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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
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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Additional Information

Gang Bao
Affiliation: Department of Mathematics, Zhejiang University, Hangzhou 310027, China and Department of Mathematics, Michigan State University, East Lansing, Michigan 48824

Hai Zhang
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824

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
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