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Journal of Mathematical Sciences

, Volume 181, Issue 3, pp 340–349 | Cite as

Modification of a method of generalized separation of variables for the solution of multidimensional integral equations

  • V. M. Biletskyy
Article

We describe a method of generalized separation of variables for the solution of multidimensional integral equations and its modification minimizing the deviation of an approximate solution from the exact one. The convergence of the modified method is proved. A comparison of methods on the basis of numerical results is presented.

Keywords

Tensor Decomposition Multidimensional Problem Generalize Separation Integral Equation Theory European Signal Processing 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    M. L. Agranovskii and R. D. Baglai, “On one decomposition in a Hilbert space and its applications,” Zh. Vychisl. Mat. Mat. Fiz., 17, No. 4, 871–878 (1977).MathSciNetGoogle Scholar
  2. 2.
    Yu. G. Balyash and N. N. Voitovich, “Variational-iterative method for the solution of multidimensional integral equations,” in: Abstracts of the XXth Republican Conf. “Integral Equations in Applied Simulation” [in Russian], Institute of Electrodynamics, Ukrainian Academy of Sciences, Part 2, Kiev (1986), p. 23.Google Scholar
  3. 3.
    Yu. G. Balyash and N. N. Voitovich, “Approximate variational-iterative separation of variables in multidimensional problems,” in: Abstracts of the All-Union Symp. on Diffraction and Propagation of Waves “Waves and Diffraction-85” [in Russian], Tbilisi University, Vol. 1, Tbilisi (1985), p 122.Google Scholar
  4. 4.
    P. Vahin, B. Ostudin, and H. Shynkarenko, Foundations of Functional Analysis [in Ukrainian], Vydavnychyi Tsentr LNU, Lviv (2005).Google Scholar
  5. 5.
    A. F. Verlan’ and I. A. Serikova, “On the problem of convergence of the variational-iterative method for the approximation of functions of two variables,” Tochn. Nadezh. Kiber. Sist., Issue 3, 10–12 (1975).Google Scholar
  6. 6.
    M. M. Voitovych and S. A. Yaroshko, “Variational-iterative method for generalized separation of variables for the solution of multidimensional integral equations,” Mat. Met. Fiz.-Mekh. Polya, 40, No. 4, 122–126 (1997).Google Scholar
  7. 7.
    N. N. Voitovich, “Synthesis of a two-dimensional antenna array with generalized separation of variables,” Radiotekh. Élektron., 33, No. 12, 2637–2640 (1988).Google Scholar
  8. 8.
    N. N. Voitovich and S. A. Yaroshko, “Numerical solution of the problem of synthesis of a two-dimensional antenna array,” Radiotekh. Élektron., 36, No. 1, 192–196 (1991).Google Scholar
  9. 9.
    P. I. Kalenyuk, Ya. E. Baranetskii, and Z. M. Nytrebych, Generalized Method of Separation of Variables [in Russian], Naukova Dumka, Kiev (1993).Google Scholar
  10. 10.
    A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis [in Russian], Nauka, Moscow (1981).Google Scholar
  11. 11.
    V. V. Pospelov, On the Approximation of a Function of Several Variables by Products of Functions of One Variable [in Russian], Preprint No. 32, Institute of Applied Mathematics, Academy of Sciences of USSR, Moscow (1978).Google Scholar
  12. 12.
    V. A. Trenogin, Functional Analysis [in Russian], Nauka, Moscow (1980).Google Scholar
  13. 13.
    Y. G. Balyash, N. N. Voitovych, and S. A. Yaroshko, “Generalized separation of variables in problems of diffraction and antenna synthesis,” in: Proceedings of the 1989-URSI International Symp. on Electromagnetic Theory (Stockholm, Sweden), Stockholm (1989), pp. 650–652.Google Scholar
  14. 14.
    V. Biletskyy, “An iterative method of generalized separation of variables for solving linear operator equations,” J. Numer. Appl. Math. (2009). (Submitted).Google Scholar
  15. 15.
    V. Biletskyy and S. Yaroshko, “A method of generalized separation of variables for solving many-dimensional linear Fredholm integral equation theory,” in: Proceeding of the XIIth International Seminar/Workshop “Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory” (Lviv, September 17–20, 2007) [in Ukrainian], Lviv (2007), pp. 94–97.Google Scholar
  16. 16.
    V. Biletskyy and S. Yaroshko, “A method of generalized separation of variables for solving three-dimensional integral equations theory,” in: Proceeding of the XI th International Seminar/Workshop “Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory” (Tbilisi, October 11–13, 2006) [in Ukrainian], Lviv–Tbilisi (2006), pp. 164–168.Google Scholar
  17. 17.
    O. Bulatsyk, B. Katsenelenbaum, Yu. Topolyuk, and N. Voitovich, Phase Optimization Problems. Applications in Wave Field Theory, Wiley–VCH, Weinheim (2010).zbMATHCrossRefGoogle Scholar
  18. 18.
    L. de Lathauwer, B. de Moor, and J. Vandewalle, “A multilinear singular value decomposition,” SIAM J. Matrix Anal. Appl., 21, 1253–1278 (2000).MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    I. Gavrilyuk, W. Hackbusch, and B. Khoromskij, “Hierarchical tensor-product approximation to the inverse and related operators for high-dimensional elliptic problems,” Computing, 74, 131–157 (2005).MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    W. Hackbusch and B. Khoromskij, “Tensor-product approximation to operators and functions in high dimensions,” J. Complexity, 23, 697–714 (2007).MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    T. Kolda and B. Bader, Tensor Decompositions and Applications, Techn. Report SAND2007-6702, Sandia Nat. Laboratories (2007).Google Scholar
  22. 22.
    C. Navasca, L. de Lathauwer, and S. Kinderman, “Swamp reducing technique for tensor decompositions,” in: Proceedings of the 16th European Signal Processing Conf. (EUSIPCO 2008), Lausanne, Switzerland, August (2008).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2012

Authors and Affiliations

  • V. M. Biletskyy
    • 1
  1. 1.LvivUkraine

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