Rate of convergence and error estimates for finite-difference schemes of solving linear ill-posed Cauchy problems of the second order

Authors

  • M.M. Kokurin Mari State University

DOI:

https://doi.org/10.26089/NumMet.v18r428

Keywords:

ill-posed Cauchy problem, Banach space, finite-difference scheme, rate of convergence, error estimate, operator calculus, sectorial operator, interpolation of Banach spaces, finite-dimensional approximation

Abstract

Finite-difference schemes of solving ill-posed Cauchy problems for linear second-order differential operator equations in Banach spaces are considered. Several time-uniform rate of convergence and error estimates are obtained for the considered schemes under the assumption that the sought solution satisfies the sourcewise condition. Necessary and sufficient conditions are found in terms of sourcewise index for a class of schemes with the power convergence rate with respect to the discretization step. A number of full discretization schemes for second-order ill-posed Cauchy problems are proposed on the basis of combining the half-discretization in time with the discrete approximation of the spaces and the operators.

Author Biography

M.M. Kokurin

References

  1. S. G. Krein, Linear Differential Equations in Banach Space (Nauka, Moscow, 1967; Amer. Math. Soc., Providence, 1971).
  2. V. K. Ivanov, I. V. Mel’nikova, and A. I. Filinkov, Differential-Operator Equations and Ill-Posed Problems (Nauka, Moscow, 1995) [in Russian].
  3. M. M. Kokurin, “Difference Schemes for Solving the Cauchy Problem for a Second-Order Operator Differential Equation,” Zh. Vychisl. Mat. Mat. Fiz. 54 (4), 569-584 (2014) [Comput. Math. Math. Phys. 54 (4), 582-597 (2014)].
  4. A. B. Bakushinskii, “Difference Schemes for the Solution of Ill-Posed Abstract Cauchy Problems,” Differ. Uravn. 7 (10), 1876-1885 (1971).
  5. A. B. Bakushinskii, “Difference Methods of Solving Ill-Posed Cauchy Problems for Evolution Equations in a Complex B-Space,” Differ. Uravn. 8 (9), 1661-1668 (1972).
  6. A. B. Bakushinskii, M. M. Kokurin, and M. Yu. Kokurin, “On a Class of Finite-Difference Schemes for Solving Ill-Posed Cauchy Problems in Banach Spaces,” Zh. Vychisl. Mat. Mat. Fiz. 52 (3), 483-498 (2012) [Comput. Math. Math. Phys. 52 (3), 411-426 (2012)].
  7. M. M. Kokurin, “Improvement of the Rate of Convergence Estimates for Some Classes of Difference Schemes for Solving an Ill-Posed Cauchy Problem,” Vychisl. Metody Programm. 14, 58-76 (2013).
  8. A. B. Bakushinskii, M. Yu. Kokurin, and V. V. Klyuchev, “Convergence Rate Estimation for Finite-Difference Methods of Solving the Ill-Posed Cauchy Problem for Second-Order Linear Differential Equations in a Banach Space,” Vychisl. Metody Programm. 11, 25-31 (2010).
  9. A. B. Bakushinskii, M. M. Kokurin, and M. Yu. Kokurin, “On a Complete Discretization Scheme for an Ill-Posed Cauchy Problem in a Banach Space,” Tr. Inst. Mat. Mekh. UrO RAN 18 (1), 96-108 (2012) [Proc. Steklov Inst. Math. (Suppl.) 280, suppl. 1, 53-65 (2013)].
  10. M. M. Kokurin, “Necessary and Sufficient Conditions for the Polynomial Convergence of the Quasi-Reversibility and Finite-Difference Methods for an Ill-Posed Cauchy Problem with Exact Data,” Zh. Vychisl. Mat. Mat. Fiz. 55 (12), 2027-2041 (2015) [Comput. Math. Math. Phys. 55 (12), 1986-2000 (2015)].
  11. M. M. Kokurin, “Polynomial Estimates for the Convergence Rate of Difference Schemes for Ill-Posed Cauchy Problems,” in Proc. Int. Workshop on Inverse and Ill-Posed Problems, Moscow, Russia, November 19-21, 2015 (Ross. Univ. Druzhby Narodov, Moscow, 2015), pp. 97-98.
  12. M. M. Kokurin, “Error Estimates of Difference Schemes for Solving Second-Order Ill-Posed Cauchy Problems,” in Proc. Int. Conf. on Contemporary Problems of Mathematical Physics and Computational Mathematics, Moscow, Russia, October 31-November 3, 2016 (MAKS Press, Moscow, 2016), p. 160.
  13. D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators (Clarendon, Oxford, 1987.
  14. S. Mizohata, Theory of Partial Differential Equations (Cambridge Univ. Press, Cambridge, 1973; Mir, Moscow, 1977).
  15. S. Agmon, “On the Eigenfunctions and on the Eigenvalues of General Elliptic Boundary Value Problems,” Comm. Pure Appl. Math. 15 (2), 119-147 (1962).
  16. H. B. Stewart, “Generation of Analytic Semigroups by Strongly Elliptic Operators,” Trans. Amer. Math. Soc. 199, 141-162 (1974).
  17. V. A. Kozlov, V. G. Maz’ya, and A. V. Fomin, “An Iterative Method for Solving the Cauchy Problem for Elliptic Equations,” Zh. Vychisl. Mat. Mat. Fiz. 31 (1), 64-74 (1991) [USSR Comput. Math. Math. Phys. 31 (1), 45-52 (1991)].
  18. J. Baumeister and A. Leitão, “On Iterative Methods for Solving Ill-Posed Problems Modeled by Partial Differential Equations,” J. Inv. Ill-Posed Problems 9 (1), 13-29 (2001).
  19. R. Lattès and J.-L. Lions, Methode de Quasi-Reversibilite et Applications (Dunod, Paris, 1967; Mir, Moscow, 1970).
  20. S. I. Piskarev, “Estimates for the Rate of Convergence in Semidiscretization of Evolution Equations,” Differ. Uravn. 19 (12), 2153-2159 (1983).
  21. S. I. Piskarev, “Solution of Second-Order Evolution Equations under Krein-Fattorini Conditions,” Differ. Uravn. 21 (9), 1604-1612 (1985).
  22. S. I. Piskarev, “Estimates of the Rate of Convergence in Solving Ill-Posed Problems for Evolution Equations,” Izv. Akad. Nauk SSSR, Ser. Mat. 51 (3), 676-687 [Math. USSR-Izv. 30 (3), 639-651 (1988)].
  23. S. I. Piskarev, Differential Equations in Banach Space and Their Approximation (Mosk. Gos. Univ., Moscow, 2005) [in Russian].
  24. V. V. Vasil’ev, S. I. Piskarev, and N. Yu. Selivanova, “Integrated Semigroups and C-Semigroups and Their Applications,” (VINITI, Moscow, 2017), Itogi Nauki Tekh., Ser. Sovrem. Mat. Pril. Temat. Obz., Vol. 131, pp. 3-109.
  25. A. Benrabah, N. Boussetila, and F. Rebbani, “Regularization Method for an Ill-Posed Cauchy Problem for Elliptic Equations,” J. Inv. Ill-Posed Problems 25 (3), 311-329 (2017).
  26. E. Gekeler, Discretization Methods for Stable Initial Value Problems (Springer, Berlin, 1984).
  27. N. S. Bakhvalov, N. P. Zhidkov, and G. M. Kobel’kov, Numerical Methods (Binom, Moscow, 2007) [in Russian].
  28. N. Dunford and J. T. Schwartz, Linear Operators. General Theory (Interscience, New York, 1958; Editorial, Moscow, 2004).
  29. M. Haase, The Functional Calculus for Sectorial Operators (Birkh854user, Basel, 2006).
  30. H. Triebel, Interpolation Theory, Function Spaces, Differential Operators (North-Holland, Amsterdam, 1978; Mir, Moscow, 1980).
  31. V. A. Trenogin, Functional Analysis (Nauka, Moscow, 1980) [in Russian].
  32. T. Kato, Perturbation Theory for Linear Operators (Springer, Berlin, 1966; Mir, Moscow, 1972).

Published

21-08-2017

How to Cite

Кокурин М.М. Rate of Convergence and Error Estimates for Finite-Difference Schemes of Solving Linear Ill-Posed Cauchy Problems of the Second Order // Numerical Methods and Programming (Vychislitel’nye Metody i Programmirovanie). 2017. 18. 322-347. doi 10.26089/NumMet.v18r428

Issue

Section

Section 1. Numerical methods and applications