Stability of inverse transport equation in diffusion scaling and fokker-planck limit

Ke Chen, Qin Li, Li Wang

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

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.

Original languageEnglish (US)
Pages (from-to)2626-2647
Number of pages22
JournalSIAM Journal on Applied Mathematics
Volume78
Issue number5
DOIs
StatePublished - Jan 1 2018

Fingerprint

Fokker-Planck
Transport Equation
Radiative transfer
Scaling
Scattering
Inverse problems
Radiative Transfer Equation
Inverse Problem
Absorption Coefficient
Diffusion equation
Recovery
Quantify
Vary
Coefficient
Term
Operator

Keywords

  • Diffusion limit
  • Fokker-Planck limit
  • Inverse problem
  • Stability
  • Transport equation

Cite this

Stability of inverse transport equation in diffusion scaling and fokker-planck limit. / Chen, Ke; Li, Qin; Wang, Li.

In: SIAM Journal on Applied Mathematics, Vol. 78, No. 5, 01.01.2018, p. 2626-2647.

Research output: Contribution to journalArticle

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N2 - 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.

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