Reconstruction of piecewise smooth wave speeds using multiple scattering
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- by Peter Caday, Maarten V. de Hoop, Vitaly Katsnelson and Gunther Uhlmann PDF
- Trans. Amer. Math. Soc. 372 (2019), 1213-1235 Request permission
Abstract:
Let $c$ be a piecewise smooth wave speed on $\mathbb {R}^n$, unknown inside a domain $\Omega$. We are given the solution operator for the scalar wave equation $(\partial _t^2-c^2\Delta )u=0$, but only outside $\Omega$ and only for initial data supported outside $\Omega$. Using our recently developed scattering control method, we prove that piecewise smooth wave speeds are uniquely determined by this map and provide a reconstruction formula. In other words, the wave imaging problem is solvable in the piecewise smooth setting under mild conditions. We also illustrate a separate method, likewise constructive, for recovering the locations of interfaces in broken geodesic normal coordinates using scattering control.References
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Additional Information
- Peter Caday
- Affiliation: Formerly of the Department of Computational and Applied Mathematics, Rice University, Houston, Texas 77251
- MR Author ID: 1093686
- Email: caday@alum.mit.edu
- Maarten V. de Hoop
- Affiliation: Departments of Computational and Applied Mathematics and Earth, Environmental, and Planetary Sciences, Rice University, Houston, Texas 77251
- MR Author ID: 311568
- Email: mdehoop@rice.edu
- Vitaly Katsnelson
- Affiliation: Department of Computational and Applied Mathematics, Rice University, Houston, Texas 77251
- MR Author ID: 894275
- Email: vk17@rice.edu
- Gunther Uhlmann
- Affiliation: Department of Mathematics, University of Washington, Seattle, Washington – and – Institute for Advanced Study, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
- MR Author ID: 175790
- Email: gunther@math.washington.edu
- Received by editor(s): January 11, 2018
- Received by editor(s) in revised form: May 9, 2018
- Published electronically: March 25, 2019
- Additional Notes: The first and third authors were supported by the Simons Foundation under the MATH $+$ X program.
The second author was partially supported by the Simons Foundation under the MATH $+$ X program, the National Science Foundation under grant DMS-1559587, and by the members of the Geo-Mathematical Group at Rice University.
The fourth author is a Walker Family Endowed Professor of Mathematics at the University of Washington, and was partially supported by NSF, a Si-Yuan Professorship at HKUST, and a FiDiPro Professorship at the Academy of Finland. - © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 372 (2019), 1213-1235
- MSC (2010): Primary 35R30; Secondary 93C20
- DOI: https://doi.org/10.1090/tran/7632
- MathSciNet review: 3968801