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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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A Lipschitz stable reconstruction formula for the inverse problem for the wave equation
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by Shitao Liu and Lauri Oksanen PDF
Trans. Amer. Math. Soc. 368 (2016), 319-335 Request permission

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 \begin{align*} |\mathcal {F}\left (\widetilde c^{-2} - c^{-2}\right )(\xi )| \le C e^{2R|\xi |}\left \|\widetilde \Lambda - \Lambda \right \|_*, \quad \xi \in \mathbb {R}^n, \end{align*} for all $\widetilde c \in U \cap V$, where $\widetilde \Lambda$ is the Dirichlet-to-Neumann map corresponding to the wave speed $\widetilde c$ and $\left |\cdot \right |_*$ 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
  • MR Author ID: 906909
  • ORCID: 0000-0002-3228-7507
  • Email: lauri.oksanen@helsinki.fi, l.oksanen@ucl.ac.uk
  • Received by editor(s): October 3, 2012
  • Received by editor(s) in revised form: November 2, 2013
  • Published electronically: April 20, 2015
  • © Copyright 2015 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 368 (2016), 319-335
  • MSC (2010): Primary 35R30
  • DOI: https://doi.org/10.1090/tran/6332
  • MathSciNet review: 3413865