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Zh. Vychisl. Mat. Mat. Fiz., 2005, Volume 45, Number 2, Pages 315–326 (Mi zvmmf708)  

This article is cited in 23 scientific papers (total in 23 papers)

Approximate inversion of matrices in the process of solving a hypersingular integral equation

I. V. Oseledets, E. E. Tyrtyshnikov

Institute of Numerical Mathematics, Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119333, Russia

Abstract: A method is proposed for approximate inversion of large matrices represented as sums of tensor products of smaller matrices. The method incorporates a modification, found by the authors, of the Newton–Hotelling–Schulz algorithm and uses a number of recently developed techniques for data compression and data structuring based on nonlinear approximations, such as tensor-product, low-rank, or wavelet approximations. The efficiency of the method is demonstrated with the help of matrices arising in the numerical solution of a hypersingular integral equation (namely, the Prandtl equation) on a square.

Key words: hypersingular integral equation, numerical method for solution, fast approximate matrix inversion, nonuniform grids.

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English version:
Computational Mathematics and Mathematical Physics, 2005, 45:2, 302–313

Bibliographic databases:

Document Type: Article
UDC: 519.642.7
Received: 01.07.2004

Citation: I. V. Oseledets, E. E. Tyrtyshnikov, “Approximate inversion of matrices in the process of solving a hypersingular integral equation”, Zh. Vychisl. Mat. Mat. Fiz., 45:2 (2005), 315–326; Comput. Math. Math. Phys., 45:2 (2005), 302–313

Citation in format AMSBIB
\by I.~V.~Oseledets, E.~E.~Tyrtyshnikov
\paper Approximate inversion of matrices in the process of solving a~hypersingular integral equation
\jour Zh. Vychisl. Mat. Mat. Fiz.
\yr 2005
\vol 45
\issue 2
\pages 315--326
\jour Comput. Math. Math. Phys.
\yr 2005
\vol 45
\issue 2
\pages 302--313

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    This publication is cited in the following articles:
    1. Olshevsky V., Oseledets I., Tyrtyshnikov E., “Tensor properties of multilevel Toeplitz and related matrices”, Linear Algebra Appl., 412:1 (2006), 1–21  crossref  mathscinet  zmath  isi  elib  scopus
    2. Hackbusch W., Khoromskij B.N., Tyrtyshnikov E.E., “Approximate iterations for structured matrices”, Numer. Math., 109:3 (2008), 365–383  crossref  mathscinet  zmath  isi  elib  scopus
    3. D. V. Savostyanov, E. E. Tyrtyshnikov, “Approximate multiplication of tensor matrices based on the individual filtering of factors”, Comput. Math. Math. Phys., 49:10 (2009), 1662–1677  mathnet  crossref  isi  elib  elib
    4. Oseledets I.V., “Approximation of matrices with logarithmic number of parameters”, Dokl. Math., 80:2 (2009), 653–654  mathnet  crossref  zmath  isi  elib  elib  scopus
    5. Zamarashkin N.L., Oseledets I.V., Tyrtyshnikov E.E., “The tensor structure of the inverse of a banded Toeplitz matrix”, Dokl. Math., 80:2 (2009), 669–670  mathnet  crossref  mathscinet  zmath  isi  elib  elib  scopus
    6. Oseledets I.V., Savostyanov D.V., Tyrtyshnikov E.E., “Linear algebra for tensor problems”, Computing, 85:3 (2009), 169–188  crossref  mathscinet  zmath  isi  elib  scopus
    7. Boikov I.V., Stasyuk B.M., Tarasov D.V., “Priblizhennoe reshenie nekotorykh klassov gipersingulyarnykh integralnykh uravnenii”, Izv. vuzov. Povolzhskii region. Fiziko-matematicheskie nauki, 2009, no. 1, 100–112
    8. O. S. Lebedeva, “Block tensor conjugate gradient-type method for Rayleigh quotient minimization in two-dimensional case”, Comput. Math. Math. Phys., 50:5 (2010), 749–765  mathnet  crossref  adsnasa  isi  elib  elib
    9. Mastronardi N., Ng M., Tyrtyshnikov E.E., “Decay in functions of multiband matrices”, SIAM J. Matrix Anal. Appl., 31:5 (2010), 2721–2737  crossref  mathscinet  zmath  isi  elib  scopus
    10. Oseledets I.V., Savostyanov D.V., Tyrtyshnikov E.E., “Cross approximation in tensor electron density computations”, Numer. Linear Algebra Appl., 17:6 (2010), 935–952  crossref  mathscinet  zmath  isi  elib  scopus
    11. Oseledets I.V., “Approximation of $2^d\times 2^d$ matrices using tensor decomposition”, SIAM J. Matrix Anal. Appl., 31:4 (2010), 2130–2145  crossref  mathscinet  zmath  isi  scopus
    12. Boykov I.V., Ventsel E.S., Boykova A.I., “An approximate solution of hypersingular integral equations”, Appl. Numer. Math., 60:6 (2010), 607–628  crossref  mathscinet  zmath  isi  elib  scopus
    13. Pan V.Y., Qian Guoliang, “Randomized preprocessing of homogeneous linear systems of equations”, Linear Algebra Appl., 432:12 (2010), 3272–3318  crossref  mathscinet  zmath  isi  elib  scopus
    14. Matthies H.G., Zander E., “Solving Stochastic Systems with Low-Rank Tensor Compression”, Linear Alg. Appl., 436:10, SI (2012), 3819–3838  crossref  mathscinet  zmath  isi  elib  scopus
    15. Oseledets I.V., Dolgov S.V., “Solution of Linear Systems and Matrix Inversion in the Tt-Format”, SIAM J. Sci. Comput., 34:5 (2012), A2718–A2739  crossref  mathscinet  zmath  isi  scopus
    16. I. A. Blatov, N. V. Rogova, “Application of semiorthogonal spline wavelets and the Galerkin method to the numerical simulation of thin wire antennas”, Comput. Math. Math. Phys., 53:5 (2013), 564–572  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    17. Soheili A.R., Sofeymani F., Petkovic M.D., “On the Computation of Weighted Moore-Penrose Inverse Using a High-Order Matrix Method”, Comput. Math. Appl., 66:11 (2013), 2344–2351  crossref  mathscinet  zmath  isi  elib  scopus
    18. Soleymani F., Stanimirovic P.S., “A Note on the Stability of a Pth Order Iteration for Finding Generalized Inverses”, Appl. Math. Lett., 28 (2014), 77–81  crossref  mathscinet  zmath  isi  elib  scopus
    19. Haghani F.Kh., Soleymani F., “A New High-Order Stable Numerical Method for Matrix Inversion”, Sci. World J., 2014, 830564  crossref  isi  scopus
    20. Hackbusch W., “Numerical Tensor Calculus”, Acta Numer., 23 (2014), 651–742  crossref  mathscinet  isi  elib  scopus
    21. Stanimirovic P.S., Soleymani F., Haghani F.Kh., “Computing Outer Inverses By Scaled Matrix Iterations”, J. Comput. Appl. Math., 296 (2016), 89–101  crossref  mathscinet  zmath  isi  elib  scopus
    22. Borle A., Lomonaco S.J., “Convergence Acceleration of Iterative Methods For Inverting Real Matrices Using Frobenius Norm Minimization”, Proceedings of 2016 18Th International Symposium on Symbolic and Numeric Algorithms For Scientific Computing (Synasc), International Symposium on Symbolic and Numeric Algorithms For Scientific Computing, eds. Davenport J., Negru V., Ida T., Jebelean T., Petcu D., Watt S., Zaharie D., IEEE, 2016, 125–132  crossref  isi
    23. Boykov I.V. Roudnev V.A. Boykova A.I. Baulina O.A., “New Iterative Method For Solving Linear and Nonlinear Hypersingular Integral Equations”, Appl. Numer. Math., 127 (2018), 280–305  crossref  mathscinet  zmath  isi  scopus
  • Журнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
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