We consider the radiative transfer equation (RTE) with two scalings in this paper: One is the diffusive scaling whose macroscopic limit is a diffusion equation, and the other is a highly forward peaked scaling, wherein the scattering term is approximated by a Fokker-Planck operator as a limit. In the inverse setting, we are concerned with reconstructing the scattering and absorption coefficients using boundary measurements. As the measurement is often polluted by errors, both experimental and computational, an important question is to quantify how the error is amplified or suppressed in the process of reconstruction. Since the solution to the forward RTE behaves differ-ently in different regimes, it is expected that stability of the inverse problem will vary accordingly. Particularly, we adopted the linearized approach and showed, in the former case, that the stability degrades when the limit is taken, following a similar approach as in [K. Chen, Q. Li and L. Wang, Inverse Problems, 34 (2018), 025004]. In the latter case, we showed that a full recovery of the scattering coefficient is less possible in the limit.
Bibliographical noteFunding Information:
∗Received by the editors November 21, 2017; accepted for publication (in revised form) August 9, 2018; published electronically October 9, 2018. http://www.siam.org/journals/siap/78-5/M115796.html Funding: The first and the second authors are supported in part by a start-up fund from UW-Madison, NSF-DMS 1619778 and 1750488, TRIPODS 1740707, and UW-data science initiatives. The third author is supported in part by a start-up fund from SUNY-Buffalo, University of Minnesota, and NSF-DMS 1620135. †Mathematics Department, University of Wisconsin–Madison, Madison, WI 53705 (ke@math. wisc.edu). ‡Mathematics Department and Wisconsin Institute of Discovery, University of Wisconsin– Madison, Madison, WI 53705 (firstname.lastname@example.org). §School of Mathematics, University of Minnesota, Twin Cities, Minneapolis, MN 55455 (wang email@example.com).
© 2018 Society for Industrial and Applied Mathematics.
- Diffusion limit
- Fokker-Planck limit
- Inverse problem
- Transport equation