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Numerical Experiments for Some Markov Models for Solving Boundary Value Problems

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Finite Difference Methods. Theory and Applications (FDM 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11386))

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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.

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References

  1. Sabelfeld, K.K.: Monte Carlo Methods in Boundary Value Problems. Springer, Heidelberg (1991)

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  2. Ermakov, S.M., Nekrutkin, V.V., Sipin, A.S.: Random Processes for Classical Equations of Mathematical Physics. Kluwer Academic Publishers, Dordrecht (1989)

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  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)

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Acknowledgments

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

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Authors and Affiliations

  1. Vologda State University, Vologda, Russia

    Alexander S. Sipin

  2. Vologda State University, IPI FRC CSC RAS; VolSC RAS, Vologda, Russia

    Alexander I. Zeifman

Authors
  1. Alexander S. Sipin
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  2. Alexander I. Zeifman
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Corresponding author

Correspondence to Alexander S. Sipin .

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Editors and Affiliations

  1. IICT, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Ivan Dimov

  2. Eötvös Loránd University, Budapest, Hungary

    István Faragó

  3. University of Rousse, Rousse, Bulgaria

    Lubin Vulkov

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Sipin, A.S., Zeifman, A.I. (2019). Numerical Experiments for Some Markov Models for Solving Boundary Value Problems. In: Dimov, I., Faragó, I., Vulkov, L. (eds) Finite Difference Methods. Theory and Applications. FDM 2018. Lecture Notes in Computer Science(), vol 11386. Springer, Cham. https://doi.org/10.1007/978-3-030-11539-5_57

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  • DOI: https://doi.org/10.1007/978-3-030-11539-5_57

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-11538-8

  • Online ISBN: 978-3-030-11539-5

  • eBook Packages: Computer ScienceComputer Science (R0)

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