June  2013, 2(2): 355-364. doi: 10.3934/eect.2013.2.355

Recovery of the sound speed and initial displacement for the wave equation by means of a single Dirichlet boundary measurement

1. 

Department of Mathematics and Statistics, University of Helsinki, FI-00014 Helsinki, Finland

Received  October 2012 Revised  February 2013 Published  March 2013

We consider an inverse problem of recovering simultaneously the sound speed and an initial condition for the wave equation from a single Dirichlet data measured on the boundary of the support of the initial condition. The problem is motived from the recently developed hybrid imaging models as well as from classical inverse hyperbolic problems with a single boundary measurement formulation. We establish uniqueness of the recovery and the proof is based on the Carleman estimate and continuous observability inequality for general Riemannian wave equations.
Citation: Shitao Liu. Recovery of the sound speed and initial displacement for the wave equation by means of a single Dirichlet boundary measurement. Evolution Equations and Control Theory, 2013, 2 (2) : 355-364. doi: 10.3934/eect.2013.2.355
References:
[1]

M. Bellassoued, Uniqueness and stability in determining the speed of propagation of second-order hyperbolic equation with variable coefficients, Appl. Anal., 83 (2004), 983-1014. doi: 10.1080/0003681042000221678.

[2]

A. Bukhgeim and M. Klibanov, Uniqueness in the large of a class of multidimensional inverse problems, Dokl. Akad. Nauk SSSR, 260 (1981), 269-272.

[3]

D. Finch, S. Patch and Rakesh, Determining a function from its mean values over a family of spheres, SIAM J. Math. Anal., 35 (2004), 1213-1240. doi: 10.1137/S0036141002417814.

[4]

S. Hristova, P. Kuchment and L. Nguyen, Reconstruction and time reversal in thermoacoustic tomography in acoustically homogeneous and inhomogeneous media, Inverse Problems, 5 (2008), 055006, 25 pp. doi: 10.1088/0266-5611/24/5/055006.

[5]

O. Imanuvilov and M. Yamamoto, Global uniqueness and stability in determining coefficients of wave equations, Comm. Partial Differential Equations, 26 (2001), 1409-1425. doi: 10.1081/PDE-100106139.

[6]

V. Isakov, "Inverse Problems for Partial Differential Equations," Second edition, Applied Mathematical Sciences, 127, Springer, New York, 2006.

[7]

M. Klibanov, "Carleman Estimates for Coefficient Inverse Problems and Numerical Applications. Inverse and Ill-Posed Problems Series," VSP, Utrecht, 2004.

[8]

P. Kuchment and L. Kunyansky, Mathematics of the thermoacoustic tomography, European J. Appl. Math., 19 (2008), 191-224. doi: 10.1017/S0956792508007353.

[9]

I. Lasiecka, R. Triggiani and X. Zhang, Nonconservative wave equations with unobserved Neumann B. C.: Global uniqueness and observability in one shot, in "Differential Geometric Methods in the Control of Partial Differential Equations" (Boulder, CO, 1999), Contemp. Math., 268, Amer. Math. Soc., Providence, RI, (2000), 227-325. doi: 10.1090/conm/268/04315.

[10]

S. Liu and R. Triggiani, Global uniqueness and stability in determining the damping coefficient of an inverse hyperbolic problem with non-homogeneous Neumann B. C. through an additional Dirichlet boundary trace, SIAM J. Math. Anal., 43 (2011), 1631-1666. doi: 10.1137/100808988.

[11]

S. Liu and R. Triggiani, Global uniqueness and stability in determining the damping and potential coefficients of an inverse hyperbolic problem, Nonlinear Anal. Real World Appl., 12 (2011), 1562-1590. doi: 10.1016/j.nonrwa.2010.10.014.

[12]

S. Liu and R. Triggiani, Recovery of damping coefficients for a system of coupled wave equations with Neumann B. C.: Uniqueness and stability, Chin. Ann. Math. Ser. B, 32 (2011), 669-698. doi: 10.1007/s11401-011-0672-1.

[13]

S. Liu and R. Triggiani, Global uniqueness and stability in determining the damping coefficient of an inverse hyperbolic problem with non-homogeneous Dirichlet B. C. through an additional localized Neumann boundary trace, Appl. Anal., 91 (2012), 1551-1581. doi: 10.1080/00036811.2011.618125.

[14]

P. Stefanov and G. Uhlmann, Thermoacoustic tomography with variable sound speed, Inverse Problems, 25 (2009), 075011, 16 pp. doi: 10.1088/0266-5611/25/7/075011.

[15]

P. Stefanov and G. Uhlmann, Recovery of a source term or a speed with one measurement and applications,, Trans. Amer. Math. Soc., (). 

[16]

R. Triggiani and P. F. Yao, Carleman estimates with no lower-order terms for general Riemannian wave equations. Global uniqueness and observability in one shot, Appl. Math. Optim., 46 (2002), 331-375. doi: 10.1007/s00245-002-0751-5.

show all references

References:
[1]

M. Bellassoued, Uniqueness and stability in determining the speed of propagation of second-order hyperbolic equation with variable coefficients, Appl. Anal., 83 (2004), 983-1014. doi: 10.1080/0003681042000221678.

[2]

A. Bukhgeim and M. Klibanov, Uniqueness in the large of a class of multidimensional inverse problems, Dokl. Akad. Nauk SSSR, 260 (1981), 269-272.

[3]

D. Finch, S. Patch and Rakesh, Determining a function from its mean values over a family of spheres, SIAM J. Math. Anal., 35 (2004), 1213-1240. doi: 10.1137/S0036141002417814.

[4]

S. Hristova, P. Kuchment and L. Nguyen, Reconstruction and time reversal in thermoacoustic tomography in acoustically homogeneous and inhomogeneous media, Inverse Problems, 5 (2008), 055006, 25 pp. doi: 10.1088/0266-5611/24/5/055006.

[5]

O. Imanuvilov and M. Yamamoto, Global uniqueness and stability in determining coefficients of wave equations, Comm. Partial Differential Equations, 26 (2001), 1409-1425. doi: 10.1081/PDE-100106139.

[6]

V. Isakov, "Inverse Problems for Partial Differential Equations," Second edition, Applied Mathematical Sciences, 127, Springer, New York, 2006.

[7]

M. Klibanov, "Carleman Estimates for Coefficient Inverse Problems and Numerical Applications. Inverse and Ill-Posed Problems Series," VSP, Utrecht, 2004.

[8]

P. Kuchment and L. Kunyansky, Mathematics of the thermoacoustic tomography, European J. Appl. Math., 19 (2008), 191-224. doi: 10.1017/S0956792508007353.

[9]

I. Lasiecka, R. Triggiani and X. Zhang, Nonconservative wave equations with unobserved Neumann B. C.: Global uniqueness and observability in one shot, in "Differential Geometric Methods in the Control of Partial Differential Equations" (Boulder, CO, 1999), Contemp. Math., 268, Amer. Math. Soc., Providence, RI, (2000), 227-325. doi: 10.1090/conm/268/04315.

[10]

S. Liu and R. Triggiani, Global uniqueness and stability in determining the damping coefficient of an inverse hyperbolic problem with non-homogeneous Neumann B. C. through an additional Dirichlet boundary trace, SIAM J. Math. Anal., 43 (2011), 1631-1666. doi: 10.1137/100808988.

[11]

S. Liu and R. Triggiani, Global uniqueness and stability in determining the damping and potential coefficients of an inverse hyperbolic problem, Nonlinear Anal. Real World Appl., 12 (2011), 1562-1590. doi: 10.1016/j.nonrwa.2010.10.014.

[12]

S. Liu and R. Triggiani, Recovery of damping coefficients for a system of coupled wave equations with Neumann B. C.: Uniqueness and stability, Chin. Ann. Math. Ser. B, 32 (2011), 669-698. doi: 10.1007/s11401-011-0672-1.

[13]

S. Liu and R. Triggiani, Global uniqueness and stability in determining the damping coefficient of an inverse hyperbolic problem with non-homogeneous Dirichlet B. C. through an additional localized Neumann boundary trace, Appl. Anal., 91 (2012), 1551-1581. doi: 10.1080/00036811.2011.618125.

[14]

P. Stefanov and G. Uhlmann, Thermoacoustic tomography with variable sound speed, Inverse Problems, 25 (2009), 075011, 16 pp. doi: 10.1088/0266-5611/25/7/075011.

[15]

P. Stefanov and G. Uhlmann, Recovery of a source term or a speed with one measurement and applications,, Trans. Amer. Math. Soc., (). 

[16]

R. Triggiani and P. F. Yao, Carleman estimates with no lower-order terms for general Riemannian wave equations. Global uniqueness and observability in one shot, Appl. Math. Optim., 46 (2002), 331-375. doi: 10.1007/s00245-002-0751-5.

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