Abstract
We study finite difference methods and the quasi-reversibility method in application to linear ill-posed Cauchy problems with a self-adjoint operator in a Hilbert space with exact data. We prove that for these problems, it is possible to strengthen our earlier results on the convergence of the mentioned methods in a general case of a Banach space. We establish close to each other necessary and sufficient conditions for the qualified convergence of methods under consideration in terms of the source-representability exponent of the desired solution. We prove that (except the trivial case) the power estimates of the convergence rate of the considered methods cannot exceed the saturation level that corresponds to this or that method.
Similar content being viewed by others
References
Bakushinskii, A.B., Kokurin, M.Yu., Kluchev, V.V. “A Rate of Convergence and Error Estimates for Difference Methods Used to Approximate Solutions to Ill-Posed Cauchy Problems in a Banach Space”, Vychisl. Metody i Programm.7, 163–171 (2006).
Ivanov, V.K., Mel’nikova, I.V., Filinkov, A.I. Differential-Operator Equations and Ill-Posed Problems (Fizmatlit, Moscow, 1995) [in Russian].
Krein, S.G. Linear Differential Equations in a Banach Space (Nauka, Moscow, 1967) [in Russian].
Bakushinskii, A.B., Kokurin, M.M., Kokurin, M.Yu. On a Class of Finite-Difference Schemes for Solving Ill-Posed Cauchy Problems in Banach Spaces, Computational Mathematics and Mathematical Physics 52(3), 411–426 (2012).
Haase, M. The Functional Calculus for Sectorial Operators (Birkhäuser, Basel, 2006).
Riesz, F., Szökefalvi-Nagy, B. Lectures on Functional Analysis (Akadémi Kiadö, Budapest, 1972; Mir, Moscow, 1979).
Trenogin, V.A. Functional Analysis (Fizmatlit, Moscow, 2007) [in Russian].
Bakushinskii, A.B. “Difference Methods of Solving Ill-Posed Cauchy Problems for Evolution Equations in a Complex B-Space”, Différents. Uravnenya8(9), 1661–1668 (1972).
Kokurin, M.M. “Optimization of Convergence Rate Estimates for Some Classes of Finite Difference Schemes for Solving Ill-Posed Cauchy Problems”, Vychisl. Metody i Programm.14, 58–76 (2013).
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”, Computational Mathematics and Mathematical Physics55(12), 1986–2000 (2015).
Samarskii, A.A., Vabishchevich, P.N. Numerical Methods for Solving Inverse Problems of Mathematical Physics (Editorial URSS, Moscow, 2004) [in Russian].
Triebel, H. Interpolation Theory, Function Spaces, Differential Operators (VEB Deutscher Verlag der Wissenschaften, Berlin, 1978; Mir, Moscow, 1980).
Lattes, R., Lions, J.-L. Quasi-Reversibility Method and Its Applications (Mir, Moscow, 1970) [in Russian].
Funding
This work was supported by the Russian Foundation for Basic Research (project 16-01-00039a), the Ministry of Education and Science of the Russian Federation within the framework of the state task (project 1.5420.2017 / 8.9), and the scholarship of the President of the Russian Federation for young scientists and postgraduate students (SP-5252.2018.5).
Author information
Authors and Affiliations
Corresponding author
Additional information
Russian Text © The Author(s), 2019, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2019, No. 10, pp. 46–61.
About this article
Cite this article
Kokurin, M.M. Conditions for the Qualified Convergence of Finite Difference Methods and the Quasi-Reversibility Method for Solving Linear Ill-Posed Cauchy Problems in a Hilbert Space. Russ Math. 63, 40–54 (2019). https://doi.org/10.3103/S1066369X19100062
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1066369X19100062