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Siberian Mathematical Journal

, Volume 44, Issue 5, pp 908–925 | Cite as

Error Estimation and Optimization of the Functional Algorithms of a Random Walk on a Grid Which Are Applied to Solving the Dirichlet Problem for the Helmholtz Equation

  • E. V. Shkarupa
Article

Abstract

We consider the algorithms of a “random walk on a grid” which are applied to global solution of the Dirichlet problem for the Helmholtz equation (the direct and conjugate methods). In the metric space C we construct some upper error bounds and obtain optimal values (in the sense of the error bound) of the parameters of the algorithms (the number of nodes and the sample size).

Monte Carlo method random walk Helmholtz equation functional algorithm error estimate optimization 

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References

  1. 1.
    Mikhailov G. A., Minimization of Computational Costs of Non-Analogue Monte Carlo Methods, World Sci., Singapore (1991). (Ser. Soviet East European Mathematics. V. 5).Google Scholar
  2. 2.
    Voytishek A. V. and Prigarin S. M., “On functional convergence of estimators and models in the Monte Carlo method,” Zh. Vychisl. Mat. i Mat. Fiz., 32, No. 10, 1641–1651 (1992).Google Scholar
  3. 3.
    Mikhailov G. A., Parametric Estimates by the Monte Carlo Method, VSP, Utrecht (1995).Google Scholar
  4. 4.
    Vo?tishek A. V., “Convergence asymptotics of discrete-stochastic numerical methods for global estimation of a solution to an integral equation of the second kind,” Sibirsk. Mat. Zh., 35, No. 4, 728–736 (1994). (See also “On the paper “Convergence asymptotics of discrete-stochastic numerical methods for global estimation of a solution to an integral equation of the second kind” Sibirsk. Mat. Zh., 37, No. 3, 714 (1996).)Google Scholar
  5. 5.
    Voytishek A. V., “On the errors of discretely stochastic procedures in estimating globally the solution of an integral equation of the second kind,” Russian J. Numer. Anal. Math. Modelling, 11, No. 1, 71–92 (1996).Google Scholar
  6. 6.
    Shkarupa E. V. and Voytishek A. V., “Optimization of discretely stochastic procedures for globally estimating the solution of an integral equation of the second kind,” Russian J. Numer. Anal. Math. Modelling, 12, No. 6, 525–546 (1997).Google Scholar
  7. 7.
    Shkarupa E. V., “Error estimation and optimization for the frequency polygon method in the C-metric,” Comput. Math. Math. Phys., 38, No. 4, 590–603 (1998).Google Scholar
  8. 8.
    Plotnikov M. Yu. and Shkarupa E. V., “Error estimation and optimization in C-space of Monte Carlo iterative solution of nonlinear integral equations,” Monte Carlo Methods and Application, 4, No. 1, 53–70 (1998).Google Scholar
  9. 9.
    Bakhvalov S. N., Numerical Methods [in Russian], Nauka, Moscow (1975).Google Scholar
  10. 10.
    Traub J. F., Wasilkowski G. W., and Wozniakowski H., Information Based Complexity, Acad. Press, New York (1988).Google Scholar
  11. 11.
    Heinrich S., “A multilevel version of the method of dependent tests,” in: Proc. Third Petersburg Workshop on Simulation, St. Petersburg, 1998, pp. 31–35.Google Scholar
  12. 12.
    Heinrich S., “Monte Carlo complexity of global solution of integral equations,” J. Complexity, 14, 151–175 (1998).Google Scholar
  13. 13.
    Mikha?lov G. A., Weighted Monte Carlo Methods [in Russian], Izdat. Sibirsk. Otdel. Ros. Akad. Nauk, Novosibirsk (2000).Google Scholar
  14. 14.
    Mikhailov G. A. and Cheshkova A. F., “Monte Carlo solution of the finite-difference Dirichlet problem for the multidimensional Helmholtz equation,” Comput. Math. Math. Phys., 38, No. 1, (1998).Google Scholar
  15. 15.
    Marchuk G. I. and Agoshkov V. I., An Introduction to Projection-Grid Methods [in Russian], Nauka, Moscow (1981).Google Scholar
  16. 16.
    Zav?yalov Yu. S., Kvasov B. I., and Miroshnichenko V. L., Methods of Spline Functions [in Russian], Nauka, Moscow (1980).Google Scholar
  17. 17.
    Leadbetter M. R., Lindgren G., and Rootzen H., Extremes and Related Properties of Random Sequences and Processes, Springer-Verlag, New York; Heidelberg; Berlin (1986).Google Scholar
  18. 18.
    Mikha?lov G. A., Optimization of Weighted Monte Carlo Methods [in Russian], Nauka, Moscow (1987).Google Scholar
  19. 19.
    Spitzer F., Principles of Random Walk [Russian translation], Mir, Moscow (1969).Google Scholar
  20. 20.
    Rozanov Yu. A., Probability, Random Processes, and Mathematical Statistics [in Russian], Nauka, Moscow (1989).Google Scholar
  21. 21.
    Prudnikov A. P., Brychkov Yu. A., and Marichev O. I., Integrals and Series [in Russian], Nauka, Moscow (1981).Google Scholar
  22. 22.
    Sabel?fel?d K. K., Monte Carlo Methods in Boundary Value Problems [in Russian], Nauka, Novosibirsk (1989).Google Scholar
  23. 23.
    Shkarupa E. V., “Error estimation and optimization of functional algorithms of the Monte Carlo method for solution of the Dirichlet problem for the Helmholtz equation,” in: Abstracts: Conference of Young Scientists of IVM and MG SO RAN, Novosibirsk, 2002, pp. 157–163.Google Scholar

Copyright information

© Plenum Publishing Corporation 2003

Authors and Affiliations

  • E. V. Shkarupa
    • 1
  1. 1.Institute of Computational Mathematics and Mathematical GeophysicsRussia

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