A New Class of Grid-Free Monte Carlo Algorithms for Elliptic Boundary Value Problems

  • R.J. Papancheva
  • I. T. Dimov
  • T. V. Gurov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2542)


In this paper we consider the following mathematical model: an elliptic boundary value problem, where the partial differential equation contains advection, diffusion, and deposition parts. A Monte Carlo (MC) method to solve this equation uses a local integral representation by the Green's function and a random process called “Walks on Balls”(WOB). A new class of grid free MC algorithms for solving the above elliptic boundary value problem is suggested and studied. We prove that the integral transformation kernel can be taken as a transition density function in the Markov chain in the case when the deposition part is equal to zero. An acceptance-rejection (AR) and an inversetransformation methods are used to sample the next point in the Markov chain. An estimate for the efficiency of the AR method is obtained.


Density Function Markov Chain Monte Carlo Transition Density Elliptic Boundary 
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  1. 1.
    Bitzadze, A.: Equations of the Mathematical Physics. Nauka, Moscow (1982)Google Scholar
  2. 2.
    Curtiss, J.: Monte Carlo methods for the iteration of the linear operators. J. Math. Phys., Vol. 32, 4 (1954), 209–232.MathSciNetzbMATHGoogle Scholar
  3. 3.
    Gurov, T., Withlock, P., Dimov, I.: A grid free Monte Carlo algorithm for solving elliptic boundary value problems. In: L. Vulkov, Waśniewski, J., Yalamov, P. (eds.): Numerical Analysis and its Applications. Lecture notes in Comp. Sci., Vol. 1988. Springer-Verlag (2001) 359–367.CrossRefGoogle Scholar
  4. 4.
    Dimov, I., Gurov, T.: Estimates of the computational complexity of iterative Monte Carlo algorithm based on Green’s function approach. Mathematics and Computers in Simulation, Vol. 47 (2-5) (1988) 183–199.CrossRefMathSciNetGoogle Scholar
  5. 5.
    Ermakov, S., Mikhailov, G.: Statistical Simulation. Nauka, Moscow (1982)zbMATHGoogle Scholar
  6. 6.
    Ermakov, S., Nekrutkin, V., Sipin, A.: Random Processes for Solving Classical Equations of the Mathematical Physics. Nauka, Moscow (1984)Google Scholar
  7. 7.
    Mikhailov, V.: Partial Differential Equations. Nauka, Moscow (1983)Google Scholar
  8. 8.
    Miranda, C.: Equasioni alle dirivate parziali di tipo ellipttico. Springer-Verlag, Berlin (1955)Google Scholar
  9. 9.
    Sobol, I.: Monte Carlo Numerical Methods. Nauka, Moscow (1973)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • R.J. Papancheva
    • 1
  • I. T. Dimov
    • 1
  • T. V. Gurov
    • 1
  1. 1.Central Laboratory for Parallel ProcessingBulgarian Academy of SciencesSofiaBulgaria

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