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

Stefanov, Plamen; Uhlmann, Gunther

doi:10.1090/S0002-9947-2013-05703-0

Recovery of a source term or a speed with one measurement and applications
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Trans. Amer. Math. Soc. 365 (2013), 5737-5758 Request permission

Abstract:

We study the problem of recovery of the source $a(t,x)F(x)$ in the wave equation in anisotropic medium with $a$ known so that $a(0,x)\not =0$, 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 $u_{tt} -c^2(x)\Delta u =0$ with one measurement. We give sharp conditions for stability as well. An application to thermoacoustic tomography is also presented.

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