Abstract
The convergence of the quasi-reversibility method and two classes of finite-difference methods for solving the ill-posed Cauchy problem for the first-order equation with a sectorial operator in a Banach space is analyzed. The necessary and sufficient conditions—close to one another—for the convergence of these methods with a rate polynomial with respect to the regularization parameter or discretization step are obtained in terms of the exponent in the source representability of the solution.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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,” Comput. Math. Math. Phys. 52 (3), 411–426 (2012).
T. Kato, Perturbation Theory for Linear Operators (Springer-Verlag, Berlin, 1966).
S. Agmon, “On the eigenfunctions and the eigenvalues of general elliptic boundary value problems,” Commun. Pure Appl. Math. 15, 119–147 (1962).
H. B. Stewart, “Generation of analytic semigroups by strongly elliptic operators under general boundary conditions,” Trans. Am. Math. Soc. 259, 299–310 (1980).
V. K. Ivanov, I. V. Mel’nikova, and A. I. Filippov, Differential-Operator Equations and Ill-Posed Problems (Nauka, Moscow, 1995) [in Russian].
A. B. Bakushinskii, M. Yu. Kokurin, and V. V. Klyuchev, “Estimation of the convergence rate and error for finite-difference methods approximating solutions of Cauchy problems in Banach spaces,” Vychisl. Metody Program. 7, 163–171 (2006).
A. B. Bakushinskii and M. Yu. Kokurin, Iterative Methods for Solving Ill-Posed Operator Equations with Smooth Operators (Editorial URSS, Moscow, 2002) [in Russian].
A. B. Bakushinskii, “Finite-Difference methods as applied to ill-posed Cauchy problems for evolution equations in complex V space,” Differ. Uravn. 8 (9), 1661–1668 (1972).
M. M. Kokurin, “On optimization of convergence rate estimate for some classes of difference schemes as applied to ill-posed Cauchy problems,” Vychisl. Metody Program. 14, 58–76 (2013).
R. Lattes and J.-L. Lions, Methode de quasi reversibilite et applications (Dunod, Paris, 1967; Mir, Moscow, 1970).
M. Haase, The Functional Calculus for Sectorial Operators (Birkhäuser, Basel, 2006).
S. G. Krein, Linear Differential Equations in Banach Spaces (Nauka, Moscow, 1971; Birkhäuser, Boston, 1982).
A. B. Bakushinsky, M. Yu. Kokurin, and S. K. Paymerov, “On error estimates of difference solution methods for ill-posed Cauchy problems in a Hilbert space,” J. Inverse Ill-Posed Probl. 16 (6), 553–565 (2008).
M. Yu. Kokurin and M. M. Kokurin, “On a class of finite difference methods for ill-posed Cauchy problems with noisy data,” J. Inverse Ill-Posed Probl. 18, 959–977 (2011).
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,” Proc. Steklov Inst. Math. 280, Suppl. 1, 53–65 (2013).
A. A. Samarskii and P. N. Vabishchevich, Numerical Methods for Inverse Problems in Mathematical Physics (Editorial URSS, Moscow, 2004) [in Russian].
H. Triebel, Interpolation Theory, Function Spaces, Differential Operators (Springer-Verlag, Berlin, 1978; Mir, Moscow, 1980).
K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations (Springer, New York, 2000).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © M.M. Kokurin, 2015, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2015, Vol. 55, No. 12, pp. 2027–2041.
Rights and permissions
About this article
Cite this article
Kokurin, M.M. 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. Comput. Math. and Math. Phys. 55, 1986–2000 (2015). https://doi.org/10.1134/S0965542515120076
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0965542515120076