##
**A stable method for linear equation in Banach spaces with smooth norms.**
*(English)*
Zbl 1450.65046

Summary: 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.

### Keywords:

linear operator equation; Banach space; numerical solution; stable method; sourcewise representability; wave equation
PDF
BibTeX
XML
Cite

\textit{A. A. Dryazhenkov} and \textit{M. M. Potapov}, Ural Math. J. 4, No. 2, 56--68 (2018; Zbl 1450.65046)

### References:

[1] | Adams R.A., Fournier J.J.F., Sobolev Spaces, Elsevier, Amsterdam, 2003, 320 pp. |

[2] | Bakushinskii A.B., “Methods for solving monotonic variational inequalities, based on the principle of iterative regularization”, USSR Computational Mathematics and Mathematical Physics, 17:6 (1977), 12-24 · Zbl 0395.49028 |

[3] | Bakushinsky A., Goncharsky A., III-Posed Problems: Theory and Applications, Kluwer Academic Publishers, Dordrecht, 1994, 258 pp. |

[4] | Brezis H., Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011, 599 pp. · Zbl 1220.46002 |

[5] | Cioranescu I., Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Kluwer Academic Publishers, Dordrecht, 1990, 260 pp. · Zbl 0712.47043 |

[6] | Dryazhenkov A.A., Potapov M.M., “Constructive observability inequalities for weak generalized solutions of the wave equation with elastic restraint”, Comput. Math. Math. Phys., 54:6 (2014), 939-952 · Zbl 1313.35208 |

[7] | Dunford N., Schwartz J.T., Linear Operators. Part I: General Theory, Interscience Publishers, New York, 1958, 872 pp. · Zbl 0084.10402 |

[8] | Ekeland I., Temam R., Convex Analysis and Variational Problems, North-Holland Publishing Company, Amsterdam, 1976, 394 pp. · Zbl 0322.90046 |

[9] | Ekeland I., Turnbull T., Infinite-Dimensional Optimization and Convexity, The University of Chicago Press, Chicago, 1983, 174 pp. · Zbl 0565.49003 |

[10] | Engl H.W., Hanke M., Neubauer A., Regularization of Inverse Problems, Kluwer Academic Publishers, Dordrecht, 1996, 322 pp. · Zbl 0859.65054 |

[11] | 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., 48:11 (2012), 1470-1476 · Zbl 1307.35153 |

[12] | Ivanov V.K., “On linear problems that are not well-posed”, Soviet Mathematics Doklady, 3 (1962), 981-983 |

[13] | Kantorovich L.V., Akilov G.P., Functional Analysis, Pergamon Press, Oxford, 1982, 604 pp. · Zbl 0484.46003 |

[14] | Krein S.G., Linear Equations in Banach Spaces, Birkhäuser, Boston, 1982, 106 pp. · Zbl 0535.47008 |

[15] | Lions J.L., “Exact controllability, stabilization and perturbations for distributed systems”, SIAM Rev., 30:1 (1988), 1-68 · Zbl 0644.49028 |

[16] | Morozov V.A., “Regularization of incorrectly posed problems and the choice of regularization parameter”, USSR Computational Mathematics and Mathematical Physics, 6:1 (1966), 242-251 · Zbl 0176.13103 |

[17] | Phillips D.L., “A technique for the numerical solution of certain integral equations of the first kind”, J. ACM, 9:1 (1962), 84-97 · Zbl 0108.29902 |

[18] | Potapov M.M., “Strong convergence of difference approximations for problems of boundary control and observation for the wave equation”, Comput. Math. Math. Phys., 38:3 (1998), 373-383 · Zbl 0948.49016 |

[19] | Potapov M.M., “A stable method for solving linear equations with nonuniformly perturbed operators”, Dokl. Math., 59:2 (1999), 286-288 · Zbl 0978.47010 |

[20] | Riesz F., Sz.-Nagy B., Functional Analysis, Blackie & Son Limited, London, 1956, 468 pp. · Zbl 0070.10902 |

[21] | Scherzer O., Grasmair M., Grossauer H., Haltmeier M., Lenzen F., Variational Methods in Imaging, Springer, New York, 2009, 320 pp. · Zbl 1177.68245 |

[22] | Schuster T., Kaltenbacher B., Hofmann B., Kazimierski K.S., Regularization Methods in Banach Spaces, De Gruyter, Berlin, 2012, 283 pp. · Zbl 1259.65087 |

[23] | Tikhonov A.N., “Solution of incorrectly formulated problems and the regularization method”, Soviet Mathematics Doklady, 4:4 (1963), 1035-1038 · Zbl 0141.11001 |

[24] | Tikhonov A.N., Arsenin V.Y., Solution of Ill-posed Problems, Winston & Sons, Washington, 1977, 258 pp. · Zbl 0354.65028 |

[25] | Tikhonov A.N., Leonov A.S., Yagola A.G., Nonlinear Ill-posed Problems, Chapman & Hall, London, 1998, 386 pp. · Zbl 0920.65038 |

[26] | Zuazua E., “Propagation, observation, and control of waves approximated by finite difference methods”, SIAM Rev., 47:2 (2005), 197-243 · Zbl 1077.65095 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.