Generalized stability estimates in inverse transport theory
Generalized stability estimates in inverse transport theory
复制标题
逆输运理论中的广义稳定性估计
DOI：
10.3934/ipi.2018003
复制
发表时间：
2017
影响因子：
1.3
通讯作者：
A. Jollivet
中科院分区：
文献类型：
--
作者：
G. Bal;A. Jollivet
逆输运理论关注的是从对所有可能的边界测量进行建模的Escherdo算子的知识中重建输运方程中的吸收和散射系数。唯一性和稳定性结果是众所周知的，并且通常是针对在\开始{document} $L^1$ \end{document}意义上测量的双线性算子的误差而获得的。我们声称，这种误差估计并不总是非常翔实。例如，检测器的任意小的模糊和未对准导致Rendo运算符的\开始{document} $O（1）$ \end{document}误差，并且因此导致关于系数的重构的\开始{document} $O（1）$ \end{document}误差预测，这是无用的。本文重新审视这种稳定性的估计，通过引入一个更宽容的度量的测量误差，即\开始{document} $1-$ \end{document} Wasserstein距离，惩罚模糊或错位的模糊内核的宽度或错位量成比例的量。在这种情况下，我们得到新的稳定性估计。我们还考虑了误差的影响，仍然在\开始{document} $1-$ \end{document} Wasserstein距离中测量，对探测源的生成。这在（激光）探头的设计中对模糊和未对准进行建模，并允许我们考虑离散源。在适当的假设下的系数，我们量化的影响，这种错误的重建。
Inverse transport theory concerns the reconstruction of the absorption and scattering coefficients in a transport equation from knowledge of the albedo operator, which models all possible boundary measurements. Uniqueness and stability results are well known and are typically obtained for errors of the albedo operator measured in the \begin{document} $L^1$ \end{document} sense. We claim that such error estimates are not always very informative. For instance, arbitrarily small blurring and misalignment of detectors result in \begin{document} $O(1)$ \end{document} errors of the albedo operator and hence in \begin{document} $O(1)$ \end{document} error predictions on the reconstruction of the coefficients, which are not useful. This paper revisit such stability estimates by introducing a more forgiving metric on the measurements errors, namely the \begin{document} $1-$ \end{document} Wasserstein distances, which penalize blurring or misalignment by an amount proportional to the width of the blurring kernel or to the amount of misalignment. We obtain new stability estimates in this setting. We also consider the effect of errors, still measured in the \begin{document} $1-$ \end{document} Wasserstein distance, on the generation of the probing source. This models blurring and misalignment in the design of (laser) probes and allows us to consider discretized sources. Under appropriate assumptions on the coefficients, we quantify the effect of such errors on the reconstructions.