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Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.

6 Issues per year


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1569-3945
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Volume 23, Issue 6

Issues

Fast Toeplitz linear system inversion for solving two-dimensional acoustic inverse problem

Sergey I. Kabanikhin
  • Institute of Computational Mathematics and Mathematical Geophysics SB RAS, Prospect Akademika Lavrentjeva, 6, Novosibirsk, 630090, Russia; and Novosibirsk State University, Pirogova St., 2, 630090, Novosibirsk, Russia
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/ Nikita S. Novikov
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  • Novosibirsk State University, Pirogova St., 2, Novosibirsk, 630090, Russia; and Institute of Computational Mathematics and Mathematical Geophysics SB RAS, Akad. Lavrentjev avenue, 6, 630090, Novosibirsk, Russia
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/ Ivan V. Oseledets / Maxim A. Shishlenin
  • Sobolev Institute of Mathematics SB RAS, Akad. Koptyug avenue, 4, Novosibirsk, 630090, Russia; and Novosibirsk State University, Pirogova St., 2, 630090, Novosibirsk, Russia
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Published Online: 2015-11-17 | DOI: https://doi.org/10.1515/jiip-2015-0083

Abstract

The coefficient inverse problem for the acoustic equation is considered. We propose the method for reconstructing the density based on the N-approximation by the finite system of one-dimensional problems and the two-dimensional M. G. Krein approach. The two-dimensional analogue of the M. G. Krein approach is applied to reduce the non-linear inverse problem to a family of linear integral equations. We consider the fast algorithm for solving the relevant linear system, based on using the block-Toeplitz structure of the matrix. The algorithm applied to the M. G. Krein equation allows to obtain the solution of the whole family of the integral equations by solving only one linear system. Results of numerical calculations are presented.

Keywords: Inverse problems; Gelfand–Levitan method; M. G. Krein equation; fast Toeplitz algorithm

MSC: 65M32; 65R20; 65F05

About the article

Received: 2015-09-01

Revised: 2015-10-27

Accepted: 2015-10-27

Published Online: 2015-11-17

Published in Print: 2015-12-01


Funding Source: RFBR

Award identifier / Grant number: 14-01-00208

Funding Source: RFBR

Award identifier / Grant number: 15-01-09230

Funding Source: the Ministry of Education and Science of the Russian Federation and the Ministry of Education and Science of the Republic of Kazakhstan

Award identifier / Grant number: MES 1760/GF: project NTP 04.03.02 “Creating methodological basis of geological-geophysical studies of focal zones UNE in igneous rocks”

Funding Source: Russian Science Foundation

Award identifier / Grant number: 14-01-000659


Citation Information: Journal of Inverse and Ill-posed Problems, Volume 23, Issue 6, Pages 687–700, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219, DOI: https://doi.org/10.1515/jiip-2015-0083.

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[1]
S.I. Kabanikhin and M.A. Shishlenin
Georesursy, 2018, Volume 20, Number 3, Page 139
[2]
D.V. Lukyanenko, M.A. Shishlenin, and V.T. Volkov
Communications in Nonlinear Science and Numerical Simulation, 2018, Volume 54, Page 233

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