Inverse problems for the stationary transport equation in the diffusion scaling. (English) Zbl 1437.35707
The paper is aimed to study 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 \(\mathrm{Ko} \leq 1\)), 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. Stability estimates for the inverse problem for RTE are obtained, and the dependence of the estimates on \(\mathrm{Ko}\). is studied. The estimates show that the stability is Lipschitz in all regimes, but the coefficient deteriorates as \(e^{\mathrm{ko}}\), making the inverse problem of RTE severely ill-posed when \(\mathrm{Ko}\) 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.
Reviewer: Elena V. Tabarintseva (Chelyabinsk)
MSC:
| 35R30 | Inverse problems for PDEs |
| 65M32 | Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs |
| 35Q79 | PDEs in connection with classical thermodynamics and heat transfer |
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