本文从$n\geq3$维的全隐式算子出发，研究了稳态线性输运方程中散射系数和吸收系数重构的稳定性。在紧致凸区域的边界上，将输入边界条件映射到输出输运解，定义了一种新的输运算子。文[M. Choulli-P. Stefanov，1996和1999]，并在[J. N. Wang，1999]，用于空间独立的散射系数。我们推广了这些结果，并证明了空间相关的散射系数的$L^1$稳定性估计。
英文摘要：
We study the stability of the reconstruction of the scattering and absorption coefficients in a stationary linear transport equation from knowledge of the full albedo operator in dimension $n\geq3$. The albedo operator is defined as the mapping from the incoming boundary conditions to the outgoing transport solution at the boundary of a compact and convex domain. The uniqueness of the reconstruction was proved in [M. Choulli-P. Stefanov, 1996 and 1999] and partial stability estimates were obtained in [J.-N. Wang, 1999] for spatially independent scattering coefficients. We generalize these results and prove an $L^1$-stability estimate for spatially dependent scattering coefficients.
We prove that every stationary state in the annihilator of all Kraus operators of a weak coupling limit type Markov generator consists of two pieces, one of them supported on the interaction-free subspace and the second one on its orthogonal complement. In particular, we apply the previous result to describe in detail the structure of a slightly modified quantum transport model due to Arefeva, Kozyrev and Volovich (modified AKV's model) studied first in Ref.\cite{gggq}, in terms of generalized annihilation and creation operators.
In this work, we study the instability of an inverse problem of radiative transport equation with angularly averaged measurement near the diffusion limit, i.e. the normalized mean free path (the Knudsen number) $0 < \eps \ll 1$. It is well-known that there is a transition of stability from Hölder type to logarithmic type with $\eps\to 0$, the theory of this transition of stability is still an open problem. In this study, we show the transition of stability by establishing the balance of two different regimes depending on the relative sizes of $\eps$ and the perturbation in measurements. When $…
We consider an inverse transport problem in fluorescence ultrasound modulated optical tomography (fUMOT) with angularly averaged illuminations and measurements. We study the uniqueness and stability of the reconstruction of the absorption coefficient and the quantum efficiency of the fluorescent probes. Reconstruction algorithms are proposed and numerical validations are performed as well. This paper is an extension of arXiv:1804.01135, where a diffusion model for this problem was considered.
Motivated by applications in quantitative photoacoustic imaging, we study inverse problems to a semilinear radiative transport equation (RTE) where we intend to reconstruct absorption coefficients in the equation from single and multiple internal data sets. We derive uniqueness and stability results for the inverse transport problem in the absence of scattering (in which case we also derive some explicit reconstruction methods) and in the presence of known scattering.
