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Inverse Problem for the Wave Equation with a White Noise Source

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Abstract

We consider a smooth Riemannian metric tensor g on \({\mathbb{R}^n}\) and study the stochastic wave equation for the Laplace-Beltrami operator \({\partial_t^2 u - \Delta_g u = F}\). Here, F = F(t, x, ω) is a random source that has white noise distribution supported on the boundary of some smooth compact domain \({M \subset \mathbb{R}^n}\). We study the following formally posed inverse problem with only one measurement. Suppose that g is known only outside of a compact subset of M int and that a solution \({u(t, x, \omega_0)}\) is produced by a single realization of the source \({F(t, x, \omega_0)}\). We ask what information regarding g can be recovered by measuring \({u(t, x, \omega_0)}\) on \({\mathbb{R}_+ \times \partial M}\) ? We prove that such measurement together with the realization of the source determine the scattering relation of the Riemannian manifold (M, g) with probability one. That is, for all geodesics passing through M, the travel times together with the entering and exit points and directions are determined. In particular, if (M, g) is a simple Riemannian manifold and g is conformally Euclidian in M, the measurement determines the metric g in M.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68, 00014, Helsinki, Finland

    Tapio Helin & Matti Lassas

  2. University College London, Gower Street, London, WC1E 6BT, UK

    Lauri Oksanen

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  1. Tapio Helin
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  2. Matti Lassas
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Helin, T., Lassas, M. & Oksanen, L. Inverse Problem for the Wave Equation with a White Noise Source. Commun. Math. Phys. 332, 933–953 (2014). https://doi.org/10.1007/s00220-014-2115-9

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