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

HTML articles powered by AMS MathViewer

- by Gang Bao and Hai Zhang;
- J. Amer. Math. Soc.
**27**(2014), 953-981 - DOI: https://doi.org/10.1090/S0894-0347-2014-00787-6
- Published electronically: March 5, 2014
- PDF | Request permission

## 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.## References

- Giovanni Alessandrini and John Sylvester,
*Stability for a multidimensional inverse spectral theorem*, Comm. Partial Differential Equations**15**(1990), no. 5, 711–736. MR**1070844**, DOI 10.1080/03605309908820705 - Salma Bougacha, Jean-Luc Akian, and Radjesvarane Alexandre,
*Gaussian beams summation for the wave equation in a convex domain*, Commun. Math. Sci.**7**(2009), no. 4, 973–1008. MR**2604628**, DOI 10.4310/CMS.2009.v7.n4.a9 - V. I. Arnol′d,
*Singularity theory*, London Mathematical Society Lecture Note Series, vol. 53, Cambridge University Press, Cambridge-New York, 1981. Selected papers; Translated from the Russian; With an introduction by C. T. C. Wall. MR**631683**, DOI 10.1017/CBO9780511662713 - V. I. Arnol′d, S. M. Guseĭn-Zade, and A. N. Varchenko,
*Singularities of differentiable maps. Vol. I*, Monographs in Mathematics, vol. 82, Birkhäuser Boston, Inc., Boston, MA, 1985. The classification of critical points, caustics and wave fronts; Translated from the Russian by Ian Porteous and Mark Reynolds. MR**777682**, DOI 10.1007/978-1-4612-5154-5 - Gang Bao, Jianliang Qian, Lexing Ying, and Hai Zhang,
*A convergent multiscale Gaussian-beam parametrix for the wave equation*, Comm. Partial Differential Equations**38**(2013), no. 1, 92–134. MR**3005548**, DOI 10.1080/03605302.2012.727130 - Gang Bao and Kihyun Yun,
*On the stability of an inverse problem for the wave equation*, Inverse Problems**25**(2009), no. 4, 045003, 7. MR**2482154**, DOI 10.1088/0266-5611/25/4/045003 - 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** - M. I. Belishev,
*Boundary control in reconstruction of manifolds and metrics (the BC method)*, Inverse Problems**13**(1997), no. 5, R1–R45. MR**1474359**, DOI 10.1088/0266-5611/13/5/002 - M. I. Belishev,
*Recent progress in the boundary control method*, Inverse Problems**23**(2007), no. 5, R1–R67. MR**2353313**, DOI 10.1088/0266-5611/23/5/R01 - 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**, DOI 10.1080/03605309208820863 - Mourad Bellassoued and David Dos Santos 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**, DOI 10.3934/ipi.2011.5.745 - Allan Greenleaf and Andreas Seeger,
*Fourier integral operators with cusp singularities*, Amer. J. Math.**120**(1998), no. 5, 1077–1119. MR**1646055**, DOI 10.1353/ajm.1998.0036 - Lars 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** - Victor Isakov,
*Inverse problems for partial differential equations*, 2nd ed., Applied Mathematical Sciences, vol. 127, Springer, New York, 2006. MR**2193218** - Alexander Katchalov, Yaroslav 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**, DOI 10.1201/9781420036220 - 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** - John M. Lee,
*Riemannian manifolds*, Graduate Texts in Mathematics, vol. 176, Springer-Verlag, New York, 1997. An introduction to curvature. MR**1468735**, DOI 10.1007/b98852 - René Michel,
*Sur la rigidité imposée par la longueur des géodésiques*, Invent. Math.**65**(1981/82), no. 1, 71–83 (French). MR**636880**, DOI 10.1007/BF01389295 - 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. - Jianliang Qian and Lexing 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**, DOI 10.1137/100787313 - Leonid Pestov and Gunther 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**, DOI 10.1090/conm/412/07779 - James 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** - 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** - V. Sharafutdinov A.,
*Ray Transform on Riemannian Manifolds, Lecture notes, University of Oulu*(1999). - Plamen Stefanov and Gunther Uhlmann,
*Stability estimates for the hyperbolic Dirichlet to Neumann map in anisotropic media*, J. Funct. Anal.**154**(1998), no. 2, 330–358. MR**1612709**, DOI 10.1006/jfan.1997.3188 - Plamen Stefanov and Gunther 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**, DOI 10.1215/S0012-7094-04-12332-2 - Plamen Stefanov and Gunther Uhlmann,
*Boundary rigidity and stability for generic simple metrics*, J. Amer. Math. Soc.**18**(2005), no. 4, 975–1003. MR**2163868**, DOI 10.1090/S0894-0347-05-00494-7 - Plamen Stefanov and Gunther Uhlmann,
*Stable determination of generic simple metrics from the hyperbolic Dirichlet-to-Neumann map*, Int. Math. Res. Not.**17**(2005), 1047–1061. MR**2145709**, DOI 10.1155/IMRN.2005.1047 - Bela Frigyik, Plamen Stefanov, and Gunther Uhlmann,
*The X-ray transform for a generic family of curves and weights*, J. Geom. Anal.**18**(2008), no. 1, 89–108. MR**2365669**, DOI 10.1007/s12220-007-9007-6 - Plamen Stefanov and Gunther 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**, DOI 10.1353/ajm.2008.0003 - Plamen Stefanov and Gunther Uhlmann,
*Linearizing non-linear inverse problems and an application to inverse backscattering*, J. Funct. Anal.**256**(2009), no. 9, 2842–2866. MR**2502425**, DOI 10.1016/j.jfa.2008.10.017 - Plamen Stefanov and Gunther 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** - 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**, DOI 10.1016/0022-247X(90)90207-V - Gunther 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**, DOI 10.1007/978-1-4684-9375-7_{1}0 - Plamen Stefanov and Gunther Uhlmann,
*The geodesic X-ray transform with fold caustics*, Anal. PDE**5**(2012), no. 2, 219–260. MR**2970707**, DOI 10.2140/apde.2012.5.219 - H. Zhang,
*On the stability/sensitivity of recovering velocity fields from boundary measurements, (PhD dissertation), Michigan State University*(2013).

## Bibliographic 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
- MR Author ID: 890053
- Email: zh.hai84@gmail.com
- 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
- © Copyright 2014
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3230816