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Sensitivity analysis of an inverse problem for the wave equation with caustics


Authors: Gang Bao and Hai Zhang
Journal: J. Amer. Math. Soc. 27 (2014), 953-981
MSC (2010): Primary 35R30; Secondary 35S30
DOI: https://doi.org/10.1090/S0894-0347-2014-00787-6
Published electronically: March 5, 2014
MathSciNet review: 3230816
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Abstract: The paper investigates the sensitivity of the inverse problem of recovering the velocity field in a bounded domain from the boundary dynamic Dirichlet-to-Neumann map (DDtN) for the wave equation. Three main results are obtained: (1) assuming that two velocity fields are non-trapping and are equal to a constant near the boundary, it is shown that the two induced scattering relations must be identical if their corresponding DDtN maps are sufficiently close; (2) a geodesic X-ray transform operator with matrix-valued weight is introduced by linearizing the operator which associates each velocity field with its induced Hamiltonian flow. A selected set of geodesics whose conormal bundle can cover the cotangent space at an interior point is used to recover the singularity of the X-ray transformed function at the point; a local stability estimate is established for this case. Although fold caustics are allowed along these geodesics, it is required that these caustics contribute to a smoother term in the transform than the point itself. The existence of such a set of geodesics is guaranteed under some natural assumptions in dimensions greater than or equal to three by the classification result on caustics and regularity theory of Fourier Integral Operators. The interior point with the above required set of geodesics is called ``fold-regular''. (3) Assuming that a background velocity field with every interior point fold-regular is fixed and another velocity field is sufficiently close to it and satisfies a certain orthogonality condition, it is shown that if the two corresponding DDtN maps are sufficiently close then they must be equal.


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  • [1] G. Alessandrini and J. Sylvester, Stability for a multidimensional inverse spectral theorem, Comm. Partial Differential Equations 15 (1990), no. 5, 711-736. MR 1070844 (91i:35198), https://doi.org/10.1080/03605309908820705
  • [2] S. Bougacha, Jean-Luc Akian, and R. Alexandre, Gaussian beams summation for the wave equation in a convex domain, Commun. Math. Sci. 7 (2009), no. 4, 973-1008. MR 2604628 (2011b:35284)
  • [3] V. I. Arnold, Singularity theory: Selected papers, London Mathematical Society Lecture Note Series, vol. 53, Cambridge University Press, Cambridge, 1981. Translated from the Russian; With an introduction by C. T. C. Wall. MR 631683 (83d:58016)
  • [4] V. I. Arnold, S. M. Guseĭn-Zade, and A. N. Varchenko, Singularities of differentiable maps, Vol. I: The classification of critical points, caustics and wave fronts, Monographs in Mathematics, vol. 82, Birkhäuser Boston Inc., Boston, MA, 1985. Translated from the Russian by Ian Porteous and Mark Reynolds. MR 777682 (86f:58018)
  • [5] G. Bao, J. Qian, L. Ying, and H. Zhang, A convergent multiscale Gaussian-beam parametrix for the wave equation, Comm. Partial Differential Equations 38 (2013), no. 1, 92-134. MR 3005548, https://doi.org/10.1080/03605302.2012.727130
  • [6] G. Bao and K. Yun, On the stability of an inverse problem for the wave equation, Inverse Problems 25 (2009), no. 4, 045003, 7. MR 2482154 (2010a:35270), https://doi.org/10.1088/0266-5611/25/4/045003
  • [7] M. I. Belishev, An approach to multidimensional inverse problems for the wave equation, Dokl. Akad. Nauk SSSR 297 (1987), no. 3, 524-527 (Russian); English transl., Soviet Math. Dokl. 36 (1988), no. 3, 481-484. MR 924687 (89c:35152)
  • [8] M. I. Belishev, Boundary control in reconstruction of manifolds and metrics (the BC method), Inverse Problems 13 (1997), no. 5, R1-R45. MR 1474359 (98k:58073), https://doi.org/10.1088/0266-5611/13/5/002
  • [9] M. I. Belishev, Recent progress in the boundary control method, Inverse Problems 23 (2007), no. 5, R1-R67. MR 2353313 (2008h:93001), https://doi.org/10.1088/0266-5611/23/5/R01
  • [10] Michael I. Belishev and Yaroslav V. Kurylev, To the reconstruction of a Riemannian manifold via its spectral data (BC-method), Comm. Partial Differential Equations 17 (1992), no. 5-6, 767-804. MR 1177292 (94a:58199), https://doi.org/10.1080/03605309208820863
  • [11] M. Bellassoued and David Dos S. Ferreira, Stability estimates for the anisotropic wave equation from the Dirichlet-to-Neumann map, Inverse Probl. Imaging 5 (2011), no. 4, 745-773. MR 2852371, https://doi.org/10.3934/ipi.2011.5.745
  • [12] A. Greenleaf and A. Seeger, Fourier integral operators with cusp singularities, Amer. J. Math. 120 (1998), no. 5, 1077-1119. MR 1646055 (99g:58120)
  • [13] L. Hörmander, The analysis of linear partial differential operators. III, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 274, Springer-Verlag, Berlin, 1985. Pseudodifferential operators. MR 781536 (87d:35002a)
  • [14] V. Isakov, Inverse problems for partial differential equations, 2nd ed., Applied Mathematical Sciences, vol. 127, Springer, New York, 2006. MR 2193218 (2006h:35279)
  • [15] A. Katchalov, Y. Kurylev, and Matti 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. MR 1889089 (2003e:58045)
  • [16] I. Lasiecka, J.-L. Lions, and R. Triggiani, Nonhomogeneous boundary value problems for second order hyperbolic operators, J. Math. Pures Appl. (9) 65 (1986), no. 2, 149-192. MR 867669 (88c:35092)
  • [17] John M. Lee, Riemannian manifolds: An introduction to curvature, Graduate Texts in Mathematics, vol. 176, Springer-Verlag, New York, 1997. MR 1468735 (98d:53001)
  • [18] R. Michel, Sur la rigidité imposée par la longueur des géodésiques, Invent. Math. 65 (1981/82), no. 1, 71-83 (French). MR 636880 (83d:58021), https://doi.org/10.1007/BF01389295
  • [19] C. Montalto, Stable determination of a simple metric, a covector field and a potential from the hyperbolic Dirichlet-to-Nuemann map, Comm. PDE., to appear.
  • [20] J. Qian and L. Ying, Fast multiscale Gaussian wavepacket transforms and multiscale Gaussian beams for the wave equation, Multiscale Model. Simul. 8 (2010), no. 5, 1803-1837. MR 2728710 (2011k:65137), https://doi.org/10.1137/100787313
  • [21] L. Pestov and G. Uhlmann, The scattering relation and the Dirichlet-to-Neumann map, Recent advances in differential equations and mathematical physics, Contemp. Math., vol. 412, Amer. Math. Soc., Providence, RI, 2006, pp. 249-262. MR 2259112 (2007k:53055), https://doi.org/10.1090/conm/412/07779
  • [22] J. Ralston, Gaussian beams and the propagation of singularities, Studies in partial differential equations, MAA Stud. Math., vol. 23, Math. Assoc. America, Washington, DC, 1982, pp. 206-248. MR 716507 (85c:35052)
  • [23] V. G. Romanov, Inverse problems of mathematical physics, VNU Science Press b.v., Utrecht, 1987. With a foreword by V. G. Yakhno; Translated from the Russian by L. Ya. Yuzina. MR 885902 (88b:35203)
  • [24] V.A. Sharafutdinov, Ray Transform on Riemannian Manifolds, Lecture notes, University of Oulu (1999).
  • [25] P. Stefanov and G. Uhlmann, Stability estimates for the hyperbolic Dirichlet to Neumann map in anisotropic media, J. Funct. Anal. 154 (1998), no. 2, 330-358. MR 1612709 (99f:35120), https://doi.org/10.1006/jfan.1997.3188
  • [26] P. Stefanov and G. Uhlmann, Stability estimates for the X-ray transform of tensor fields and boundary rigidity, Duke Math. J. 123 (2004), no. 3, 445-467. MR 2068966 (2005h:53130), https://doi.org/10.1215/S0012-7094-04-12332-2
  • [27] P. Stefanov and G. Uhlmann, Boundary rigidity and stability for generic simple metrics, J. Amer. Math. Soc. 18 (2005), no. 4, 975-1003. MR 2163868 (2006h:53031), https://doi.org/10.1090/S0894-0347-05-00494-7
  • [28] P. Stefanov and G. Uhlmann, Stable determination of generic simple metrics from the hyperbolic Dirichlet-to-Neumann map, Int. Math. Res. Not. 17 (2005), 1047-1061. MR 2145709 (2006a:58030), https://doi.org/10.1155/IMRN.2005.1047
  • [29] B. Frigyik, P. Stefanov, and G. Uhlmann, The X-ray transform for a generic family of curves and weights, J. Geom. Anal. 18 (2008), no. 1, 89-108. MR 2365669 (2008j:53128), https://doi.org/10.1007/s12220-007-9007-6
  • [30] P. Stefanov and G. Uhlmann, Integral geometry on tensor fields on a class of non-simple Riemannian manifolds, Amer. J. Math. 130 (2008), no. 1, 239-268. MR 2382148 (2009e:53051), https://doi.org/10.1353/ajm.2008.0003
  • [31] P. Stefanov and G. Uhlmann, Linearizing non-linear inverse problems and an application to inverse backscattering, J. Funct. Anal. 256 (2009), no. 9, 2842-2866. MR 2502425 (2010f:47028), https://doi.org/10.1016/j.jfa.2008.10.017
  • [32] P. Stefanov and G. Uhlmann, Local lens rigidity with incomplete data for a class of non-simple Riemannian manifolds, J. Differential Geom. 82 (2009), no. 2, 383-409. MR 2520797 (2011d:53081)
  • [33] Zi Qi Sun, On continuous dependence for an inverse initial-boundary value problem for the wave equation, J. Math. Anal. Appl. 150 (1990), no. 1, 188-204. MR 1059582 (91i:35024), https://doi.org/10.1016/0022-247X(90)90207-V
  • [34] G. Uhlmann, The Cauchy data and the scattering relation, Geometric methods in inverse problems and PDE control, IMA Vol. Math. Appl., vol. 137, Springer, New York, 2004, pp. 263-287. MR 2169908 (2006f:58032), https://doi.org/10.1007/978-1-4684-9375-7_10
  • [35] P. Stefanov and G. Uhlmann, The geodesic X-ray transform with fold caustics, Anal. PDE 5 (2012), no. 2, 219-260. MR 2970707, https://doi.org/10.2140/apde.2012.5.219
  • [36] H. Zhang, On the stability/sensitivity of recovering velocity fields from boundary measurements, (PhD dissertation), Michigan State University (2013).

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Additional Information

Gang Bao
Affiliation: Department of Mathematics, Zhejiang University, Hangzhou 310027, China and Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
Email: drbaogang@gmail.com

Hai Zhang
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
Email: zh.hai84@gmail.com

DOI: https://doi.org/10.1090/S0894-0347-2014-00787-6
Received by editor(s): December 17, 2012
Received by editor(s) in revised form: July 20, 2013
Published electronically: March 5, 2014
Additional Notes: The research was supported in part by the NSF grants DMS-0968360, DMS-1211292, the ONR grant N00014-12-1-0319, a Key Project of the Major Research Plan of NSFC (No. 91130004), and a special research grant from Zhejiang University
Article copyright: © Copyright 2014 American Mathematical Society
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