Algorithm construction and numerical solution based on the gradient method of one inverse problem for the acoustics equation
DOI:
https://doi.org/10.15587/1729-4061.2022.253568Keywords:
inverse problems, continuation problem, acoustic equation, numerical experiment, Landweber methodAbstract
The paper considers the problem of continuation of solutions of hyperbolic equations from a part of the domain boundary. These problems include the Cauchy problem for a hyperbolic equation with data on a timelike surface. In the inverse problems, the inhomogeneities are located at some depth under the medium layer, the parameters of which are known. In this case, an important tool for practitioners are the problems of continuation of geophysical fields from the Earth's surface towards the lay of inhomogeneities. In equations of mathematical physics, solution of the continuation problem from part of the boundary is in many cases strongly ill-posed problems in classes of functions of finite smoothness. The ill-posedness of this problem is considered, that is, the example of Hadamard, a Cauchy problem for a hyperbolic equation, is given. The physical formulation of the continuation problem is considered and reduced to the inverse problem. The definition of the generalized solution is formulated and the correctness of the direct problem is presented in the form of a theorem. The inverse problem is reduced to the problem of minimizing the objective functional. The objective functional is minimized by the Landweber method. By the increment of the functional, we consider the perturbed problem for the direct problem. We multiply the equation of the perturbed problem by some function and integrate by parts, we obtain the formulation of the conjugate problem. After that, we get the gradient of the functional. The algorithm for solving the inverse problem is listed. A finite-difference algorithm for the numerical solution of the problem is presented. The numerical solution of the direct problem is performed by the method of inversion of difference schemes. The results of numerical calculations are presented
Supporting Agency
- This research is funded by the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan (Grant No. AP09058430 Development of numerical methods for solving Navier-Stokes equations combining fictitious domains and conjugate equations).
References
- Kabanikhin, S., Nurseitov, D., Sholpanbaev, B. (2014). Zadacha prodolzheniya elektromagnitnogo polya v napravlenii k neodnorodnostyam. Sibirskie elektronnye matematicheskie izvestiya, 11, 85–102.
- Pierce, A. (2019). Acoustics: An Introduction to Its Physical Principles and Applications. Springer, 768. doi: https://doi.org/10.1007/978-3-030-11214-1
- Koskov, N., Koskov, B. (2007). Geofizicheskie issledovaniya skvazhin i interpretatsiya dannykh GIS. Perm': Izd. Permskogo gosudarstvennogo tekhnicheskogo universiteta, 307.
- Tsai, C.-C., Lin, C.-H. (2022). Review and Future Perspective of Geophysical Methods Applied in Nearshore Site Characterization. Journal of Marine Science and Engineering, 10 (3), 344. doi: https://doi.org/10.3390/jmse10030344
- Obornev, E. A., Obornev, I. E., Rodionov, E. A., Shimelevich, M. I. (2020). Application of Neural Networks in Nonlinear Inverse Problems of Geophysics. Computational Mathematics and Mathematical Physics, 60 (6), 1025–1036. doi: https://doi.org/10.1134/s096554252006007x
- Yaman, F., Yakhno, V. G., Potthast, R. (2013). A Survey on Inverse Problems for Applied Sciences. Mathematical Problems in Engineering, 2013, 1–19. doi: https://doi.org/10.1155/2013/976837
- Kasenov, S., Nurseitova, A., Nurseitov, D. (2016). A conditional stability estimate of continuation problem for the Helmholtz equation. doi: https://doi.org/10.1063/1.4959733
- Kozlov, V. A., Maz'ya, V. G., Fomin, A. V. (1991). Ob odnom iteratsionnom metode resheniya zadachi Koshi dlya ellipticheskikh uravneniy. Zhurnal vychislitel'noy matematiki i matematicheskoy fiziki, 31 (1), 45–52.
- Safarov, Z. S., Durdiev, D. K. (2018). Inverse Problem for an Integro-Differential Equation of Acoustics. Differential Equations, 54 (1), 134–142. doi: https://doi.org/10.1134/s0012266118010111
- Blaunstein, N, Yakubov, V. (2018). Electromagnetic and Acoustic Wave Tomography: Direct and Inverse Problems in Practical Applications. CRC Press, 380. doi: https://doi.org/10.1201/9780429488276
- Ibraheem, K. I., Khudhur, H. M. (2022). Optimization algorithm based on the Euler method for solving fuzzy nonlinear equations. Eastern-European Journal of Enterprise Technologies, 1 (4 (115)), 13–19. doi: https://doi.org/10.15587/1729-4061.2022.252014
- Kabanikhin, S. (2011). Inverse and Ill-posed Problems: Theory and Applications. De Gruyter. doi: https://doi.org/10.1515/9783110224016
- Kabanikhin, S., Nurseitova, A., Kasenov, S. (2021). Stability estimation of the generalized solution to the direct problem for the acoustic equation. Journal of Physics: Conference Series, 2092 (1), 012005. doi: https://doi.org/10.1088/1742-6596/2092/1/012005
- Shishlenin, M. A., Kasenov, S. E., Askerbekova, Z. A. (2019). Numerical Algorithm for Solving the Inverse Problem for the Helmholtz Equation. Computational and Information Technologies in Science, Engineering and Education, 197–207. doi: https://doi.org/10.1007/978-3-030-12203-4_20
- Kabanikhin, S. I., Scherzer, O., Shishlenin, M. A. (2003). Iteration methods for solving a two dimensional inverse problem for a hyperbolic equation. Journal of Inverse and Ill-Posed Problems, 11 (1), 87–109. doi: https://doi.org/10.1515/156939403322004955
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2022 Syrym Kasenov, Janar Askerbekova, Aigerim Tleulesova
This work is licensed under a Creative Commons Attribution 4.0 International License.
The consolidation and conditions for the transfer of copyright (identification of authorship) is carried out in the License Agreement. In particular, the authors reserve the right to the authorship of their manuscript and transfer the first publication of this work to the journal under the terms of the Creative Commons CC BY license. At the same time, they have the right to conclude on their own additional agreements concerning the non-exclusive distribution of the work in the form in which it was published by this journal, but provided that the link to the first publication of the article in this journal is preserved.
A license agreement is a document in which the author warrants that he/she owns all copyright for the work (manuscript, article, etc.).
The authors, signing the License Agreement with TECHNOLOGY CENTER PC, have all rights to the further use of their work, provided that they link to our edition in which the work was published.
According to the terms of the License Agreement, the Publisher TECHNOLOGY CENTER PC does not take away your copyrights and receives permission from the authors to use and dissemination of the publication through the world's scientific resources (own electronic resources, scientometric databases, repositories, libraries, etc.).
In the absence of a signed License Agreement or in the absence of this agreement of identifiers allowing to identify the identity of the author, the editors have no right to work with the manuscript.
It is important to remember that there is another type of agreement between authors and publishers – when copyright is transferred from the authors to the publisher. In this case, the authors lose ownership of their work and may not use it in any way.
