Recovery of a source term or a speed with one measurement and applications

We study the problem of recovery of the source a(t,x)F(x)a(t,x)F(x) in the wave equation in anisotropic medium with aa known so that a(0,x)≠0a(0,x)\not =0, with a single measurement. We use Carleman estimates combined with geometric arguments and give sharp conditions for uniqueness. We also study t...

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Bibliographic Details
Published in:Transactions of the American Mathematical Society 2013-11, Vol.365 (11), p.5737-5758
Main Authors: Stefanov, Plamen, Uhlmann, Gunther
Format: Article
Language:English
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Research article
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Summary:We study the problem of recovery of the source a(t,x)F(x)a(t,x)F(x) in the wave equation in anisotropic medium with aa known so that a(0,x)≠0a(0,x)\not =0, with a single measurement. We use Carleman estimates combined with geometric arguments and give sharp conditions for uniqueness. We also study the non-linear problem of recovery of the sound speed in the equation utt−c2(x)Δu=0u_{tt} -c^2(x)\Delta u =0 with one measurement. We give sharp conditions for stability as well. An application to thermoacoustic tomography is also presented.
ISSN:0002-9947
1088-6850
DOI:10.1090/S0002-9947-2013-05703-0