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Sparsified Randomization algorithms for low rank approximations and applications to integral equations and inhomogeneous random field simulation

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  • Sabelfeld, K.K.
  • Mozartova, N.S.

Abstract

Sparsified Randomization Monte Carlo (SRMC) algorithms introduced in our recent paper [60] for solving systems of linear algebraic equations are extended to construct the SVD-based randomized low rank approximations for large matrices. We suggest some efficient implementations of SRMC based on low rank approximations, and give different applications. In particular, an important application we present in this paper is a fast simulation algorithm for a randomized approximation of non-homogeneous random fields based on a discrete version of the Karhunen-Loéve expansion. We present two examples of non-homogeneous random field simulation which include a long-correlated Lorenzian random field and the fractional Wiener process. Another application we deal in this paper concerns the randomized solvers for large linear systems. We suggest a hybrid method which combines SRMC with an algorithm for solving boundary integral equations based on a separation representation of the kernel. This method is illustrated in this paper by solving a 2D boundary integral equation from potential theory governing the Dirichlet problem for the Laplace equation.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:matcom:v:82:y:2011:i:2:p:295-317
    DOI: 10.1016/j.matcom.2011.08.002
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    References listed on IDEAS

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    1. 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.
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    Cited by:

    1. Shalimova Irina A. & Sabelfeld Karl K., 2017. "Stochastic polynomial chaos expansion method for random Darcy equation," Monte Carlo Methods and Applications, De Gruyter, vol. 23(2), pages 101-110, June.
    2. 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.
    3. Grigoriu Mircea, 2017. "Monte Carlo algorithm for vector-valued Gaussian functions with preset component accuracies," Monte Carlo Methods and Applications, De Gruyter, vol. 23(3), pages 165-188, September.
    4. Grigoriu Mircea, 2014. "An efficient Monte Carlo solution for problems with random matrices," Monte Carlo Methods and Applications, De Gruyter, vol. 20(2), pages 121-136, June.
    5. Corlay Sylvain & Pagès Gilles, 2015. "Functional quantization-based stratified sampling methods," Monte Carlo Methods and Applications, De Gruyter, vol. 21(1), pages 1-32, March.
    6. Sabelfeld, Karl K., 2018. "Stochastic projection methods and applications to some nonlinear inverse problems of phase retrieving," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 143(C), pages 169-175.
    7. Sabelfeld Karl K., 2016. "Vector Monte Carlo stochastic matrix-based algorithms for large linear systems," Monte Carlo Methods and Applications, De Gruyter, vol. 22(3), pages 259-264, September.
    8. Shalimova Irina A. & Sabelfeld Karl K., 2014. "Stochastic polynomial chaos based algorithm for solving PDEs with random coefficients," Monte Carlo Methods and Applications, De Gruyter, vol. 20(4), pages 279-289, December.

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