element method. Instn. Eng. 59(2), 395–409 (1975)
Brandt, A.: Multi-level adaptive solutions to boundary value problems. Math. Comp. 31, 333–390 (1977)
Niederreiter, H.: Quasi-Monte Carlo methods and pseudo-random numbers. Bull. Amer. Math. Soc. 84, 957–1041 (1978)
Baclawski, K., Donsker, M.D., Kac, M.: Probability, Number Theory, and Statistical Physics, Selected Papers, pp 268–280. The MIT Press, Cambridge (1979)
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)
Liszka, T., Orkisz, J.: The finite difference method at arbitrary irregular grids and its application in applied mechanics. Comput. Struct. 11, 83–95 (1980)
Booth, T.E.: Exact Monte Carlo solutions of elliptic partial differential equations. J. Comput. Phys. 39, 396–404 (1981)
Lancaster, P., Salkauskas, K.: Surfaces generated by moving least-squares method. Math. Comput. 155(37), 141–158 (1981)
Anderson, D.A., Tannenhill, J., Fletcher, R.H.: Computational Fluid Mechanics and Heat Transfer. McGraw-Hill, Washington (1984)
Ghoniem, A.F., Sherman, F.S.: Grid-free simulation of diffusion using random walk methods. J. Comp. Phys. 61, 1–37 (1985)
Eckhardt, R., Ulam, S., von Neumann, J.: The Monte Carlo method Los Alamos Science, Special Issue (15) , pp. 131–137 (1987)
Atkinson, K.E.: An introduction to numerical analysis. Wiley Ed., New York (1988)
Ermakov, S.M., Nekrutkin, V.V., Sipin, A.S.: Random Processes for Classical Equations of Mathematical Physics. Kluwer Academic Publishers, Dordrecht (1989)
Goldberg, D.: Genetic Algorithms in Search, Optimization and Machine Learning. Addison-Wesley Professional, Reading (1989)
Lancaster, P., Salkauskas, K.: Curve and Surface Fitting. Academic Press Inc, Cambridge (1990)
Mascagni, M.: High dimensional numerical integration and massively parallel computing. Contemp. Math. 1, 115:53–73 (1991)
Sabelfeld, K.K.: Monte Carlo Methods in Boundary Value Problems. Springer, Berlin (1991)
Dimov, I., Tonev, O.: Random walk on distant mesh points Monte Carlo methods. J. Stat. Phys. 70. Plenum Publishing Corporation (1993)
Liu, W.K., Jun, S., Zhang, Y.F.: Reproducing kernel particle methods. Int. J. Numer. Methods Eng. 20, 1081–1106 (1995)
Mikhailov, G.A.: New Monte Carlo Methods with Estimating Derivatives. V. S. P. Publishers, London (1995)
Orkisz, J.: Finite difference method (part III), Handbook of Computational Solid Mechanics, pp 336–431. Springer, Berlin (1998)
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)
Li, S., Liu, W.K.: Meshfree and particle methods and their applications. Appl. Mech. Rev. 55, 1–34 (2002)
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)
Liu, G.R.: Mesh Free Methods: Moving Beyond the Finite Element Method. CRC Press, Boca Raton (2003)
Moller, B., Beer, M.: Fuzzy Randomness, Uncertainty in Civil Engineering and Computational Mechanics. Springer, Berlin (2004)
Zienkiewicz, O.C., Taylor, R.L.: Finite Element Method its Basis and Fundamentals. Elsevier, Amsterdam (2005)
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)
Ripley, B.D.: Pattern Recognition and Neural Networks. Cambridge University Press, Cambridge (2007)
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)
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)
Sadiku, M.N.: Monte Carlo Methods for Electromagnetics. Taylor (2009)
Milstein, G., Tretyakov, M.V.: Stochastic Numerics for Mathematical Physics. Springer, Berlin (2010)
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)
Milewski, S.: Meshless finite difference method with higher order approximation - applications in mechanics. Arch. Comput. Meth. Eng. 19(1), 1–49 (2012)
Milewski, S.: Selected computational aspects of the meshless finite difference method. Numer. Algorithms 63(1), 107–126 (2013)
Yu, W., Wang, X.: Advanced Field-Solver Techniques for RC Extraction of Integrated Circuits. Springer, Berlin (2014)
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)
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)
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)
Keady, K.P., Cleveland, M.A.: An improved random walk algorithm for the implicit Monte Carlo method. J. Comput. Phys. 328, 160–176 (2017)
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)
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)
Funding
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).
Author information
Authors and Affiliations
Additional information
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), 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.
About this article
Cite this article
Milewski, S. A Matlab software for approximate solution of 2D elliptic problems by means of the meshless Monte Carlo random walk method. Numer Algor 83, 565–591 (2020). https://doi.org/10.1007/s11075-019-00694-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-019-00694-x