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 & 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.  doi: 10.1080/0003681042000221678.  Google Scholar

[2]

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

[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.  doi: 10.1137/S0036141002417814.  Google Scholar

[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).  doi: 10.1088/0266-5611/24/5/055006.  Google Scholar

[5]

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

[6]

V. Isakov, "Inverse Problems for Partial Differential Equations,", Second edition, 127 (2006).   Google Scholar

[7]

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

[8]

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

[9]

I. Lasiecka, R. Triggiani and X. Zhang, Nonconservative wave equations with unobserved Neumann B. C.: Global uniqueness and observability in one shot,, in, 268 (2000), 227.  doi: 10.1090/conm/268/04315.  Google Scholar

[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.  doi: 10.1137/100808988.  Google Scholar

[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.  doi: 10.1016/j.nonrwa.2010.10.014.  Google Scholar

[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.  doi: 10.1007/s11401-011-0672-1.  Google Scholar

[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.  doi: 10.1080/00036811.2011.618125.  Google Scholar

[14]

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

[15]

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

[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.  doi: 10.1007/s00245-002-0751-5.  Google Scholar

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.  doi: 10.1080/0003681042000221678.  Google Scholar

[2]

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

[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.  doi: 10.1137/S0036141002417814.  Google Scholar

[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).  doi: 10.1088/0266-5611/24/5/055006.  Google Scholar

[5]

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

[6]

V. Isakov, "Inverse Problems for Partial Differential Equations,", Second edition, 127 (2006).   Google Scholar

[7]

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

[8]

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

[9]

I. Lasiecka, R. Triggiani and X. Zhang, Nonconservative wave equations with unobserved Neumann B. C.: Global uniqueness and observability in one shot,, in, 268 (2000), 227.  doi: 10.1090/conm/268/04315.  Google Scholar

[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.  doi: 10.1137/100808988.  Google Scholar

[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.  doi: 10.1016/j.nonrwa.2010.10.014.  Google Scholar

[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.  doi: 10.1007/s11401-011-0672-1.  Google Scholar

[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.  doi: 10.1080/00036811.2011.618125.  Google Scholar

[14]

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

[15]

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

[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.  doi: 10.1007/s00245-002-0751-5.  Google Scholar

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