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Stochastic boundary collocation and spectral methods for solving PDEs

Author

Listed:
  • Sabelfeld Karl

    (Institute of Computational Mathematics and Mathematical Geophysics, Russian Acad. Sci., Lavrentieva str., 6, 630090 Novosibirsk, Russia)

  • Mozartova Nadezhda

    (Novosibirsk State University, Pirogova str., 2, 630090 Novosibirsk, Russia)

Abstract

We develop a stochastic boundary method (SBM) which can be considered as a randomized version of the method of fundamental solutions (MFS). We suggest solving the large system of linear equations for the weights in the expansion over the fundamental solutions by a randomized SVD method introduced by Sabelfeld and Mozartova (2011). In addition, we also deal with solving inhomogeneous problems where we use the integral representation through the Green integral formula. The relevant volume integrals are calculated by a Monte Carlo integration technique which uses the symmetry of the Green function. We construct also a stochastic boundary method based on the spectral inversion of the Poisson formula representing the solution in a disc. This is done for the Laplace equation, and the system of elasticity equations. We stress that the stochastic boundary method proposed is of high generality; it can be applied to any bounded and unbounded domain with any boundary condition provided the existence and uniqueness of the solution are proven. We present a series of numerical results which illustrate the performance of the suggested methods.

Suggested Citation

  • Sabelfeld Karl & Mozartova Nadezhda, 2012. "Stochastic boundary collocation and spectral methods for solving PDEs," Monte Carlo Methods and Applications, De Gruyter, vol. 18(3), pages 217-263, September.
    • Handle: RePEc:bpj:mcmeap:v:18:y:2012:i:3:p:217-263:n:2
    DOI: 10.1515/mcma-2012-0008
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    References listed on IDEAS

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    1. ,, 2003. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 19(4), pages 691-705, August.
    2. ,, 2003. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 19(1), pages 225-228, February.
    3. Sabelfeld, K.K. & Mozartova, N.S., 2011. "Sparsified Randomization algorithms for low rank approximations and applications to integral equations and inhomogeneous random field simulation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 82(2), pages 295-317.
    4. Sabelfeld K. & Mozartova N., 2009. "Sparsified Randomization Algorithms for large systems of linear equations and a new version of the Random Walk on Boundary method," Monte Carlo Methods and Applications, De Gruyter, vol. 15(3), pages 257-284, January.
    5. ,, 2003. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 19(5), pages 879-883, October.
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    7. ,, 2003. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 19(2), pages 411-413, April.
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