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.
Bibliographical noteFunding Information:
The work of the first author was partially supported by a start-up grant from the University of Minnesota and the NSF grant DMS-1714490. The work of the second author was partially supported by NSF grant DMS1619778 and TRIPODS 1740707. The work of the third author was partially supported by NSF grant DMS-1265958 and a Si-Yuan Professorship at HKUST.
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- Knudsen number
- Radiative transfer equation
- Stability estimate