The Conjugate Gradient Method for the Dirichlet Problem and Its Modifications

  • V. I. Vasil’evEmail author
  • A. M. KardashevskyEmail author
  • V. V. PopovEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11386)


In this paper we consider a numerical solution of the non-classical Dirichlet problem and its modifications for the second-order two-dimensional hyperbolic equations. In order to determine the missing initial condition using an additional condition specified at the final time, an iterative method of conjugate gradient is used. A direct problem is numerically realized at each iteration. The efficiency of the proposed computational algorithm is confirmed by calculations for model two-dimensional problems.


Hyperbolic equation Inverse problem Dirichlet problem Finite difference method Iterative method Conjugate gradients method Random errors 



Supported by mega-grant of the Russian Federation Government (14.Y26.31.0013) and Grant of the Russian Foundation for Basic Research (17-01-00732).


  1. 1.
    Kabanikhin, S.I.: Inverse and Ill-Posed Problems. Theory and Applications. De Gruyter, Berlin (2011)CrossRefGoogle Scholar
  2. 2.
    Lavrent’ev, M.M., Romanov, V.G., Shishatskii, S.P.: Ill-Posed Problems of Mathematical Physics and Analysis. American Mathematical Society, Providence (1986)CrossRefGoogle Scholar
  3. 3.
    Samarskii, A.A., Vabishchevich, P.N.: Numerical Methods for Solving Inverse Problems of Mathematical Physics. De Gruyter, Berlin (2007)CrossRefGoogle Scholar
  4. 4.
    Kabanikhin, S.I., Bektemesov, M.A., Nurseitov, D.B., Krivorotko, O.I., Alimova, A.N.: An optimization method in the Dirichlet problems for the wave equation. J. Inverse Ill-Posed Probl. 2(20), 193–211 (2012)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Kabanikhin, S.I., Krivorotko, O.I.: Chislennyj metod reshenia zadachi Dirichle dlia volnovogo uravnenia. Sib. J. Ind. Math. 15(4), 90–101 (2012)Google Scholar
  6. 6.
    Samarskii, A.A., Vabishchevich, P.N., Vasil’ev, V.I.: Iterative solution of a retrospective inverse problem of heat conduction. Matematicheskoe Modelirovanie 9(5), 119–127 (1997)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Vasil’ev, V.I., Popov, V.V., Eremeeva, M.S., Kardashevsky, A.M.: Iterative solution of a nonclassical problem for the equation of string vibrations. Vest. Mosk. Gos. Univ. im. N. E. Baumana, Estest. Nauki. 3, 77–87 (2015)Google Scholar
  8. 8.
    Vabishchevich, P.N., Vasil’ev, V.I.: Iteracionnoe reshenie zadachi Dirichle dlia giperbolicheskogo uravnenia, Grid methods for boundary value problems and applications. In: Proceedings of the Tenth International Conference, pp. 162–166. Publishing House of Kazan University, Kazan (2014)Google Scholar
  9. 9.
    Vasil’ev, V.I., Kardashevsky A.M.: Iteracionnoe reshenie nekotoryx obratnyx zadach dlia giperbolicheskix uravnenij vtorogo poriadka. In: Proceedings of the International Conference “Actual Problems of Computational and Applied Mathematics”, Novosibirsk, pp. 150–156 (2015)Google Scholar
  10. 10.
    Samarskii, A.A.: The Theory of Difference Schemes. Marcel Dekker, New York (2001)CrossRefGoogle Scholar
  11. 11.
    Saad, U.: Iterative Methods for Sparse Linear Systems, 2nd edn. SIAM (2003)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.North-Eastern Federal UniversityYakutskRussia

Personalised recommendations