Andrey A. Dryazhenkov     (Lomonosov Moscow State University, Leninskie Gory, Moscow, 119991, Russian Federation)
Mikhail M. Potapov     (Lomonosov Moscow State University, Leninskie Gory, Moscow, 119991, Russian Federation)


A stable method for numerical solution of a linear operator equation in reflexive Banach spaces is proposed. The operator and the right-hand side of the equation are assumed to be known approximately. The corresponding error levels may remain unknown. Approximate operators and their conjugate ones must possess the property of strong pointwise convergence. The exact normal solution is assumed to be sourcewise representable and some upper estimate for the norm of its source element must be known. The norm in the Banach space of solutions is supposed to satisfy the following smoothness-type condition: some function of the norm must be differentiable. Under these conditions a stability of the method with respect to nonuniform perturbations in operator is shown and the strong convergence to the normal solution is proved. A boundary control problem for the one-dimensional wave equation is considered as an example of possible application. The results of the model numerical experiments are presented.



Linear operator equation, Banach space, Numerical solution, Stable method, Sourcewise representability, Wave equation

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Adams R.A., Fournier J.J.F. Sobolev Spaces. Amsterdam: Elsevier, 2003. 320 p.

Bakushinskii A.B. Methods for solving monotonic variational inequalities, based on the principle of iterative regularization. USSR Computational Mathematics and Mathematical Physics, 1977. Vol. 17, No. 6. P. 12–24.

Bakushinsky A., Goncharsky A. III-Posed Problems: Theory and Applications. Dordrecht: Kluwer Academic Publishers, 1994. 258 p. DOI: 10.1007/978-94-011-1026-6

Brezis H. Functional Analysis, Sobolev Spaces and Partial Differential Equations. New York: Springer, 2011. 599 p. DOI: 10.1007/978-0-387-70914-7

Cioranescu I. Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Dordrecht: Kluwer Academic Publishers, 1990. 260 p. DOI: 10.1007/978-94-009-2121-4

Dryazhenkov A.A., Potapov M.M. Constructive observability inequalities for weak generalized solutions of the wave equation with elastic restraint. Comput. Math. Math. Phys., 2014. Vol. 54, No. 6. P. 939–952. DOI: 10.1134/S0965542514060062

Dunford N., Schwartz J.T. Linear Operators. Part I: General Theory. New York: Interscience Publishers, 1958. 872 p.

Ekeland I., Temam R. Convex Analysis and Variational Problems. Amsterdam: North-Holland Publishing Company, 1976. 394 p. DOI: 10.1137/1.9781611971088

Ekeland I., Turnbull T. Infinite-Dimensional Optimization and Convexity. Chicago: The University of Chicago Press, 1983. 174 p.

Engl H.W., Hanke M., Neubauer A. Regularization of Inverse Problems. Dordrecht: Kluwer Academic Publishers, 1996. 322 p.

Il’in V.A., Kuleshov A.A. On some properties of generalized solutions of the wave equation in the classes \(L_p\) and \(W_p^1\) for \(p \geq 1\). Differ. Equ., 2012. Vol. 48, No. 11. P. 1470–1476. DOI: 10.1134/S0012266112110043

Ivanov V.K. On linear problems that are not well-posed. Soviet Mathematics Doklady, 1962. Vol. 3. P. 981–983.

Kantorovich L.V., Akilov G.P. Functional Analysis. Oxford: Pergamon Press, 1982. 604 p. DOI: 10.1016/C2013-0-03044-7

Krein S.G. Linear Equations in Banach Spaces. Boston: Birkhäuser, 1982. 106 p. DOI: 10.1007/978-1-4684-8068-9

Lions J.-L. Exact controllability, stabilization and perturbations for distributed systems. SIAM Rev., 1988. Vol. 30, No. 1. P. 1–68. DOI: 10.1137/1030001

Morozov V.A. Regularization of incorrectly posed problems and the choice of regularization parameter. USSR Computational Mathematics and Mathematical Physics, 1966. Vol. 6, No. 1. P. 242–251. DOI: 10.1016/0041-5553(66)90046-2

Phillips D.L. A technique for the numerical solution of certain integral equations of the first kind. J. ACM, 1962. Vol. 9, No. 1. P. 84–97. DOI: 10.1145/321105.321114

Potapov M.M. Strong convergence of difference approximations for problems of boundary control and observation for the wave equation. Comput. Math. Math. Phys., 1998. Vol. 38, No. 3. P. 373–383.

Potapov M.M. A stable method for solving linear equations with nonuniformly perturbed operators. Dokl. Math., 1999. Vol. 59, No. 2. P. 286–288.

Riesz F., Sz.-Nagy B. Functional Analysis. London: Blackie & Son Limited, 1956. 468 p.

Scherzer O., Grasmair M., Grossauer H., Haltmeier M., Lenzen F. Variational Methods in Imaging. New York: Springer, 2009. 320 p. DOI: 10.1007/978-0-387-69277-7

Schuster T., Kaltenbacher B., Hofmann B., Kazimierski K.S. Regularization Methods in Banach Spaces. Berlin: De Gruyter, 2012. 283 p.

Tikhonov A.N. Solution of incorrectly formulated problems and the regularization method. Soviet Mathematics Doklady, 1963. Vol. 4, No. 4. P. 1035–1038.

Tikhonov A.N., Arsenin V.Y. Solution of Ill-posed Problems. Washington: Winston & Sons, 1977. 258 p.

Tikhonov A.N., Leonov A.S., Yagola A.G. Nonlinear Ill-posed Problems. London: Chapman & Hall, 1998. 386 p.

Zuazua E. Propagation, observation, and control of waves approximated by finite difference methods. SIAM Rev., 2005. Vol. 47, No. 2. P. 197–243. DOI: 10.1137/S0036144503432862

DOI: http://dx.doi.org/10.15826/umj.2018.2.007

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