Stability of stationary inverse transport equation in diffusion scaling. (12th January 2018)

• Record Type: Journal Article
• Authors: Chen, Ke; Li, Qin; Wang, Li
• Is Part Of: Inverse problems. Volume 34:Number 2(2018:Feb.)
• Publication Date: 2018-01-12
• Subjects: radiative transfer -- diffusion approximation -- stability -- asymptotic limit; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/1361-6420/aa990c
• Languages: English
• ISSNs: 0266-5611

Abstract

Abstract: We consider the inverse problem of reconstructing the optical parameters for the stationary radiative transfer equation (RTE) from velocity-averaged measurement. The RTE often contains multiple scales, characterized by the magnitude of a dimensionless parameter—the Knudsen number ( K n ). In the diffusive scaling ( K n ≪ 1 ), the stationary RTE is well approximated by an elliptic equation in the forward setting. However, the inverse problem for the elliptic equation is acknowledged to be severely ill-posed, as compared to the well-posedness of the inverse transport equation, which raises the question of how uniqueness is lost as K n → 0 . We tackle this problem by examining the stability of the inverse problem with varying K n . We show that the discrepancy in two measurements is amplified in the reconstructed parameters at the order of K n p ( p = 1 o r 2 ), and as a result leads to ill-posedness in the zero limit of K n . Our results apply to both continuous and discrete settings. Lastly, some numerical tests are performed to validate these theoretical findings.