Geodesic
Numerical solution to a three-dimensional coefficient inverse problem for the wave equation with integral data in a cylindrical domain
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 22 (2019) no. 4, pp. 381-397.

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A three-dimensional coefficient inverse problem for the wave equation (with losses) in a cylindrical domain is under consideration. The data for its solution are special time integrals of the wave field measured in a cylindrical layer. We present and substantiate an efficient algorithm for solving such a three-dimensional problem based on the fast Fourier transform. The algorithm proposed makes possible to obtain a solution on grids of $512\times 512\times 512$ size in a time of about $1.4$ hours on a typical PC without parallelizing the calculations. The results of the numerical experiments for solving the corresponding model inverse problems are presented. 
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A. B. Bakushinsky; A. S. Leonov. Numerical solution to a three-dimensional coefficient inverse problem for the wave equation with integral data in a cylindrical domain. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 22 (2019) no. 4, pp. 381-397. http://geodesic.mathdoc.fr/item/SJVM_2019_22_4_a0/

[1] D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer, Berlin, 1998 | MR | Zbl

[2] L. Pestov, “On determining an absorption coefficient and a speed of sound in the wave equation by the BC method”, J. of Inverse and Ill-posed Problems, 22:2 (2014), 245–250 | DOI | MR | Zbl

[3] A. Agaltsov, R. Novikov, “Riemann-Hilbert problem approach for two-dimensional flow inverse scattering”, J. of Mathematical Physics, 55:1 (2014), 103502, 25 pp. | DOI | MR | Zbl

[4] V. A. Burov, D. I. Zotov, O. D. Rumyanceva, “Vosstanovlenie prostranstvennykh raspredeleniy skorosti zvuka i pogloscheniya v fantomakh myagkikh biotkaney po eksperimental'nym dannym ul'trazvukovogo tomografirovaniya”, Akust. zhurn., 61:2 (2015), 254–273 | DOI

[5] M. I. Belishev, I. B. Ivanov, I. V. Kubyshkin, V. S. Semenov, “Numerical testing in determination of sound speed from a part of boundary by the BC-method”, J. Inverse Ill-Posed Probl., 24:2 (2016), 159–180 | DOI | MR | Zbl

[6] L. Beilina, M. V. Klibanov, Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems, Springer, New York:, 2012 | Zbl

[7] L. Beilina, N. T. Th'anh, M. V. Klibanov, M. A. Fiddy, “Reconstruction from blind experimental data for an inverse problem for a hyperbolic equation”, Inverse Problems, 30:24, 025002 | MR | Zbl

[8] A. B. Bakushinsky, M. Yu. Kokurin, Iterative Methods for Approximate Solution of Inverse Problems, Mathematics and Its Applications, 577, Springer, Dordrecht, 2004 | MR | Zbl

[9] A. B. Bakushinskiy, M. Yu. Kokurin, Iteracionnye metody resheniya nekorrektnykh operatornykh uravneniy s gladkimi operatorami, Editorial URSS, M., 2002

[10] A. V. Goncharskiy, S. Yu. Romanov, “O dvukh podkhodakh k resheniyu koefficientnykh obratnykh zadach dlya volnovykh uravneniy”, Zhurn. vychisl. matem. i mat. fiziki, 52:2 (2012), 263–269 | MR | Zbl

[11] A. V. Goncharskiy, S. Yu. Romanov, “Superkomp'yuternye tekhnologii v razrabotke metodov resheniya obratnykh zadach v UZI-tomografii”, Vychislitel'nye metody i programmirovanie: novye vychislitel'nye tekhnologii, 13:1 (2012), 235–238

[12] M. M. Lavrent'ev, “Ob odnoy obratnoy zadache dlya volnovogo uravneniya”, Dokl. AN SSSR, 157:3 (1964), 520–521 | Zbl

[13] A. B. Bakushinskiy, A. I. Kozlov, M. Yu. Kokurin, “Ob odnoy obratnoy zadache dlya trekhmernogo volnovogo uravneniya”, Zhurn. vychisl. matem. i mat. fiziki, 43:8 (2003), 1201–1209 | MR | Zbl

[14] A. B. Bakushinsky, A. S. Leonov, “Fast numerical method of solving 3D coefficient inverse problem for wave equation with integral data”, J. Inv. Ill-Posed Probl., 26:4 (2018), 477–492 | DOI | MR | Zbl

[15] A. N. Tikhonov, A. A. Samarskiy, Uravneniya matematicheskoy fiziki, Nauka, M., 1966 | MR

[16] Bakushinsky A., Goncharsky A., Ill-Posed Problems: Theory and Applications, Kluwer Academic Publishers, Dordrecht, 1994 | MR

[17] G. M. Vaynikko, A. Yu. Veretennikov, Iteracionnye procedury v nekorrektnykh zadachakh, Nauka, M., 1986 | MR

[18] V. A. Morozov, Regulyarnye metody resheniya nekorrektno postavlennykh zadach, Nauka, M., 1987

[19] Tikhonov A., Goncharsky A., Stepanov V., A. Yagola, Numerical methods for the solution of ill-posed problems, Kluwer, Dordrecht, 1995 | MR | MR | Zbl

[20] H. W. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems, Kluwer, Dordrecht, 1996 | MR | Zbl

[21] Tikhonov A. N., Leonov A. S., Yagola A. G., Non-Linear Ill-Posed Problems, Chapmen and Hall, London, 1998 | MR

[22] A. S. Leonov, Reshenie nekorrektno postavlennykh obratnykh zadach. Ocherk teorii, prakticheskie algoritmy i demonstracii v MATLAB, Izd-e 2, Librokom, M., 2013