# American Institute of Mathematical Sciences

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 & 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. doi: 10.3934/ipi.2011.5.745. Google Scholar [2] M. I. Belishev, An approach to multidimensional inverse problems for the wave equation,, Dokl. Akad. Nauk SSSR, 297 (1987), 524. Google Scholar [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. doi: 10.1080/03605309208820863. Google Scholar [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. doi: 10.1090/S0894-0347-2014-00787-6. Google Scholar [5] C. Croke and P. Herreros, Lens rigidity with trapped geodesics in two dimensions,, Asian Journal of Mathematics, (2016), 47. doi: 10.4310/AJM.2016.v20.n1.a3. Google Scholar [6] C. Croke, Boundary and lens rigidity of finite quotients,, Proc. Amer. Math. Soc., 133 (2005), 3663. doi: 10.1090/S0002-9939-05-07927-X. Google Scholar [7] _______, Scattering rigidity with trapped geodesics,, Ergodic Theory Dynam. Systems, 34 (2014), 826. doi: 10.1017/etds.2012.164. Google Scholar [8] A. Katchalov, Y. Kurylev and M. Lassas, Inverse Boundary Spectral Problems,, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, (2001). doi: 10.1201/9781420036220. Google Scholar [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. Google Scholar [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. doi: 10.1090/tran/6332. Google Scholar [11] M. Lassas, V. Sharafutdinov and G. Uhlmann, Semiglobal boundary rigidity for Riemannian metrics,, Math. Ann., 325 (2003), 767. doi: 10.1007/s00208-002-0407-4. Google Scholar [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. doi: 10.1080/03605302.2013.843429. Google Scholar [13] R. G. Muhometov, On a problem of reconstructing Riemannian metrics,, Sibirsk. Mat. Zh., 22 (1981), 119. Google Scholar [14] L. Pestov and G. Uhlmann, Two dimensional compact simple Riemannian manifolds are boundary distance rigid,, Ann. of Math. (2), 161 (2005), 1093. doi: 10.4007/annals.2005.161.1093. Google Scholar [15] M. Salo, Stability for solutions of wave equations with $C^{1,1}$ coefficients,, Inverse Probl. Imaging, 1 (2007), 537. doi: 10.3934/ipi.2007.1.537. Google Scholar [16] P. Stefanov and G. Uhlmann, Boundary rigidity and stability for generic simple metrics,, J. Amer. Math. Soc., 18 (2005), 975. doi: 10.1090/S0894-0347-05-00494-7. Google Scholar [17] _______, Stable determination of generic simple metrics from the hyperbolic Dirichlet-to-Neumann map,, Int. Math. Res. Not., 17 (2005), 1047. Google Scholar [18] _______, Local lens rigidity with incomplete data for a class of non-simple Riemannian manifolds,, J. Differential Geom., 82 (2009), 383. Google Scholar [19] _______, Thermoacoustic tomography with variable sound speed,, Inverse Problems, 25 (2009). doi: 10.1088/0266-5611/25/7/075011. Google Scholar [20] _______, Thermoacoustic tomography arising in brain imaging,, Inverse Problems, 27 (2011). doi: 10.1088/0266-5611/27/4/045004. Google Scholar [21] P. Stefanov, G. Uhlmann and A. Vasy, Boundary rigidity with partial data,, J. Amer. Math. Soc., 29 (2016), 299. doi: 10.1090/jams/846. Google Scholar [22] P. Stefanov and Y. Yang, The inverse problem for the Dirichlet-to-Neumann map on Lorentzian manifolds,, , (2016). Google Scholar [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. doi: 10.1080/03605309508821117. Google Scholar [24] M. Zworski, Semiclassical Analysis,, Graduate Studies in Mathematics, (2012). doi: 10.1090/gsm/138. Google Scholar

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. doi: 10.3934/ipi.2011.5.745. Google Scholar [2] M. I. Belishev, An approach to multidimensional inverse problems for the wave equation,, Dokl. Akad. Nauk SSSR, 297 (1987), 524. Google Scholar [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. doi: 10.1080/03605309208820863. Google Scholar [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. doi: 10.1090/S0894-0347-2014-00787-6. Google Scholar [5] C. Croke and P. Herreros, Lens rigidity with trapped geodesics in two dimensions,, Asian Journal of Mathematics, (2016), 47. doi: 10.4310/AJM.2016.v20.n1.a3. Google Scholar [6] C. Croke, Boundary and lens rigidity of finite quotients,, Proc. Amer. Math. Soc., 133 (2005), 3663. doi: 10.1090/S0002-9939-05-07927-X. Google Scholar [7] _______, Scattering rigidity with trapped geodesics,, Ergodic Theory Dynam. Systems, 34 (2014), 826. doi: 10.1017/etds.2012.164. Google Scholar [8] A. Katchalov, Y. Kurylev and M. Lassas, Inverse Boundary Spectral Problems,, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, (2001). doi: 10.1201/9781420036220. Google Scholar [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. Google Scholar [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. doi: 10.1090/tran/6332. Google Scholar [11] M. Lassas, V. Sharafutdinov and G. Uhlmann, Semiglobal boundary rigidity for Riemannian metrics,, Math. Ann., 325 (2003), 767. doi: 10.1007/s00208-002-0407-4. Google Scholar [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. doi: 10.1080/03605302.2013.843429. Google Scholar [13] R. G. Muhometov, On a problem of reconstructing Riemannian metrics,, Sibirsk. Mat. Zh., 22 (1981), 119. Google Scholar [14] L. Pestov and G. Uhlmann, Two dimensional compact simple Riemannian manifolds are boundary distance rigid,, Ann. of Math. (2), 161 (2005), 1093. doi: 10.4007/annals.2005.161.1093. Google Scholar [15] M. Salo, Stability for solutions of wave equations with $C^{1,1}$ coefficients,, Inverse Probl. Imaging, 1 (2007), 537. doi: 10.3934/ipi.2007.1.537. Google Scholar [16] P. Stefanov and G. Uhlmann, Boundary rigidity and stability for generic simple metrics,, J. Amer. Math. Soc., 18 (2005), 975. doi: 10.1090/S0894-0347-05-00494-7. Google Scholar [17] _______, Stable determination of generic simple metrics from the hyperbolic Dirichlet-to-Neumann map,, Int. Math. Res. Not., 17 (2005), 1047. Google Scholar [18] _______, Local lens rigidity with incomplete data for a class of non-simple Riemannian manifolds,, J. Differential Geom., 82 (2009), 383. Google Scholar [19] _______, Thermoacoustic tomography with variable sound speed,, Inverse Problems, 25 (2009). doi: 10.1088/0266-5611/25/7/075011. Google Scholar [20] _______, Thermoacoustic tomography arising in brain imaging,, Inverse Problems, 27 (2011). doi: 10.1088/0266-5611/27/4/045004. Google Scholar [21] P. Stefanov, G. Uhlmann and A. Vasy, Boundary rigidity with partial data,, J. Amer. Math. Soc., 29 (2016), 299. doi: 10.1090/jams/846. Google Scholar [22] P. Stefanov and Y. Yang, The inverse problem for the Dirichlet-to-Neumann map on Lorentzian manifolds,, , (2016). Google Scholar [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. doi: 10.1080/03605309508821117. Google Scholar [24] M. Zworski, Semiclassical Analysis,, Graduate Studies in Mathematics, (2012). doi: 10.1090/gsm/138. Google Scholar
 [1] Michael Anderson, Atsushi Katsuda, Yaroslav Kurylev, Matti Lassas and Michael Taylor. Metric tensor estimates, geometric convergence, and inverse boundary problems. Electronic Research Announcements, 2003, 9: 69-79. [2] Tan Bui-Thanh, Omar Ghattas. A scalable algorithm for MAP estimators in Bayesian inverse problems with Besov priors. Inverse Problems & Imaging, 2015, 9 (1) : 27-53. doi: 10.3934/ipi.2015.9.27 [3] Victor Isakov, Joseph Myers. On the inverse doping profile problem. Inverse Problems & Imaging, 2012, 6 (3) : 465-486. doi: 10.3934/ipi.2012.6.465 [4] Amadeu Delshams, Josep J. Masdemont, Pablo Roldán. Computing the scattering map in the spatial Hill's problem. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 455-483. doi: 10.3934/dcdsb.2008.10.455 [5] Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271 [6] Nguyen Huy Tuan, Mokhtar Kirane, Long Dinh Le, Van Thinh Nguyen. On an inverse problem for fractional evolution equation. Evolution Equations & Control Theory, 2017, 6 (1) : 111-134. doi: 10.3934/eect.2017007 [7] Russell Johnson, Luca Zampogni. On the inverse Sturm-Liouville problem. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 405-428. doi: 10.3934/dcds.2007.18.405 [8] Mikko Orispää, Markku Lehtinen. Fortran linear inverse problem solver. Inverse Problems & Imaging, 2010, 4 (3) : 485-503. doi: 10.3934/ipi.2010.4.485 [9] A. Doubov, Enrique Fernández-Cara, Manuel González-Burgos, J. H. Ortega. A geometric inverse problem for the Boussinesq system. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1213-1238. doi: 10.3934/dcdsb.2006.6.1213 [10] Alexandr Mikhaylov, Victor Mikhaylov. Dynamic inverse problem for Jacobi matrices. Inverse Problems & Imaging, 2019, 13 (3) : 431-447. doi: 10.3934/ipi.2019021 [11] Chunhui Qiu, Rirong Yuan. On the Dirichlet problem for fully nonlinear elliptic equations on annuli of metric cones. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5707-5730. doi: 10.3934/dcds.2017247 [12] Ihsane Bikri, Ronald B. Guenther, Enrique A. Thomann. The Dirichlet to Neumann map - An application to the Stokes problem in half space. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 221-230. doi: 10.3934/dcdss.2010.3.221 [13] Hermann Gross, Sebastian Heidenreich, Mark-Alexander Henn, Markus Bär, Andreas Rathsfeld. Modeling aspects to improve the solution of the inverse problem in scatterometry. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 497-519. doi: 10.3934/dcdss.2015.8.497 [14] Pedro Caro. On an inverse problem in electromagnetism with local data: stability and uniqueness. Inverse Problems & Imaging, 2011, 5 (2) : 297-322. doi: 10.3934/ipi.2011.5.297 [15] Peter Monk, Virginia Selgas. Sampling type methods for an inverse waveguide problem. Inverse Problems & Imaging, 2012, 6 (4) : 709-747. doi: 10.3934/ipi.2012.6.709 [16] Michele Di Cristo. Stability estimates in the inverse transmission scattering problem. Inverse Problems & Imaging, 2009, 3 (4) : 551-565. doi: 10.3934/ipi.2009.3.551 [17] Mohsen Tadi. A computational method for an inverse problem in a parabolic system. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 205-218. doi: 10.3934/dcdsb.2009.12.205 [18] Victor Isakov. On uniqueness in the inverse conductivity problem with local data. Inverse Problems & Imaging, 2007, 1 (1) : 95-105. doi: 10.3934/ipi.2007.1.95 [19] Luca Rondi. On the regularization of the inverse conductivity problem with discontinuous conductivities. Inverse Problems & Imaging, 2008, 2 (3) : 397-409. doi: 10.3934/ipi.2008.2.397 [20] Fang Zeng, Pablo Suarez, Jiguang Sun. A decomposition method for an interior inverse scattering problem. Inverse Problems & Imaging, 2013, 7 (1) : 291-303. doi: 10.3934/ipi.2013.7.291

2018 Impact Factor: 1.469