Inverse problems for the stationary transport equation in the diffusion scaling

Ru Yu Lai, Qin Li, Gunther Uhlmann

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Abstract

We consider the inverse problem of reconstructing the optical parameters of the radiative transfer equation (RTE) from boundary measurements in the diffusion limit. In the diffusive regime (the Knudsen number Kn ≦ 1), the forward problem for the stationary RTE is well approximated by an elliptic equation. However, the connection between the inverse problem for the RTE and the inverse problem for the elliptic equation has not been fully developed. This problem is particularly interesting because the former one is mildly ill-posed, with a Lipschitz type stability estimate, while the latter is well known to be severely ill-posed with a logarithmic type stability estimate. In this paper, we derive stability estimates for the inverse problem for RTE and examine its dependence on Kn. We show that the stability is Lipschitz in all regimes, but the coefficient deteriorates as e 1 Kn, making the inverse problem of RTE severely ill-posed when Kn is small. In this way we connect the two inverse problems. Numerical results agree with the analysis of worsening stability as the Knudsen number gets smaller.

Original languageEnglish (US)
Pages (from-to)2340-2358
Number of pages19
JournalSIAM Journal on Applied Mathematics
Volume79
Issue number6
DOIs
StatePublished - Jan 1 2019

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Keywords

  • Knudsen number
  • Radiative transfer equation
  • Stability estimate

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