November  2016, 10(4): 1141-1147. doi: 10.3934/ipi.2016035

On the stable recovery of a metric from the hyperbolic DN map with incomplete data

1. 

Department of Mathematics, Purdue University, West Lafayette, IN 47907, United States

2. 

Department of Mathematics, University of Washington, Seattle, WA 9819A, United States

3. 

Department of Mathematics, Stanford University, Stanford, CA 94305, United States

Received  May 2015 Revised  July 2016 Published  October 2016

We show that given two hyperbolic Dirichlet to Neumann maps associated to two Riemannian metrics of a Riemannian manifold with boundary which coincide near the boundary are close then the lens data of the two metrics is the same. As a consequence, we prove uniqueness of recovery a conformal factor (sound speed) locally under some conditions on the latter.
Citation: Plamen Stefanov, Gunther Uhlmann, Andras Vasy. On the stable recovery of a metric from the hyperbolic DN map with incomplete data. Inverse Problems and Imaging, 2016, 10 (4) : 1141-1147. doi: 10.3934/ipi.2016035
References:
[1]

M. Bellassoued and D. D. 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.

[2]

M. I. Belishev, An approach to multidimensional inverse problems for the wave equation, Dokl. Akad. Nauk SSSR, 297 (1987), 524-527.

[3]

M. I. Belishev and Y. V. Kurylev, To the reconstruction of a Riemannian manifold via its spectral data (BC-method), Comm. Partial Differential Equations, 17 (1992), 767-804. doi: 10.1080/03605309208820863.

[4]

G. Bao and H. Zhang, Sensitivity analysis of an inverse problem for the wave equation with caustics, J. Amer. Math. Soc., 27 (2014), 953-981. doi: 10.1090/S0894-0347-2014-00787-6.

[5]

C. Croke and P. Herreros, Lens rigidity with trapped geodesics in two dimensions, Asian Journal of Mathematics, 20 (2016), 47-58, http://www.math.upenn.edu/~ccroke/dvi-papers/SurfaceLensRigidity.pdf. doi: 10.4310/AJM.2016.v20.n1.a3.

[6]

C. Croke, Boundary and lens rigidity of finite quotients, Proc. Amer. Math. Soc., 133 (2005), 3663-3668 (electronic). doi: 10.1090/S0002-9939-05-07927-X.

[7]

_______, Scattering rigidity with trapped geodesics, Ergodic Theory Dynam. Systems, 34 (2014), 826-836. doi: 10.1017/etds.2012.164.

[8]

A. Katchalov, Y. Kurylev and M. Lassas, Inverse Boundary Spectral Problems, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 123, Chapman & Hall/CRC, Boca Raton, FL, 2001. doi: 10.1201/9781420036220.

[9]

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

[10]

S. Liu and L. Oksanen, A lipschitz stable reconstruction formula for the inverse problem for the wave equation, Trans. Amer. Math. Soc., 368 (2016), 319-335. doi: 10.1090/tran/6332.

[11]

M. Lassas, V. Sharafutdinov and G. Uhlmann, Semiglobal boundary rigidity for Riemannian metrics, Math. Ann., 325 (2003), 767-793. doi: 10.1007/s00208-002-0407-4.

[12]

C. Montalto, Stable determination of a simple metric, a covector field and a potential from the hyperbolic Dirichlet-to-Neumann map, Comm. Partial Differential Equations, 39 (2014), 120-145. doi: 10.1080/03605302.2013.843429.

[13]

R. G. Muhometov, On a problem of reconstructing Riemannian metrics, Sibirsk. Mat. Zh., 22 (1981), 119-135, 237.

[14]

L. Pestov and G. Uhlmann, Two dimensional compact simple Riemannian manifolds are boundary distance rigid, Ann. of Math. (2), 161 (2005), 1093-1110. doi: 10.4007/annals.2005.161.1093.

[15]

M. Salo, Stability for solutions of wave equations with $C^{1,1}$ coefficients, Inverse Probl. Imaging, 1 (2007), 537-556. doi: 10.3934/ipi.2007.1.537.

[16]

P. Stefanov and G. Uhlmann, Boundary rigidity and stability for generic simple metrics, J. Amer. Math. Soc., 18 (2005), 975-1003. doi: 10.1090/S0894-0347-05-00494-7.

[17]

_______, Stable determination of generic simple metrics from the hyperbolic Dirichlet-to-Neumann map, Int. Math. Res. Not., 17 (2005), 1047-1061.

[18]

_______, Local lens rigidity with incomplete data for a class of non-simple Riemannian manifolds, J. Differential Geom., 82 (2009), 383-409.

[19]

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

[20]

_______, Thermoacoustic tomography arising in brain imaging, Inverse Problems, 27 (2011), 045004, 26pp. doi: 10.1088/0266-5611/27/4/045004.

[21]

P. Stefanov, G. Uhlmann and A. Vasy, Boundary rigidity with partial data, J. Amer. Math. Soc., 29 (2016), 299-332. doi: 10.1090/jams/846.

[22]

P. Stefanov and Y. Yang, The inverse problem for the Dirichlet-to-Neumann map on Lorentzian manifolds, arXiv:1607.08690, 2016.

[23]

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: 10.1080/03605309508821117.

[24]

M. Zworski, Semiclassical Analysis, Graduate Studies in Mathematics, vol. 138, American Mathematical Society, Providence, RI, 2012. doi: 10.1090/gsm/138.

show all references

References:
[1]

M. Bellassoued and D. D. 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.

[2]

M. I. Belishev, An approach to multidimensional inverse problems for the wave equation, Dokl. Akad. Nauk SSSR, 297 (1987), 524-527.

[3]

M. I. Belishev and Y. V. Kurylev, To the reconstruction of a Riemannian manifold via its spectral data (BC-method), Comm. Partial Differential Equations, 17 (1992), 767-804. doi: 10.1080/03605309208820863.

[4]

G. Bao and H. Zhang, Sensitivity analysis of an inverse problem for the wave equation with caustics, J. Amer. Math. Soc., 27 (2014), 953-981. doi: 10.1090/S0894-0347-2014-00787-6.

[5]

C. Croke and P. Herreros, Lens rigidity with trapped geodesics in two dimensions, Asian Journal of Mathematics, 20 (2016), 47-58, http://www.math.upenn.edu/~ccroke/dvi-papers/SurfaceLensRigidity.pdf. doi: 10.4310/AJM.2016.v20.n1.a3.

[6]

C. Croke, Boundary and lens rigidity of finite quotients, Proc. Amer. Math. Soc., 133 (2005), 3663-3668 (electronic). doi: 10.1090/S0002-9939-05-07927-X.

[7]

_______, Scattering rigidity with trapped geodesics, Ergodic Theory Dynam. Systems, 34 (2014), 826-836. doi: 10.1017/etds.2012.164.

[8]

A. Katchalov, Y. Kurylev and M. Lassas, Inverse Boundary Spectral Problems, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 123, Chapman & Hall/CRC, Boca Raton, FL, 2001. doi: 10.1201/9781420036220.

[9]

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

[10]

S. Liu and L. Oksanen, A lipschitz stable reconstruction formula for the inverse problem for the wave equation, Trans. Amer. Math. Soc., 368 (2016), 319-335. doi: 10.1090/tran/6332.

[11]

M. Lassas, V. Sharafutdinov and G. Uhlmann, Semiglobal boundary rigidity for Riemannian metrics, Math. Ann., 325 (2003), 767-793. doi: 10.1007/s00208-002-0407-4.

[12]

C. Montalto, Stable determination of a simple metric, a covector field and a potential from the hyperbolic Dirichlet-to-Neumann map, Comm. Partial Differential Equations, 39 (2014), 120-145. doi: 10.1080/03605302.2013.843429.

[13]

R. G. Muhometov, On a problem of reconstructing Riemannian metrics, Sibirsk. Mat. Zh., 22 (1981), 119-135, 237.

[14]

L. Pestov and G. Uhlmann, Two dimensional compact simple Riemannian manifolds are boundary distance rigid, Ann. of Math. (2), 161 (2005), 1093-1110. doi: 10.4007/annals.2005.161.1093.

[15]

M. Salo, Stability for solutions of wave equations with $C^{1,1}$ coefficients, Inverse Probl. Imaging, 1 (2007), 537-556. doi: 10.3934/ipi.2007.1.537.

[16]

P. Stefanov and G. Uhlmann, Boundary rigidity and stability for generic simple metrics, J. Amer. Math. Soc., 18 (2005), 975-1003. doi: 10.1090/S0894-0347-05-00494-7.

[17]

_______, Stable determination of generic simple metrics from the hyperbolic Dirichlet-to-Neumann map, Int. Math. Res. Not., 17 (2005), 1047-1061.

[18]

_______, Local lens rigidity with incomplete data for a class of non-simple Riemannian manifolds, J. Differential Geom., 82 (2009), 383-409.

[19]

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

[20]

_______, Thermoacoustic tomography arising in brain imaging, Inverse Problems, 27 (2011), 045004, 26pp. doi: 10.1088/0266-5611/27/4/045004.

[21]

P. Stefanov, G. Uhlmann and A. Vasy, Boundary rigidity with partial data, J. Amer. Math. Soc., 29 (2016), 299-332. doi: 10.1090/jams/846.

[22]

P. Stefanov and Y. Yang, The inverse problem for the Dirichlet-to-Neumann map on Lorentzian manifolds, arXiv:1607.08690, 2016.

[23]

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: 10.1080/03605309508821117.

[24]

M. Zworski, Semiclassical Analysis, Graduate Studies in Mathematics, vol. 138, American Mathematical Society, Providence, RI, 2012. doi: 10.1090/gsm/138.

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