Direct Method for Identification of Two Coefficients of Acoustic Equation

Novikov, Nikita S.; Shishlenin, Maxim A.

doi:10.3390/math11133029

Cited by 2 publications

(1 citation statement)

References 41 publications

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“…The Gelfand-Levitan-Krein method was applied for solving acoustic [98][99][100], elasticity [101] and seismic [102,103] coefficient inverse problems. We also should mention series of works of A.V.…”

Section: Inverse Problems For Hyperbolic Equationsmentioning

confidence: 99%

Spectral, Scattering and Dynamics: Gelfand–Levitan–Marchenko–Krein Equations

Kabanikhin,

Shishlenin,

Novikov

et al. 2023

Mathematics

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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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Section: Inverse Problems For Hyperbolic Equationsmentioning

confidence: 99%

Spectral, Scattering and Dynamics: Gelfand–Levitan–Marchenko–Krein Equations

Kabanikhin,

Shishlenin,

Novikov

et al. 2023

Mathematics

Self Cite

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Numerical Solution of the Cauchy Problem for the Helmholtz Equation Using Nesterov’s Accelerated Method

Kasenov,

Tleulesova,

Sarsenbayeva

et al. 2024

Mathematics

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In this paper, the Cauchy problem for the Helmholtz equation, also known as the continuation problem, is considered. The continuation problem is reduced to a boundary inverse problem for a well-posed direct problem. A generalized solution to the direct problem is obtained and an estimate of its stability is given. The inverse problem is reduced to an optimization problem solved using the gradient method. The convergence of the Landweber method with respect to the functionals is compared with the convergence of the Nesterov method. The calculation of the gradient in discrete form, which is often used in the numerical solutions of the inverse problem, is described. The formulation of the conjugate problem in discrete form is presented. After calculating the gradient, an algorithm for solving the inverse problem using the Nesterov method is constructed. A computational experiment for the boundary inverse problem is carried out, and the results of the comparative analysis of the Landweber and Nesterov methods in a graphical form are presented.

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Direct Method for Identification of Two Coefficients of Acoustic Equation

Cited by 2 publications

References 41 publications

Spectral, Scattering and Dynamics: Gelfand–Levitan–Marchenko–Krein Equations

Spectral, Scattering and Dynamics: Gelfand–Levitan–Marchenko–Krein Equations

Numerical Solution of the Cauchy Problem for the Helmholtz Equation Using Nesterov’s Accelerated Method

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