Geodesic
Stability estimates for solutions to inverse extremal problems for the Helmholtz equation
Sibirskij žurnal industrialʹnoj matematiki, Tome 16 (2013) no. 2, pp. 14-25.

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Inverse problems for the Helmholtz equation of the acoustic scattering on a three-dimensional inclusion are considered. Using an optimization method, we reduce these problems to inverse extremal problems in which the role of controls is played by a variable refraction index and boundary source density. Solvability of these problems is proved and some optimality systems are obtained that describe necessary optimality conditions. Basing on the analysis of the optimality systems, sufficient conditions on the input data are deduced that guarantee the uniqueness and stability of optimal solutions.
Keywords: Helmholtz equation, scattering problem, inhomogeneous medium, inverse problem, uniqueness, stability.
Mots-clés : multiplicative control 
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G. V. Alekseev; A. V. Lobanov. Stability estimates for solutions to inverse extremal problems for the Helmholtz equation. Sibirskij žurnal industrialʹnoj matematiki, Tome 16 (2013) no. 2, pp. 14-25. http://geodesic.mathdoc.fr/item/SJIM_2013_16_2_a1/

[1] Cummer S. A., Schurig D., “One path to acoustic cloaking”, New J. Phys., 9 (2007), 45 | DOI

[2] Cummer S. A., Popa B. I., Schurig D. et al., “Scattering theory derivation of a 3D acoustic cloaking shell”, Phys. Rev. Letters, 100 (2008), 024301 | DOI

[3] Romanov V. G., “Obratnaya zadacha difraktsii dlya uravnenii akustiki”, Dokl. AN, 431 (2010), 319–322 | MR | Zbl

[4] Alekseev G. V., Romanov V. G., “Ob odnom klasse nerasseivayuschikh akusticheskikh obolochek dlya modeli anizotropnoi akustiki”, Sib. zhurn. industr. matematiki, 14:2(46) (2011), 15–20 | Zbl

[5] Dubinov A. E., Mytareva L. A., “Maskirovka materialnykh tel metodom volnovogo obtekaniya”, Uspekhi fiz. nauk, 180:5 (2010), 475–501 | DOI

[6] Alekseev G. V., Brizitskii R. V., “Teoreticheskii analiz ekstremalnykh zadach granichnogo upravleniya dlya uravnenii Maksvella”, Sib. zhurn industr. matematiki, 14:1(45) (2011), 3–16 | Zbl

[7] Alekseev G. V., Brizitskii R. V., Romanov V. G., “Otsenki ustoichivostireshenii zadach granichnogo upravleniya dlya uravnenii Maksvella pri smeshannykh granichnykh usloviyakh”, Dokl. AN, 447:1 (2012), 7–12 | MR | Zbl

[8] Alekseev G. V., “Optimizatsiya v zadachakh maskirovki materialnykh tel metodom volnovogo obtekaniya”, Dokl. AN, 449:6 (2013), 652–656 | DOI

[9] Alekseev G. V., “Cloaking via impedance boundary condition for the 2-D Helmholtz equation”, Appl. Analysis, 2013 | DOI

[10] Beilina L., Klibanov M. V., “A globally convergent numerical method for a coefficient inverse problem”, SIAM J. Sci. Comput., 31 (2008), 478–509 | DOI | MR | Zbl

[11] Beilina L., Klibanov M. V., “A new approximate mathematical model for global convergence for a coefficient inverse problem with backscattering data”, J. Inverse Ill-Posed Problems, 20 (2012), 513–565 | MR

[12] Colton D., Kress R., Inverse Acoustic and Electromagnetic Scattering Theory, Springer-Verl., Berlin, 2013 | MR | Zbl

[13] Alekseev G. V., “Zadachi upravleniya dlya statsionarnykh uravnenii magnitnoi gidrodinamiki”, Dokl. AN, 395:3 (2004), 322–325 | MR | Zbl

[14] Ioffe A. D., Tikhomirov V. M., Teoriya ekstremalnykh zadach, Nauka, M., 1974 | MR | Zbl