May  2015, 9(2): 301-315. doi: 10.3934/ipi.2015.9.301

A control approach to recover the wave speed (conformal factor) from one measurement

1. 

Department of Pediatrics - Cardiology, Baylor College of Medicine, Houston, TX

Received  January 2014 Revised  January 2015 Published  March 2015

In this paper we consider the problem of recovering the conformal factor in a conformal class of Riemannian metrics from the boundary measurement of one wave field. More precisely, using boundary control operators, we derive an explicit equation satisfied by the contrast between two conformal factors (or wave speeds). This equation is Fredholm and generically invertible provided that the domain of interest is properly illuminated at an initial time. We also show locally Lipschitz stability estimates.
Citation: Sebastian Acosta. A control approach to recover the wave speed (conformal factor) from one measurement. Inverse Problems & Imaging, 2015, 9 (2) : 301-315. doi: 10.3934/ipi.2015.9.301
References:
[1]

G. Bal and G. Uhlmann, Inverse diffusion theory of photoacoustics,, Inverse Problems, 26 (2010). doi: 10.1088/0266-5611/26/8/085010. Google Scholar

[2]

C. Bardos, Problèmes aux limites pour les équations aux dérivées partielles du premier ordre à coefficients réels; théorèmes d'approximation; application à l'équation de transport,, Ann. Sci. École Norm. Sup., 3 (1970), 185. Google Scholar

[3]

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. doi: 10.1137/0330055. Google Scholar

[4]

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

[5]

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

[6]

M. Bellassoued and D. Dos Santos Ferreira, Stability estimates for the anisotropic wave equation from the Dirichlet-to-Neumann map,, Inverse Probl. Imaging, 5 (2011), 745. doi: 10.3934/ipi.2011.5.745. Google Scholar

[7]

M. Bellassoued and M. Yamamoto, Logarithmic stability in determination of a coefficient in an acoustic equation by arbitrary boundary observation,, J. Math. Pures Appl. (9), 85 (2006), 193. doi: 10.1016/j.matpur.2005.02.004. Google Scholar

[8]

M. Bellassoued and M. Yamamoto, Determination of a coefficient in the wave equation with a single measurement,, Appl. Anal., 87 (2008), 901. doi: 10.1080/00036810802369249. Google Scholar

[9]

K. D. Blazek, C. Stolk and W. W. Symes, A mathematical framework for inverse wave problems in heterogeneous media,, Inverse Problems, 29 (2013). doi: 10.1088/0266-5611/29/6/065001. Google Scholar

[10]

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

[11]

N. Burq, Contrôlabilité exacte des ondes dans des ouverts peu réguliers,, Asymptot. Anal., 14 (1997), 157. Google Scholar

[12]

J. Chen and Y. Yang, Quantitative photo-acoustic tomography with partial data,, Inverse Problems, 28 (2012). doi: 10.1088/0266-5611/28/11/115014. Google Scholar

[13]

J. Chen and Y. Yang, Inverse problem of electro-seismic conversion,, Inverse Problems, 29 (2013). doi: 10.1088/0266-5611/29/11/115006. Google Scholar

[14]

C. B. Croke, I. Lasiecka, G. Uhlmann and M. S. Vogelius (eds.), Geometric Methods in Inverse Problems and PDE Control,, The IMA Volumes in Mathematics and its Applications, (2004). doi: 10.1007/978-1-4684-9375-7. Google Scholar

[15]

K.-J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations,, With contributions by S. Brendle, (2000). Google Scholar

[16]

L. C. Evans, Partial Differential Equations,, Graduate Studies in Mathematics, (1998). Google Scholar

[17]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Reprint of the 1998 edition, (1998). Google Scholar

[18]

R. Glowinski, J.-L. Lions and J. He, Exact and Approximate Controllability for Distributed Parameter Systems. A Numerical Approach,, Encyclopedia of Mathematics and its Applications, (2008). doi: 10.1017/CBO9780511721595. Google Scholar

[19]

O. Y. 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

[20]

O. Y. Imanuvilov and M. Yamamoto, Determination of a coefficient in an acoustic equation with a single measurement,, Inverse Problems, 19 (2003), 157. doi: 10.1088/0266-5611/19/1/309. Google Scholar

[21]

V. Isakov, Inverse Problems for Partial Differential Equations,, 2nd edition, (2006). Google Scholar

[22]

D. Jerison and C. E. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains,, J. Funct. Anal., 130 (1995), 161. doi: 10.1006/jfan.1995.1067. Google Scholar

[23]

T. Kato, Perturbation Theory for Linear Operators,, Reprint of the 1980 edition, (1980). Google Scholar

[24]

M. V. Klibanov, Inverse problems and Carleman estimates,, Inverse Problems, 8 (1992), 575. doi: 10.1088/0266-5611/8/4/009. Google Scholar

[25]

M. V. Klibanov and M. Yamamoto, Lipschitz stability of an inverse problem for an acoustic equation,, Appl. Anal., 85 (2006), 515. doi: 10.1080/00036810500474788. Google Scholar

[26]

I. Lasiecka, J.-L. Lions and R. Triggiani, Nonhomogeneous boundary value problems for second order hyperbolic operators,, J. Math. Pures Appl. (9), 65 (1986), 149. Google Scholar

[27]

J.-L. Lions, Exact controllability, stabilization and perturbations for distributed systems,, SIAM Rev., 30 (1988), 1. doi: 10.1137/1030001. Google Scholar

[28]

J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Vol. I-II,, Translated from the French by P. Kenneth, (1972). Google Scholar

[29]

S. Liu and L. Oksanen, A Lipschitz stable reconstruction formula for the inverse problem for the wave equation,, Accepted in Trans. Amer. Math. Soc., (). Google Scholar

[30]

S. Liu, Recovery of the sound speed and initial displacement for the wave equation by means of a single Dirichlet boundary measurement,, Evol. Equ. Control Theory, 2 (2013), 355. doi: 10.3934/eect.2013.2.355. Google Scholar

[31]

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

[32]

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

[33]

S. Liu and R. Triggiani, Recovering damping and potential coefficients for an inverse non-homogeneous second-order hyperbolic problem via a localized Neumann boundary trace,, Discrete Contin. Dyn. Syst., 33 (2013), 5217. doi: 10.3934/dcds.2013.33.5217. Google Scholar

[34]

W. McLean, Strongly Elliptic Systems and Boundary Integral Equations,, Cambridge University Press, (2000). Google Scholar

[35]

J.-P. Puel and M. Yamamoto, Generic well-posedness in a multidimensional hyperbolic inverse problem,, J. Inverse Ill-Posed Probl., 5 (1997), 55. doi: 10.1515/jiip.1997.5.1.55. Google Scholar

[36]

J.-P. Puel and M. Yamamoto, On a global estimate in a linear inverse hyperbolic problem,, Inverse Problems, 12 (1996), 995. doi: 10.1088/0266-5611/12/6/013. Google Scholar

[37]

Rakesh and W. W. Symes, Uniqueness for an inverse problem for the wave equation,, Comm. Partial Differential Equations, 13 (1988), 87. doi: 10.1080/03605308808820539. Google Scholar

[38]

M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations,, 2nd edition, (2004). Google Scholar

[39]

P. Stefanov and G. Uhlmann, Stability estimates for the hyperbolic Dirichlet to Neumann map in anisotropic media,, J. Funct. Anal., 154 (1998), 330. doi: 10.1006/jfan.1997.3188. Google Scholar

[40]

P. Stefanov and G. Uhlmann, Stable determination of generic simple metrics from the hyperbolic Dirichlet-to-Neumann map,, Int. Math. Res. Not., (2005), 1047. doi: 10.1155/IMRN.2005.1047. Google Scholar

[41]

P. Stefanov and G. Uhlmann, Recovery of a source term or a speed with one measurement and applications,, Trans. Amer. Math. Soc., 365 (2013), 5737. doi: 10.1090/S0002-9947-2013-05703-0. Google Scholar

[42]

Z. Q. Sun, On continuous dependence for an inverse initial-boundary value problem for the wave equation,, J. Math. Anal. Appl., 150 (1990), 188. doi: 10.1016/0022-247X(90)90207-V. Google Scholar

[43]

J. Sylvester and G. Uhlmann, Inverse problems in anisotropic media,, in Inverse Scattering and Applications (Amherst, (1990), 105. doi: 10.1090/conm/122/1135861. Google Scholar

[44]

G. Uhlmann, Inverse scattering in anisotropic media,, in Surveys on Solution Methods for Inverse Problems, (2000), 235. Google Scholar

[45]

M. Yamamoto, Uniqueness and stability in multidimensional hyperbolic inverse problems,, J. Math. Pures Appl., 78 (1999), 65. doi: 10.1016/S0021-7824(99)80010-5. Google Scholar

show all references

References:
[1]

G. Bal and G. Uhlmann, Inverse diffusion theory of photoacoustics,, Inverse Problems, 26 (2010). doi: 10.1088/0266-5611/26/8/085010. Google Scholar

[2]

C. Bardos, Problèmes aux limites pour les équations aux dérivées partielles du premier ordre à coefficients réels; théorèmes d'approximation; application à l'équation de transport,, Ann. Sci. École Norm. Sup., 3 (1970), 185. Google Scholar

[3]

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. doi: 10.1137/0330055. Google Scholar

[4]

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

[5]

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

[6]

M. Bellassoued and D. Dos Santos Ferreira, Stability estimates for the anisotropic wave equation from the Dirichlet-to-Neumann map,, Inverse Probl. Imaging, 5 (2011), 745. doi: 10.3934/ipi.2011.5.745. Google Scholar

[7]

M. Bellassoued and M. Yamamoto, Logarithmic stability in determination of a coefficient in an acoustic equation by arbitrary boundary observation,, J. Math. Pures Appl. (9), 85 (2006), 193. doi: 10.1016/j.matpur.2005.02.004. Google Scholar

[8]

M. Bellassoued and M. Yamamoto, Determination of a coefficient in the wave equation with a single measurement,, Appl. Anal., 87 (2008), 901. doi: 10.1080/00036810802369249. Google Scholar

[9]

K. D. Blazek, C. Stolk and W. W. Symes, A mathematical framework for inverse wave problems in heterogeneous media,, Inverse Problems, 29 (2013). doi: 10.1088/0266-5611/29/6/065001. Google Scholar

[10]

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

[11]

N. Burq, Contrôlabilité exacte des ondes dans des ouverts peu réguliers,, Asymptot. Anal., 14 (1997), 157. Google Scholar

[12]

J. Chen and Y. Yang, Quantitative photo-acoustic tomography with partial data,, Inverse Problems, 28 (2012). doi: 10.1088/0266-5611/28/11/115014. Google Scholar

[13]

J. Chen and Y. Yang, Inverse problem of electro-seismic conversion,, Inverse Problems, 29 (2013). doi: 10.1088/0266-5611/29/11/115006. Google Scholar

[14]

C. B. Croke, I. Lasiecka, G. Uhlmann and M. S. Vogelius (eds.), Geometric Methods in Inverse Problems and PDE Control,, The IMA Volumes in Mathematics and its Applications, (2004). doi: 10.1007/978-1-4684-9375-7. Google Scholar

[15]

K.-J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations,, With contributions by S. Brendle, (2000). Google Scholar

[16]

L. C. Evans, Partial Differential Equations,, Graduate Studies in Mathematics, (1998). Google Scholar

[17]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Reprint of the 1998 edition, (1998). Google Scholar

[18]

R. Glowinski, J.-L. Lions and J. He, Exact and Approximate Controllability for Distributed Parameter Systems. A Numerical Approach,, Encyclopedia of Mathematics and its Applications, (2008). doi: 10.1017/CBO9780511721595. Google Scholar

[19]

O. Y. 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

[20]

O. Y. Imanuvilov and M. Yamamoto, Determination of a coefficient in an acoustic equation with a single measurement,, Inverse Problems, 19 (2003), 157. doi: 10.1088/0266-5611/19/1/309. Google Scholar

[21]

V. Isakov, Inverse Problems for Partial Differential Equations,, 2nd edition, (2006). Google Scholar

[22]

D. Jerison and C. E. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains,, J. Funct. Anal., 130 (1995), 161. doi: 10.1006/jfan.1995.1067. Google Scholar

[23]

T. Kato, Perturbation Theory for Linear Operators,, Reprint of the 1980 edition, (1980). Google Scholar

[24]

M. V. Klibanov, Inverse problems and Carleman estimates,, Inverse Problems, 8 (1992), 575. doi: 10.1088/0266-5611/8/4/009. Google Scholar

[25]

M. V. Klibanov and M. Yamamoto, Lipschitz stability of an inverse problem for an acoustic equation,, Appl. Anal., 85 (2006), 515. doi: 10.1080/00036810500474788. Google Scholar

[26]

I. Lasiecka, J.-L. Lions and R. Triggiani, Nonhomogeneous boundary value problems for second order hyperbolic operators,, J. Math. Pures Appl. (9), 65 (1986), 149. Google Scholar

[27]

J.-L. Lions, Exact controllability, stabilization and perturbations for distributed systems,, SIAM Rev., 30 (1988), 1. doi: 10.1137/1030001. Google Scholar

[28]

J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Vol. I-II,, Translated from the French by P. Kenneth, (1972). Google Scholar

[29]

S. Liu and L. Oksanen, A Lipschitz stable reconstruction formula for the inverse problem for the wave equation,, Accepted in Trans. Amer. Math. Soc., (). Google Scholar

[30]

S. Liu, Recovery of the sound speed and initial displacement for the wave equation by means of a single Dirichlet boundary measurement,, Evol. Equ. Control Theory, 2 (2013), 355. doi: 10.3934/eect.2013.2.355. Google Scholar

[31]

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

[32]

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

[33]

S. Liu and R. Triggiani, Recovering damping and potential coefficients for an inverse non-homogeneous second-order hyperbolic problem via a localized Neumann boundary trace,, Discrete Contin. Dyn. Syst., 33 (2013), 5217. doi: 10.3934/dcds.2013.33.5217. Google Scholar

[34]

W. McLean, Strongly Elliptic Systems and Boundary Integral Equations,, Cambridge University Press, (2000). Google Scholar

[35]

J.-P. Puel and M. Yamamoto, Generic well-posedness in a multidimensional hyperbolic inverse problem,, J. Inverse Ill-Posed Probl., 5 (1997), 55. doi: 10.1515/jiip.1997.5.1.55. Google Scholar

[36]

J.-P. Puel and M. Yamamoto, On a global estimate in a linear inverse hyperbolic problem,, Inverse Problems, 12 (1996), 995. doi: 10.1088/0266-5611/12/6/013. Google Scholar

[37]

Rakesh and W. W. Symes, Uniqueness for an inverse problem for the wave equation,, Comm. Partial Differential Equations, 13 (1988), 87. doi: 10.1080/03605308808820539. Google Scholar

[38]

M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations,, 2nd edition, (2004). Google Scholar

[39]

P. Stefanov and G. Uhlmann, Stability estimates for the hyperbolic Dirichlet to Neumann map in anisotropic media,, J. Funct. Anal., 154 (1998), 330. doi: 10.1006/jfan.1997.3188. Google Scholar

[40]

P. Stefanov and G. Uhlmann, Stable determination of generic simple metrics from the hyperbolic Dirichlet-to-Neumann map,, Int. Math. Res. Not., (2005), 1047. doi: 10.1155/IMRN.2005.1047. Google Scholar

[41]

P. Stefanov and G. Uhlmann, Recovery of a source term or a speed with one measurement and applications,, Trans. Amer. Math. Soc., 365 (2013), 5737. doi: 10.1090/S0002-9947-2013-05703-0. Google Scholar

[42]

Z. Q. Sun, On continuous dependence for an inverse initial-boundary value problem for the wave equation,, J. Math. Anal. Appl., 150 (1990), 188. doi: 10.1016/0022-247X(90)90207-V. Google Scholar

[43]

J. Sylvester and G. Uhlmann, Inverse problems in anisotropic media,, in Inverse Scattering and Applications (Amherst, (1990), 105. doi: 10.1090/conm/122/1135861. Google Scholar

[44]

G. Uhlmann, Inverse scattering in anisotropic media,, in Surveys on Solution Methods for Inverse Problems, (2000), 235. Google Scholar

[45]

M. Yamamoto, Uniqueness and stability in multidimensional hyperbolic inverse problems,, J. Math. Pures Appl., 78 (1999), 65. doi: 10.1016/S0021-7824(99)80010-5. Google Scholar

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