Geodesic
An optimization approach to solving a~discrete inverse problem for a~one-dimensional hyperbolic equation
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 4 (2001) no. 3, pp. 243-258.

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In this paper, an optimization approach to solving a discrete problem of determining the coefficient of a one-dimensional hyperbolic equation in integral formulation is studied. The properties of the solutions to the inverse and direct discrete problems are investigated. Estimates both for the cost function and its gradient are obtained. Also, the convergence of the steepest descent method minimizing the misfit functional is proved. 
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K. T. Iskakov. An optimization approach to solving a~discrete inverse problem for a~one-dimensional hyperbolic equation. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 4 (2001) no. 3, pp. 243-258. http://geodesic.mathdoc.fr/item/SJVM_2001_4_3_a2/

[1] Tikhonov A. N., “O reshenii nekorrektno postavlennykh zadach i metode regulyarizatsii”, Dokl. AN SSSR, 151:3 (1963), 501–504 | MR | Zbl

[2] Alekseev A. S., “Obratnye dinamicheskie zadachi seismiki”, Nekotorye metody i algoritmy interpretatsii geofizicheskikh dannykh, Nauka, M., 1967, 9–84

[3] Bamberger A., Chavent G., Lailly P., “About the stability of the inverse problem in 1-D wave equations. Application to the interpretation of seismic profiles”, Appl. Math. Optim., 5 (1979), 1–47 | DOI | MR | Zbl

[4] Dmitriev V. I., Fedorova E. A., “Obratnaya zadacha chastotnogo elektromagnitnogo zondirovaniya sloistykh sred”, Vychislitelnaya matematika i matematicheskoe obespechenie EVM, Izd-vo MGU, M., 1985, 159–165

[5] Nyambaa Sh., Cheverda V. A., Optimizatsionnyi metod resheniya obratnoi zadachi elektrorazvedki na postoyannom toke dlya vertikalno-neodnorodnykh sred, Preprint / AN SSSR. Sib. otd-nie. VTs; 794, Novosibirsk, 1988 | MR

[6] Iskakov K. T., Kabanikhin S. I., “The solution of one-dimensional inverse problem of geoelectrics by the method of conjugate gradients”, Russian J. Theor. Appl. Mechanics, 2:3 (1992), 197–221 | MR

[7] Romanov V. G., Kabanikhin S. I., Obratnye zadachi geoelektriki, Nauka, M., 1991 | MR

[8] Kabanikhin S. I., “Numerical analysis of inverse problems”, J. of Inverse and Ill-Posed Problems, 3:4 (1995), 278–304 | DOI | MR | Zbl

[9] Kabanikhin S. I., Bakanov G. V., “The optimization method for solving the discrete inverse problem for hyperbolic equation”, J. of Inverse and Ill-Posed Problems, 4:6 (1996), 513–530 | DOI | MR | Zbl

[10] Karchevsky A. L., “Properties of the misfit functional for a nonlinear one-dimensional coefficient of hyperbolic inverse problem”, J. of Inverse and Ill-Posed Problems, 5:2 (1997), 139–165 | DOI | MR

[11] Karchevsky A. L., “Finite-Difference Coefficient of Inverse Problem and Properties of the Misfit Functional”, J. of Inverse and Ill-Posed Problems, 6:5 (1998), 433–454 | MR

[12] Karchevskii A. L., “O povedenii funktsionala nevyazki dlya odnomernoi giperbolicheskoi obratnoi zadachi”, Sib. zhurn. vychisl. matematiki, 2:2 (1999), 137–160 | MR

[13] Iskakov K. T., Kabanikhin S. I., Ob odnom podkhode k obratnym zadacham dlya giperbolicheskikh uravnenii, Metodicheskie ukazaniya po spetskursu, izd-vo Karagandinskogo gos. universiteta, Karaganda, 1999

[14] Iskakov K. T., Kabanikhin S. I., Obobschennoe reshenie obratnoi zadachi dlya odnomernogo uravneniya giperbolicheskogo tipa, Preprint, NGU, izd-vo NII Diskretnoi matematiki i informatiki, Novosibirsk, 2000 | MR

[15] Iskakov K. T., Kabanikhin S. I., Obobschennoe reshenie obratnoi zadachi dlya uravneniya akustiki, Preprint / NGU, izd-vo NII Diskretnoi matematiki i informatiki; 37, Novosibirsk, 2000 | MR

[16] Azamatov J. S., Kabanikhin S. I., “Nonlinear Operator Equations. $L_2$-theory”, J. of Inverse and Ill-Posed Problems, 7:6 (1999), 497–529 | MR