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Monte Carlo solvers of large linear systems with Toeplitz matrices, preconditioning, iterative refinement with applications to integral equations and acoustic inverse problem

Author

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  • Sabelfeld Karl K.

    (Institute of Computational Mathematics and Mathematical Geophysics, Russian Academy of Sciences; and Sobolev Institute of Mathematics, Russian Academy of Sciences, Novosibirsk, Russia)

  • Shafigulin Igor

    (Institute of Computational Mathematics and Mathematical Geophysics, Russian Academy of Sciences; and Novosibirsk State University, Novosibirsk, Russia)

Abstract

This study deals with randomized algorithms and random projection methods for solving systems of linear algebraic equations with Toeplitz matrices. A preconditioning of such systems with circulant matrices is used that improves the convergence of the stochastic projection method. The developed stochastic algorithms are applied to first kind boundary integral equations for the Laplace, screened Poisson, and Helmholtz equations. Another application concerns the inverse problem for a wave equation where the task is to recover the unknown coefficient of this equation. A series of computer simulations are carried out to analyze the efficiency of the developed algorithm.

Suggested Citation

  • Sabelfeld Karl K. & Shafigulin Igor, 2025. "Monte Carlo solvers of large linear systems with Toeplitz matrices, preconditioning, iterative refinement with applications to integral equations and acoustic inverse problem," Monte Carlo Methods and Applications, De Gruyter, vol. 31(3), pages 207-224.
    • Handle: RePEc:bpj:mcmeap:v:31:y:2025:i:3:p:207-224:n:1003
    DOI: 10.1515/mcma-2025-2012
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