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Numerical Experiments for Some Markov Models for Solving Boundary Value Problems

  • Alexander S. SipinEmail author
  • Alexander I. Zeifman
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11386)

Abstract

The main purpose of this work is the analysis of some stochastic algorithms to determine values of harmonic functions at points of a bounded domain of Euclidean space. To solve the Dirichlet problem we use a Random Walk on Spheres algorithm. The Neumann problem is solved by means of integral equations of potential theory.

We compare Monte Carlo and quasi-Monte Carlo versions of these algorithms numerically. The desired value of the harmonic function is represented as the sum of a series of integrals on hypercubes whose dimension grows. Therefore, the asymptotic formulas for discrepancy cannot be used for estimation of the error of quasi-Monte Carlo algorithm. New results are obtained about the influence of the smoothness of the domain boundary on the accuracy of calculations.

Keywords

Boundary value problem quasi-Monte Carlo method Monte Carlo method 

Notes

Acknowledgments

Supported by the Russian Foundation for Basic Research, projects No. 17-01-00267, 18-47-350002.

References

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    Sabelfeld, K.K.: Monte Carlo Methods in Boundary Value Problems. Springer, Heidelberg (1991)CrossRefGoogle Scholar
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    Ermakov, S.M., Nekrutkin, V.V., Sipin, A.S.: Random Processes for Classical Equations of Mathematical Physics. Kluwer Academic Publishers, Dordrecht (1989)CrossRefGoogle Scholar
  3. 3.
    Niederreiter, H.: Random Number Generation and Quasi-Monte Carlo Methods. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 63. Society for Industrial and Applied Mathematics, Philadelphia (1992)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Vologda State UniversityVologdaRussia
  2. 2.Vologda State University, IPI FRC CSC RAS; VolSC RASVologdaRussia

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