Perturbation Theory and the Banach – Steinhaus Theorem for Regularization of the Linear Equations of the First Kind
The regularizing equations with a vector parameter of regularization are constructed for the linear equations with closed operator acting in Banach spaces. Range of the operator can be an open, and the homogeneous equation may have a non-trivial solution. It is assumed that only approximations of operator and source are known. The conditions of solution uniqueness for the auxiliary regularized equation are derived. The convergence of regularized solution to B-normal solution of the exact equation is proved. The bounds estimates are derived for both deterministic and stochastic cases. The choice of the stabilizing operator and vector regularization parameter are provided. The method is applied to the problem of stable differentiation.
1. Govurin M.K. Lectures on Computational Methods. Moscow, Nauka Publ., 1971.
2. Gukovskii S. A. Regularization of the Construction of a Nonsingular Solution of a Linear Equations of the First Kind with Degeneracy. (in Russian)Izv. Vyssh. Uchebn. Zaved. Mat., 1980, no 1, pp. 71-74.
3. Ivanov V.K., Vasin V.V., Tanana V.P. The Theory of Linear Ill-posed Problems and their Applications (in Russian). Moscow, Nauka, 1978.
4. Lavrentiev M.M. Some Improperly Posed Problems in Mathematical Physics. Springer, Berlin, 1967.
5. Latt`es R., Lions J.-L. The method of quasi-inversibility: application to partial differential equations. NY, American Elsevier, 1969.
6. Loginov B.V., Sidorov N.A. Calculation of Eigenvalues and Eigenvectors of Bounded Operators by the False-Perturbation Method. Mathematical notes of the Academy of Sciences of the USSR, 1976, vol. 19, issue 1, p. 62–64.
7. Maslov V.P. The Existence of a Solution of an Ill-Posed Problem is Equivalent to the Convergence of a Regularization Process. (in Russian) Uspekhi Mat. Nauk, 1968, vol. 23, no 3 (141), pp. 183-184.
8. Samarskii A.A. Introduction to Finite Difference Schemes. Moscow, Nauka Publ., 1971.
9. Sidorov N.A., Trenogin V.A. Linear Equations Regularization using the Perturbation Theory. Diff. Eqs., 1980, vol. 16, no 11, pp. 2038-2049.
10. Sidorov N.A., Trenogin V.A. A Certain Approach the Problem of Regularization of the Basis of the Perturbation of Linear Operators. Mathematical notes of the Academy of Sciences of the USSR, 1976, vol. 20, no 5, p. 976-979.
11. Sidorov N.A. Calculation of Eigenvalues and Eigenvectors of Linear Operators by the Theory of Perturbations (in Russian). Differential Equations, 1978, vol. 14, no 8, p. 1522-1525.
12. Sidorov N. A. General Issues of Regularisation in the Problems of the Theory of Branching. Irkutsk, Irkutsk State University Publ., 1982. 312 p.
13. Sidorov N.A., Leont’ev R.Yu., Dreglya A.I. On Small Solutions of Nonlinear Equations with Vector Parameter in Sectorial Neighborhood. Mathematical Notes, feb. 2012, vol. 91, no 1-2, pp. 90-104.
14. Sidorov N.A. Explicit and Implicit Parametrisation of the Construction of Branching Solutions by Iterative Methods. Sbornik: Mathematics. 1995, vol. 186, no. 2, pp. 297-310.
15. Sidorov N.A., Sidorov D.N. Solving the Hammerstein Integral Equation in Irregular Case by Successive Approximations. Siberian Mathematical Journal, March 2010, vol. 51, no 2, pp. 325-329.
16. Sidorov N. A., Sidorov D. N., Krasnik A.V. On Solution of the Volterra Operator-Integral Equations in Irregular Case using Successive Approximations. Differential Equations, 2010, vol. 46, no 6, pp.874-882.
17. Sidorov D.N., Tynda A. N., Muftahov I.R. Numerical Solution of Volterra Integral Equations of the First Kind with Piecewise Continuous Kernel. Bul. of the South Ural State University. Ser. “Math. Model., Programming and Comp. Software”, 2014, vol. 7, no 3, pp. 107-115.
18. Stechkin S.B. The Best Approximation of Linear Operators. Mat. Notes. 1967, vol. 1, no 2, pp. 137-148.
19. Tikhonov A.N. Solution of Ill-Posed Problems. Winston. New York, 1977.
20. Tikhonov A.N., Ivanov V.K., Lavrent’ev. Ill-posed Problems. In the book Partial Differential Eqs, Moscow, Nauka, 1970, pp. 224-239.
21. Trenogin V.A. Functional analysis. Moscow, Nauka, 1980. 496 p.
22. Yagola A.G. Inverse Problems and Methods of Their Solution. Applications to Geophysics. (in Russian) Binom Publ. Ser. Mathematical Modelling, 2014. 216 p.
23. H¨amarik U., Kaltenbacher B., Kangro U., Resmerita E. Regularization by Discretization in Banach Spaces. ArXiv, Numerical Analysis, arXiv:1506.05425, 2015, pp. 1-36.
24. H`ao N. D., Chuonga L.N., Lesnic D. Heuristic regularization methods for numerical differentiation. Computers and Mathematics with Applications, 2012, vol. 63, pp. 816-826.
25. Marchuk G.I. Perturbation theory and the statement of inverse problems. Lecture Notes in Computer Science. Vol. 4: 5th Conf. on Optimization Tech., 1973, pp. 159-166.
26. Muftahov I.R., Sidorov D.N., Sidorov N.A. On perturbation method for the first kind equations: regularization and applications. Bul. of the South Ural State University. Ser. “Math. Model., Programming and Comp. Software”, 2015, vol.8, no 2, pp. 69-80.
27. Ramm A. G., Smirnova A. B. On Stable Numerical Differentiation. Mathematics of Computation, 2001, vol. 70, no. 235, pp. 1131-1153.
28. Sidorov D. Integral Dynamical Integral Dynamical Models: Singularities, Signals and Control Ed. by L. O. Chua, Singapore. London, World Scientific Publ., 2014, vol. 87 of World Scientific Series on Nonlinear Science, Series A, 243 p.
29. Sidorov N., Loginov B., Sinitsyn A., Falaleev M. Lyapunov-Schmidt Methods in Nonlinear Analysis and Applications. Dortrecht, Kluwer Academic Publ., 2002. 548 p.
30. Sizikov V.S. Further Development of the New Version of a Posteriori Choosing Regularization Parameter in Ill-Posed Problems. Intl. J. of Artificial Intelligence, 2015, vol. 13, no 1, pp. 184–199.
31. Trenogin V.А., Sidorov N.A. Regularization of computation of branching solution of nonlinear equations. Lecture Notes in Mathematics, 1977, vol. 594, pp. 491-506.
32. Wu J. K., Long J., He F., He Q. L. Numerical differentiation based algorithm for power measurement. 5th Intl IEEE Conf. in Power Electronics and Drive Systems, 2003, vol. 1, pp. 302-307.