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A Lipschitz stable reconstruction formula for the inverse problem for the wave equation


Authors: Shitao Liu and Lauri Oksanen
Journal: Trans. Amer. Math. Soc. 368 (2016), 319-335
MSC (2010): Primary 35R30
DOI: https://doi.org/10.1090/tran/6332
Published electronically: April 20, 2015
MathSciNet review: 3413865
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Abstract: We consider the problem to reconstruct a wave speed $ c \in C^\infty (M)$ in a domain $ M \subset \mathbb{R}^n$ from acoustic boundary measurements modelled by the hyperbolic Dirichlet-to-Neumann map $ \Lambda $. We introduce a reconstruction formula for $ c$ that is based on the Boundary Control method and incorporates features also from the complex geometric optics solutions approach. Moreover, we show that the reconstruction formula is locally Lipschitz stable for a low frequency component of $ c^{-2}$ under the assumption that the Riemannian manifold $ (M, c^{-2} dx^2)$ has a strictly convex function with no critical points. That is, we show that for all bounded $ C^2$ neighborhoods $ U$ of $ c$, there is a $ C^1$ neighborhood $ V$ of $ c$ and constants $ C, R > 0$ such that

$\displaystyle \vert\mathcal {F}\left (\tilde c^{-2} - c^{-2}\right )(\xi )\vert... ...left \Vert\tilde \Lambda - \Lambda \right \Vert _*, \quad \xi \in \mathbb{R}^n,$    

for all $ \tilde c \in U \cap V$, where $ \tilde \Lambda $ is the Dirichlet-to-Neumann map corresponding to the wave speed $ \tilde c$ and $ \left \vert\cdot \right \vert _*$ is a norm capturing certain regularity properties of the Dirichlet-to-Neumann maps.

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Additional Information

Shitao Liu
Affiliation: Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68 FI-00014 Helsinki, Finland
Address at time of publication: Department of Mathematical Sciences, Clemson University, Clemson, South Carolina 29634
Email: shitao.liu@helsinki.fi, liul@clemson.edu

Lauri Oksanen
Affiliation: Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68 FI-00014 Helsinki, Finland
Address at time of publication: Department of Mathematics, University College London, Gower Street, WC1E 6BT, London, United Kingdom
Email: lauri.oksanen@helsinki.fi, l.oksanen@ucl.ac.uk

DOI: https://doi.org/10.1090/tran/6332
Keywords: Inverse problems, wave equation, Lipschitz stability
Received by editor(s): October 3, 2012
Received by editor(s) in revised form: November 2, 2013
Published electronically: April 20, 2015
Article copyright: © Copyright 2015 American Mathematical Society