November  2010, 4(4): 703-712. doi: 10.3934/ipi.2010.4.703

Numerical recovering of a density by the BC-method

1. 

Ugra Research Institute of Information Technologies, Khanty-Mansiysk, 628011, Russian Federation, Russian Federation, Russian Federation

Received  March 2009 Revised  June 2010 Published  September 2010

In this paper we develop the numerical algorithm for solving the inverse problem for the wave equation by the Boundary Control method. The results of numerical experiments are presented.
Citation: Leonid Pestov, Victoria Bolgova, Oksana Kazarina. Numerical recovering of a density by the BC-method. Inverse Problems and Imaging, 2010, 4 (4) : 703-712. doi: 10.3934/ipi.2010.4.703
References:
[1]

C. Bardos, G. Lebeau and J. Rauch., Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary. SIAM J. Control Optim., 30 (1992), 1024-1065. doi: doi:10.1137/0330055.

[2]

M. I. Belishev, Boundary control in reconstruction of manifolds and metrics (the BC-method), Inverse Problems, 13 (1997), R1-R45. doi: doi:10.1088/0266-5611/13/5/002.

[3]

M. I. Belishev, Recent progress in the boundary control method, Inverse Problems, 23 (2007), R1-R67. doi: doi:10.1088/0266-5611/23/5/R01.

[4]

M. I. Belishev and V. Yu. Gotlib, Dynamical variant of the BC-method: theory and numerical testing, J. Inverse Ill-Posed Problems, 7 (1999), 221-240. doi: doi:10.1515/jiip.1999.7.3.221.

[5]

A. P. Calderón, On an inverse boundary value problem, In: "Seminar on Numerical Analysis and Its Applications to Continuum Physics", Rió de Janeiro, 1980, 65-73.

[6]

Y. Kurylev and M. Lassas, Inverse Problems and Index Formula for Dirac Operators, Adv. Math., 221 (2009), 170-216. doi: doi:10.1016/j.aim.2008.12.001.

[7]

I. Lasiecka, J.-L. Lions, and R. Triggiani, Non homogeneous boundary value problems for second order hyperbolic operators, J. Math. Pures Appl., 65 (1986), 149-192.

[8]

J.-L. Lions, "Contrôle Optimale de Systèmes Gouvernés par des Équations aux Dérivées partielles", Dunod, Paris, 1968.

[9]

J.-L. Lions and E. Magenes, "Problèmes aux limites non homogènes et applications", v. 1, 2, 3, Dunod, Paris, 1968.

[10]

L. N. Pestov, On reconstruction of the speed of sound from a part of boundary, J. Inverse Ill-Posed Problems, 7 (1999), 481–486. doi: doi:10.1515/jiip.1999.7.5.481.

[11]

D.Tataru, Unique continuation for solutions to PDE's; between Hörmander's theorem and Holmgren's theorem, Comm. Partial Differential Equations, 20 (1995), 855-884. doi: doi:10.1080/03605309508821117.

show all references

References:
[1]

C. Bardos, G. Lebeau and J. Rauch., Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary. SIAM J. Control Optim., 30 (1992), 1024-1065. doi: doi:10.1137/0330055.

[2]

M. I. Belishev, Boundary control in reconstruction of manifolds and metrics (the BC-method), Inverse Problems, 13 (1997), R1-R45. doi: doi:10.1088/0266-5611/13/5/002.

[3]

M. I. Belishev, Recent progress in the boundary control method, Inverse Problems, 23 (2007), R1-R67. doi: doi:10.1088/0266-5611/23/5/R01.

[4]

M. I. Belishev and V. Yu. Gotlib, Dynamical variant of the BC-method: theory and numerical testing, J. Inverse Ill-Posed Problems, 7 (1999), 221-240. doi: doi:10.1515/jiip.1999.7.3.221.

[5]

A. P. Calderón, On an inverse boundary value problem, In: "Seminar on Numerical Analysis and Its Applications to Continuum Physics", Rió de Janeiro, 1980, 65-73.

[6]

Y. Kurylev and M. Lassas, Inverse Problems and Index Formula for Dirac Operators, Adv. Math., 221 (2009), 170-216. doi: doi:10.1016/j.aim.2008.12.001.

[7]

I. Lasiecka, J.-L. Lions, and R. Triggiani, Non homogeneous boundary value problems for second order hyperbolic operators, J. Math. Pures Appl., 65 (1986), 149-192.

[8]

J.-L. Lions, "Contrôle Optimale de Systèmes Gouvernés par des Équations aux Dérivées partielles", Dunod, Paris, 1968.

[9]

J.-L. Lions and E. Magenes, "Problèmes aux limites non homogènes et applications", v. 1, 2, 3, Dunod, Paris, 1968.

[10]

L. N. Pestov, On reconstruction of the speed of sound from a part of boundary, J. Inverse Ill-Posed Problems, 7 (1999), 481–486. doi: doi:10.1515/jiip.1999.7.5.481.

[11]

D.Tataru, Unique continuation for solutions to PDE's; between Hörmander's theorem and Holmgren's theorem, Comm. Partial Differential Equations, 20 (1995), 855-884. doi: doi:10.1080/03605309508821117.

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