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Inverse problems in transport theory. (English) Zbl 1083.45008

Uhlmann, Gunther (ed.), Inside out: Inverse problems and applications. Cambridge: Cambridge University Press (ISBN 0-521-82469-9/hbk). Math. Sci. Res. Inst. Publ. 47, 111-131 (2003).
Summary: We study an inverse problem for the transport equation in a bounded domain in \(\mathbb{R}^n\). Given the incoming flux on the boundary, we measure the outgoing one. The inverse problem is to recover the absorption coefficient \(\sigma_a(x)\) and the collision kernel \(k(x,v',t)\) from this data. This paper is a survey of recent results about general \(k\)’s without assuming that \(k\) depends on a reduced number of variables. We present uniqueness results in dimensions \(n\geq 3\) for the time dependent and the stationary problem, and in the time dependent case we study the inverse scattering problem as well. The proofs are constructive and lead to direct procedures for recovering \(\sigma_a\) and \(k\). For \(n=2\) the problem of recovering \(k\) is formally determined and we prove uniqueness for small \(k\) and a stability estimate.
For the entire collection see [Zbl 1034.78003].

MSC:

45Q05 Inverse problems for integral equations
45M10 Stability theory for integral equations
45K05 Integro-partial differential equations
82C70 Transport processes in time-dependent statistical mechanics
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