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 and 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), 085010, 20pp. doi: 10.1088/0266-5611/26/8/085010.

[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), 185-233.

[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), 1024-1065. doi: 10.1137/0330055.

[4]

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

[5]

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

[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), 745-773. doi: 10.3934/ipi.2011.5.745.

[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), 193-224. doi: 10.1016/j.matpur.2005.02.004.

[8]

M. Bellassoued and M. Yamamoto, Determination of a coefficient in the wave equation with a single measurement, Appl. Anal., 87 (2008), 901-920. doi: 10.1080/00036810802369249.

[9]

K. D. Blazek, C. Stolk and W. W. Symes, A mathematical framework for inverse wave problems in heterogeneous media, Inverse Problems, 29 (2013), 065001, 37pp. doi: 10.1088/0266-5611/29/6/065001.

[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), 269-272.

[11]

N. Burq, Contrôlabilité exacte des ondes dans des ouverts peu réguliers, Asymptot. Anal., 14 (1997), 157-191.

[12]

J. Chen and Y. Yang, Quantitative photo-acoustic tomography with partial data, Inverse Problems, 28 (2012), 115014, 15pp. doi: 10.1088/0266-5611/28/11/115014.

[13]

J. Chen and Y. Yang, Inverse problem of electro-seismic conversion, Inverse Problems, 29 (2013), 115006, 15pp. doi: 10.1088/0266-5611/29/11/115006.

[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, 137, Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4684-9375-7.

[15]

K.-J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt, Graduate Texts in Mathematics, 194, Springer-Verlag, New York, 2000.

[16]

L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998.

[17]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.

[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, 117, Cambridge University Press, Cambridge, 2008. doi: 10.1017/CBO9780511721595.

[19]

O. Y. 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.

[20]

O. Y. Imanuvilov and M. Yamamoto, Determination of a coefficient in an acoustic equation with a single measurement, Inverse Problems, 19 (2003), 157-171. doi: 10.1088/0266-5611/19/1/309.

[21]

V. Isakov, Inverse Problems for Partial Differential Equations, 2nd edition, Applied Mathematical Sciences, 127, Springer, New York, 2006.

[22]

D. Jerison and C. E. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal., 130 (1995), 161-219. doi: 10.1006/jfan.1995.1067.

[23]

T. Kato, Perturbation Theory for Linear Operators, Reprint of the 1980 edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995.

[24]

M. V. Klibanov, Inverse problems and Carleman estimates, Inverse Problems, 8 (1992), 575-596. doi: 10.1088/0266-5611/8/4/009.

[25]

M. V. Klibanov and M. Yamamoto, Lipschitz stability of an inverse problem for an acoustic equation, Appl. Anal., 85 (2006), 515-538. doi: 10.1080/00036810500474788.

[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), 149-192.

[27]

J.-L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev., 30 (1988), 1-68. doi: 10.1137/1030001.

[28]

J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Vol. I-II, Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181, Springer-Verlag, New York-Heidelberg, 1972.

[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.

[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), 355-364. doi: 10.3934/eect.2013.2.355.

[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), 1631-1666. doi: 10.1137/100808988.

[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), 1551-1581. doi: 10.1080/00036811.2011.618125.

[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), 5217-5252. doi: 10.3934/dcds.2013.33.5217.

[34]

W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000.

[35]

J.-P. Puel and M. Yamamoto, Generic well-posedness in a multidimensional hyperbolic inverse problem, J. Inverse Ill-Posed Probl., 5 (1997), 55-83. doi: 10.1515/jiip.1997.5.1.55.

[36]

J.-P. Puel and M. Yamamoto, On a global estimate in a linear inverse hyperbolic problem, Inverse Problems, 12 (1996), 995-1002. doi: 10.1088/0266-5611/12/6/013.

[37]

Rakesh and W. W. Symes, Uniqueness for an inverse problem for the wave equation, Comm. Partial Differential Equations, 13 (1988), 87-96. doi: 10.1080/03605308808820539.

[38]

M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, 2nd edition, Texts in Applied Mathematics 13, Springer-Verlag, New York, 2004.

[39]

P. Stefanov and G. Uhlmann, Stability estimates for the hyperbolic Dirichlet to Neumann map in anisotropic media, J. Funct. Anal., 154 (1998), 330-358. doi: 10.1006/jfan.1997.3188.

[40]

P. Stefanov and G. Uhlmann, Stable determination of generic simple metrics from the hyperbolic Dirichlet-to-Neumann map, Int. Math. Res. Not., (2005), 1047-1061. doi: 10.1155/IMRN.2005.1047.

[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), 5737-5758. doi: 10.1090/S0002-9947-2013-05703-0.

[42]

Z. Q. Sun, On continuous dependence for an inverse initial-boundary value problem for the wave equation, J. Math. Anal. Appl., 150 (1990), 188-204. doi: 10.1016/0022-247X(90)90207-V.

[43]

J. Sylvester and G. Uhlmann, Inverse problems in anisotropic media, in Inverse Scattering and Applications (Amherst, MA, 1990), Contemp. Math., Amer. Math. Soc., 122, Providence, RI, 1991, 105-117. doi: 10.1090/conm/122/1135861.

[44]

G. Uhlmann, Inverse scattering in anisotropic media, in Surveys on Solution Methods for Inverse Problems, Springer, Vienna, 2000, 235-251.

[45]

M. Yamamoto, Uniqueness and stability in multidimensional hyperbolic inverse problems, J. Math. Pures Appl., 78 (1999), 65-98. doi: 10.1016/S0021-7824(99)80010-5.

show all references

References:
[1]

G. Bal and G. Uhlmann, Inverse diffusion theory of photoacoustics, Inverse Problems, 26 (2010), 085010, 20pp. doi: 10.1088/0266-5611/26/8/085010.

[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), 185-233.

[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), 1024-1065. doi: 10.1137/0330055.

[4]

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

[5]

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

[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), 745-773. doi: 10.3934/ipi.2011.5.745.

[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), 193-224. doi: 10.1016/j.matpur.2005.02.004.

[8]

M. Bellassoued and M. Yamamoto, Determination of a coefficient in the wave equation with a single measurement, Appl. Anal., 87 (2008), 901-920. doi: 10.1080/00036810802369249.

[9]

K. D. Blazek, C. Stolk and W. W. Symes, A mathematical framework for inverse wave problems in heterogeneous media, Inverse Problems, 29 (2013), 065001, 37pp. doi: 10.1088/0266-5611/29/6/065001.

[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), 269-272.

[11]

N. Burq, Contrôlabilité exacte des ondes dans des ouverts peu réguliers, Asymptot. Anal., 14 (1997), 157-191.

[12]

J. Chen and Y. Yang, Quantitative photo-acoustic tomography with partial data, Inverse Problems, 28 (2012), 115014, 15pp. doi: 10.1088/0266-5611/28/11/115014.

[13]

J. Chen and Y. Yang, Inverse problem of electro-seismic conversion, Inverse Problems, 29 (2013), 115006, 15pp. doi: 10.1088/0266-5611/29/11/115006.

[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, 137, Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4684-9375-7.

[15]

K.-J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt, Graduate Texts in Mathematics, 194, Springer-Verlag, New York, 2000.

[16]

L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998.

[17]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.

[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, 117, Cambridge University Press, Cambridge, 2008. doi: 10.1017/CBO9780511721595.

[19]

O. Y. 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.

[20]

O. Y. Imanuvilov and M. Yamamoto, Determination of a coefficient in an acoustic equation with a single measurement, Inverse Problems, 19 (2003), 157-171. doi: 10.1088/0266-5611/19/1/309.

[21]

V. Isakov, Inverse Problems for Partial Differential Equations, 2nd edition, Applied Mathematical Sciences, 127, Springer, New York, 2006.

[22]

D. Jerison and C. E. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal., 130 (1995), 161-219. doi: 10.1006/jfan.1995.1067.

[23]

T. Kato, Perturbation Theory for Linear Operators, Reprint of the 1980 edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995.

[24]

M. V. Klibanov, Inverse problems and Carleman estimates, Inverse Problems, 8 (1992), 575-596. doi: 10.1088/0266-5611/8/4/009.

[25]

M. V. Klibanov and M. Yamamoto, Lipschitz stability of an inverse problem for an acoustic equation, Appl. Anal., 85 (2006), 515-538. doi: 10.1080/00036810500474788.

[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), 149-192.

[27]

J.-L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev., 30 (1988), 1-68. doi: 10.1137/1030001.

[28]

J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Vol. I-II, Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181, Springer-Verlag, New York-Heidelberg, 1972.

[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.

[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), 355-364. doi: 10.3934/eect.2013.2.355.

[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), 1631-1666. doi: 10.1137/100808988.

[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), 1551-1581. doi: 10.1080/00036811.2011.618125.

[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), 5217-5252. doi: 10.3934/dcds.2013.33.5217.

[34]

W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000.

[35]

J.-P. Puel and M. Yamamoto, Generic well-posedness in a multidimensional hyperbolic inverse problem, J. Inverse Ill-Posed Probl., 5 (1997), 55-83. doi: 10.1515/jiip.1997.5.1.55.

[36]

J.-P. Puel and M. Yamamoto, On a global estimate in a linear inverse hyperbolic problem, Inverse Problems, 12 (1996), 995-1002. doi: 10.1088/0266-5611/12/6/013.

[37]

Rakesh and W. W. Symes, Uniqueness for an inverse problem for the wave equation, Comm. Partial Differential Equations, 13 (1988), 87-96. doi: 10.1080/03605308808820539.

[38]

M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, 2nd edition, Texts in Applied Mathematics 13, Springer-Verlag, New York, 2004.

[39]

P. Stefanov and G. Uhlmann, Stability estimates for the hyperbolic Dirichlet to Neumann map in anisotropic media, J. Funct. Anal., 154 (1998), 330-358. doi: 10.1006/jfan.1997.3188.

[40]

P. Stefanov and G. Uhlmann, Stable determination of generic simple metrics from the hyperbolic Dirichlet-to-Neumann map, Int. Math. Res. Not., (2005), 1047-1061. doi: 10.1155/IMRN.2005.1047.

[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), 5737-5758. doi: 10.1090/S0002-9947-2013-05703-0.

[42]

Z. Q. Sun, On continuous dependence for an inverse initial-boundary value problem for the wave equation, J. Math. Anal. Appl., 150 (1990), 188-204. doi: 10.1016/0022-247X(90)90207-V.

[43]

J. Sylvester and G. Uhlmann, Inverse problems in anisotropic media, in Inverse Scattering and Applications (Amherst, MA, 1990), Contemp. Math., Amer. Math. Soc., 122, Providence, RI, 1991, 105-117. doi: 10.1090/conm/122/1135861.

[44]

G. Uhlmann, Inverse scattering in anisotropic media, in Surveys on Solution Methods for Inverse Problems, Springer, Vienna, 2000, 235-251.

[45]

M. Yamamoto, Uniqueness and stability in multidimensional hyperbolic inverse problems, J. Math. Pures Appl., 78 (1999), 65-98. doi: 10.1016/S0021-7824(99)80010-5.

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