Spline-regularization method for solving inverse problems: theory and algorithms

A. I. Grebennikov

Matematičeskoe modelirovanie, Tome 14 (2002) no. 5, pp. 35-50.

Résumé

The regularization method, based on spline approximation for solving operator equations of the first kind on discrete data, is considered. The technique of spline approximation consists in preliminary smoothing the noised right-hand side of the equation by Recursive Smoothing method and in the subsequent application of a collocation scheme. Here the new theoretical results on justification of existence and uniqueness of collocation spline are given. In particular, the justifications of some new collocation schemes for singular two-dimensional integral equations, typical for electroencephalography inverse problems, are obtained. The effective algorithms for proposed regularization method are constructed and realized as software package in MATLAB system. It can be used for the applied electrodynamics and aerodynamics problems, electroencephalography inverse problems, heat-conduction coefficient inverse problems and for the identifying of characteristics of the porous media of confined aquifers.

Bibliographie

[1] V. A. Morozov, Regularization Methods for Ill-Posed Problems, CRC Press, London, 1993 | MR | Zbl

[2] A. L. Grebennikov, Spline method and solving ill-posed problems of approximation theory, Moscow State Univ. Publ. House, M., 1983

[3] A. L. Grebennikov, Algorithms and programs for approximation of functions of one and several variables by splines and applications, Moscow State Univ. Publ. House, M., 1987 | Zbl

[4] A. I. Grebennikov, “Spline – approximating method of the solution the ill-conditioned problems”, Docl. Acad. Nauk of USSR, 298:3 (1988), 533–537 | MR | Zbl

[5] A. L. Grebennikov, “Spline approximation method for restoring functions”, Sov. J. Numer. Anal. Mathem. Modelling, 4:4 (1989), 1–15 | MR

[6] A. L. Grebennikov, “Solving integral equations with a singularity in the kernel by spline approximation method”, Sov. J. Numer. Anal. Mathem. Modelling, 5:3 (1990), 199–208 | DOI | MR | Zbl

[7] A. L. Grebennikov, “Algorithms for experimental data analysis allowing for additional a priori information”, Vychislitel'nye Metody. Sistemy Obrabotki na EVM, Plenum Publishing Corporation, 1991, 148–152

[8] V. A. Morozov, A. L. Grebennikov, Methods of Solving Ill-Posed Problems: Algorithmic Aspect, Moscow State Univ. Publ. House, Moscow, 1992

[9] A. N. Tikhonov, V. I. Dmitriev, “Method of calculation of the current density in the systems of linear vibrators and its polar pattern”, Numerical methods and programming, 10, Moscow State Univ. Publ. House, M., 1968, 3–8

[10] V. I. Dmitriev, E. V. Zaharov, “On numerical solution of some integral equations of the first kind”, Numerical methods and programming, 10, Moscow State Univ. Publ. House, M., 1968, 49–54 | MR

[11] S. M. Belotserkovsky, I. K. Lifanov, Method of Discrete Vortices, CRC Press, 1993 | MR

[12] I. K. Lifanov, Method of singular integral equations and numerical experiment, Yanus, M., 1995 | MR | Zbl

[13] V. N. Sobolev, Equations of mathematical physics, Nauka, M., 1966

[14] R. Kress, Linear Integral Equations, Springer-Verlag, Berlin Heidelberg, 1989 | MR | Zbl

[15] D. Kolton, R. Kress, Integral equations methods in scattering theory, Kruger Publishing Company, Malabar Florida, 1992

[16] B. G. Gabdulkhaev, Direct methods for Solving Singular Integral Equations of the First Kind. Numerical Analysis, Kazan Univ. Publ. House, Kazan, 1994 | MR

[17] V. A. Tsetsokho, “On collocation method for solving the integral equations of the first kind with the weak singularity in the kernel in the case of non-closed contours”, Inverse problems and interpretation of the geophysics observations, VC SO AN SSSR, Novosibirsk, 1983, 82–94

[18] A. I. Grebennikov, “Regularizating algorithms for solving incorrect problems by splines”, Methods and algorithms in numerical analysis, Moscow State Univ. Publ. House, M., 1984, 128–140 | MR

[19] A. I. Grebennikov, “Recursive smoothing method and spline-algorithms of handling of experimental data”, Numerical analysis: methods, algorithms, applications, Moscow State Univ. Publ. House, 1985, 82–95 | MR

[20] A. I. Grebennikov, Spline method in numerical analysis, Moscow State Univ. Publ. House, M., 1979 | MR

[21] I. J. Schoenberg, “Contributions to problem of approximation of equidistant data by analytic functions”, Quart. Appl. Math., 1946, 45–99 | MR | Zbl

[22] C. De Boor, “On calculation with $\mathrm{B}$-splines”, J. Approxim. Theory, 6:1 (1972), 50–62 | DOI | MR | Zbl

[23] A. N. Tikhonov, V. And. Arsenin, Solutions of ill-posed problems, Winston and Sons, Washington, 1977 | MR | Zbl

[24] S. G. Mikhlin, Linear Partial Differential Equations, Vyschaya Schkola, 1977 | MR

[25] E. V. Zakharov, Yu. V. Pimenov, Numerical analysis of radio-waves diffraction, Radio and Communication, M., 1982 | Zbl

[26] A. I. Grebennikov, “Spline-collocation method for solving some integral equation problems of diffraction of electromagnetic waves”, Using ECM for solving problems of mathematical physics, Moscow State Univ. Publ. House, 1985, 43–56 | MR

[27] A. I. Grebennikov, “Spline algorithms for data processing and solving some inverse problems”, Recent Advances in Numerical Methods and Applications. II, Proceedings of the Fourth International Conference NMA'98 (Sofia, Bulgaria), World Scientific, Singapore, 1999, 375–383 | MR | Zbl

[28] A. I. Grebennikov, “On solving some inverse problems of electroencephalography on spline – approximation method”, Collection of the Papers of the International Symposium on Optimization of Calculating, Kiev, 1999, 117–121

[29] A. I. Grebennikov, “Simulation by integral equations and numerical resolving some inverse problems of electroencephalography. Integral and Differential Equations”, Abstracts of the international conference (Tcheliabinsk, 22–26 June of 1999), Tcheliabinsk State Univ. Publ. House, 1999, 35

[30] A. I. Grebennikov, A. Fraguela, “Statement and numerical analysis of some inverse problems of electroencephalography”, Numerical Analysis: theory, applications, programs, Moscow State Univ. Publ. House, M., 1999, 28–46

[31] L. A. Lusternik, V. N. Sobolev, Elements of functional analysis, Nauka, M., 1970

[32] M. M. Lavrentiev, V. G. Romanov, S. P. Shishatsky, Ill-posed problems of the mathematical physics and analysis, Nauka, M., 1980 | MR

[33] O. M. Alifanov, E. A. Artyukhin, S. V. Rumiantsev, Extreme methods for solving illposed problems with applications to inverse heat transfer problems, Begel House Inc., New-York-Wallimgford, 1995 | Zbl

[34] A. M. Denisov, Introduction in the Theory of the Inverse Problems, Moscow State Univ. Publ. House, M., 1994

[35] S. Gomez, A. Perez, R. M. Alvarez, “Multiscale Optimization for Aquifer Parameter Identification with Noisy Data”, Computational Methods in Water Resources XII, XII International Conference, v. 2, 1998

[36] A. I. Grebennikov, “Fast and stable algorithms of solving some inverse problems Algorithmic and numerical analysis of ill-posed problems”, Abstracts of Russian conference dedicated to the memory of V. K. Ivanov, Ekaterinburg, 1995, 53

[37] A. I. Grebennikov, “Numerical restoration of coefficients of differential equations. Algorithmic analysis of ill-posed problems”, Abstracts of Russian conference dedicated to the memory of V. K. Ivanov, Ekaterinburg, 1998, 75

[38] A. I. Grebennikov, “Fast spline-algorithms for multidimensional data processing and solving coefficient inverse problems for heat-conduction equation”, Numerical analysis: methods and algorithms, MSU, Moscow, 1998, 62–71

[39] A. I. Grebennikov, “Spline approximation method for solving unstable problems and applications”, XXXII National Congress of the Mexican Mathematical Society (Guadalajara, 1999), Aportaciones Mat. Comun., 27, Soc. Mat. Mexicana, Mexico, 2000, 53–70 | MR | Zbl