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Relevant bibliographies by topics / Gelfand–Levitan method

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Academic literature on the topic 'Gelfand–Levitan method'

Author: Grafiati

Published: 5 June 2025

Last updated: 24 June 2025

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Journal articles on the topic "Gelfand–Levitan method"

1

Temirbekov, N. М., S. I. Kabanikhin, L. N. Тemirbekova, and Zh E. Demeubayeva. "Gelfand-Levitan integral equation for solving coefficient inverse problem." Bulletin of the National Engineering Academy of the Republic of Kazakhstan 85, no. 3 (2022): 158–67. http://dx.doi.org/10.47533/2020.1606-146x.184.

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In this paper, numerical methods for solving multidimensional equations of hyperbolic type by the Gelfand-Levitan method are proposed and implemented. The Gelfand-Levitan method is one of the most widely used in the theory of inverse problems and consists in reducing a nonlinear inverse problem to a one-parameter family of linear Fredholm integral equations of the first and second kind. In the class of generalized functions, the initial-boundary value problem for a multidimensional hyperbolic equation is reduced to the Goursat problem. Discretization and numerical implementation of the direct Goursat problem are obtained to obtain additional information for solving a multidimensional inverse problem of hyperbolic type. For the numerical solution, a sequence of Goursat problems is used for each giveny. A comparative analysis of numerical experiments of the two-dimensional Gelfand-Levitan equation is performed. Numerical experiments are presented in the form of tables and figures for various continuous functions q(x, y).

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2

Bakanov, G. B., and S. K. Meldebekova. "Relationship between solutions of discrete inverse and auxiliary problems for a hyperbolic equation." Q A Iasaýı atyndaǵy Halyqaralyq qazaq-túrіk ýnıversıtetіnіń habarlary (fızıka matematıka ınformatıka serııasy) 31, no. 4 (2024): 19–27. https://doi.org/10.47526/2024-4/2524-0080.13.

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One-dimensional and multidimensional methods for solving inverse problems for a hyperbolic equation by the Gelfand-Levitan method lead to the numerical solution of Fredholm integral equations of the first and second kind. This paper examines the connection between a discrete inverse problem and solutions to a discrete auxiliary problem for a hyperbolic equation studied by the Gelfand-Levitan method. First, we present the formulations of the discrete inverse problem and the discrete auxiliary problem for the hyperbolic equation, as well as the properties of the grid functions that are solutions to these problems. Lemmas are proved about the structure of the grid function determined by the solution of the auxiliary discrete problem, and the connection of this grid function with the desired grid function, which is the solution to the discrete inverse problem. When proving the lemmas, the properties of the solution to the discrete auxiliary problem and the discrete analogue of the Dirac delta function are taken into account. A theorem showing the existence of a discrete inverse problem solution and its uniqueness is proved.

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3

Kabanikhin, Sergey I., Nikita S. Novikov, and Maxim A. Shishlenin. "Gelfand-Levitan-Krein method in one-dimensional elasticity inverse problem." Journal of Physics: Conference Series 2092, no. 1 (2021): 012022. http://dx.doi.org/10.1088/1742-6596/2092/1/012022.

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Abstract In this article we propose the numerical solution of the one dimensional inverse coefficient problem for seismic equation. We use a dynamical version of Gelfand-Levitan-Krein approach for reducing a nonlinear inverse problem for recovering the shear wave’s velocity and the density of the medium to two sequences of the linear integral equations. We propose numerical algorithm for solving these equations based on a fast inversion of a Toeplitz matrix. The proposed numerical methods base on the structure of the problem and therefore improve the efficiency of the algorithms, compared with standard approaches. We present numerical results for solving considered integral equations.

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4

Kabanikhin, Sergey, Maxim Shishlenin, Nikita Novikov, and Nikita Prokhoshin. "Spectral, Scattering and Dynamics: Gelfand–Levitan–Marchenko–Krein Equations." Mathematics 11, no. 21 (2023): 4458. http://dx.doi.org/10.3390/math11214458.

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In this paper, we consider the Gelfand–Levitan–Marchenko–Krein approach. It is used for solving a variety of inverse problems, like inverse scattering or inverse problems for wave-type equations in both spectral and dynamic formulations. The approach is based on a reduction of the problem to the set of integral equations. While it is used in a wide range of applications, one of the most famous parts of the approach is given via the inverse scattering method, which utilizes solving the inverse problem for integrating the nonlinear Schrodinger equation. In this work, we present a short historical review that reflects the development of the approach, provide the variations of the method for 1D and 2D problems and consider some aspects of numerical solutions of the corresponding integral equations.

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5

Becker, Kyle M., and George V. Frisk. "Application of the Gelfand–Levitan method to geoacoustic inversion in shallow water." Journal of the Acoustical Society of America 108, no. 5 (2000): 2536. http://dx.doi.org/10.1121/1.4743394.

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6

Chaturvedi, S., and K. Raghunathan. "Relation between the Gelfand-Levitan procedure and the method of supersymmetric partners." Journal of Physics A: Mathematical and General 19, no. 13 (1986): L775—L778. http://dx.doi.org/10.1088/0305-4470/19/13/004.

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7

HORVÁTH, MIKLÓS, and BARNABÁS APAGYI. "SOLUTION OF THE INVERSE SCATTERING PROBLEM AT FIXED ENERGY FOR POTENTIALS BEING ZERO BEYOND A FIXED RADIUS." Modern Physics Letters B 22, no. 23 (2008): 2137–49. http://dx.doi.org/10.1142/s0217984908016923.

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Based on the relation between the m-function and the spectral function we construct an inverse quantum scattering procedure at fixed energy which can be applied to spherical radial potentials vanishing beyond a fixed radius a. To solve the Gelfand–Levitan–Marchenko integral equation for the transformation kernel, we determine the input symmetrical kernel by using a minimum norm method with moments defined by the input set of scattering phase shifts. The method applied to the box and Gauss potentials needs further practical developments regarding the treatment of bound states.

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8

Novikov, Nikita, and Maxim Shishlenin. "Direct Method for Identification of Two Coefficients of Acoustic Equation." Mathematics 11, no. 13 (2023): 3029. http://dx.doi.org/10.3390/math11133029.

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We consider the coefficient inverse problem for the 2D acoustic equation. The problem is recovering the speed of sound in the medium (which depends only on the depth) and the density (function of both variables). We describe the method, based on the Gelfand–Levitan–Krein approach, which allows us to obtain both functions by solving two sets of integral equations. The main advantage of the proposed approach is that the method does not use the multiple solution of direct problems, and thus has quite low CPU time requirements. We also consider the variation of the method for the 1D case, where the variation of the wave equation is considered. We illustrate the results with numerical experiments in the 1D and 2D case and study the efficiency and stability of the approach.

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9

Romanov, V. G. "On Justification of the Gelfand–Levitan–Krein Method for a Two-Dimensional Inverse Problem." Siberian Mathematical Journal 62, no. 5 (2021): 908–24. http://dx.doi.org/10.1134/s003744662105013x.

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10

Romanov, V. G. "On justification of the Gelfand–Levitan–Krein method for a two-dimensional inverse problem." Sibirskii matematicheskii zhurnal 62, no. 5 (2021): 1124–42. http://dx.doi.org/10.33048/smzh.2021.62.513.

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Book chapters on the topic "Gelfand–Levitan method"

1

Sarwar, A. K. M., and K. W. Holladay. "An Efficient Algorithm for Impedance Reconstruction by the Modified Gelfand-Levitan Inverse Method." In Theory and Practice of Geophysical Data Inversion. Vieweg+Teubner Verlag, 1992. http://dx.doi.org/10.1007/978-3-322-89417-5_12.

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Conference papers on the topic "Gelfand–Levitan method"

1

Winick, Kim A., and Jose E. Roman. "Dispersion compensation and pulse compression using waveguide grating filters." In OSA Annual Meeting. Optica Publishing Group, 1991. http://dx.doi.org/10.1364/oam.1991.my7.

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In long distance optical communication systems, fiber chromatic dispersion degrades the system’s transmission capabilities by limiting the achievable bit rates and repeater spacings. One way to solve this problem is to perform dispersion compensation, or equivalently pulse compression, in the communications network. Recently, chromatic dispersion compensation in a 5-Gbit/s transmission system has been achieved with a fiber Fabry-Perot optical equalizer. The amount of dispersion compensated by a single Fabry-Perot equalizer is, however, limited. Linearly chirped waveguide grating filters can also be used for dispersion compensation as reported by Ouel lette.1 In this paper, we derive the grating parameters needed to obtain nearly optimum dispersion compensation. The resulting aperiodic grating filter has both a flat amplitude and a quadratic phase response over the pulse bandwidth. The response corresponds to the ideal transfer function for the compression linearly chirped pulses. The filter is analytically designed by cascading a Butterworth filter with an all-pass network. The pole placement of the all-pass network is selected to give the desired quadratic phase response. The grating parameters, corresponding to this filter, are obtained by solving the inverse problem for contradirectional waveguide coupling using the Gelfand-Levitan-Marchenko method.2

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