Multiscale computational methods in kinetic theory and optimal transport

具有多尺度的动力学方程出现在各种应用中，如稀薄气体动力学，等离子体物理学，半导体和生物学;由于小尺度的刚度，它们经常引入严重的数值挑战。最优交通在图像配准、视频恢复、城市交通、动力学理论和许多其他方面起着基础性的作用。然而，它的数值方法还没有达到他们的全部能力，以满足最苛刻的实际应用。该项目旨在推进多尺度计算方法-特别是渐近保持（AP）计划-在动力学方程的新前景，包括多阶段和分数渐近极限，并通过先进的优化技术开发快速并行算法的最佳运输。具体而言，本项目将研究以下几个方面的问题：（1）从理论上研究半导体Boltzmann方程的AP格式，并将其推广到隐式/高阶格式，以捕捉宏观模型的层次结构;（2）将分数阶扩散极限动力学方程的AP格式推广到包括各向异性散射、简并碰撞、和Levy-Fokker-Planck相互作用（将讨论云中非经典光子输运的应用）;（3）开发最优输运问题的有效算法并进行收敛性分析，并将其应用于实际问题，特别是恐慌情况下的人群动力学。随着人们对多尺度动力学方程和最优输运的兴趣越来越大，这里开发的计算方法将超越本提案中的特定应用。半导体器件中的电子输运动力学是物理学和工程学中的主要关注点之一;从该提案中开发的方法将同样适用于更广泛的背景下，如气体放电和多群辐射转移。导致分数扩散的非经典输运在等离子体物理学和经济学中引起了广泛的关注;它现在已被应用于气候科学中，以模拟云中的光子输运，以及在犯罪学中模拟住宅小区的热点。最优交通已成为图像处理、城市交通、计算机视觉等领域的一个有用工具;快速并行算法的发展将大大推进这些领域，而在人群建模中的应用对于更好地准备安全的群体事件至关重要。

Kinetic equations with multiple scales arise in diverse applications such as rarefied gas dynamics, plasma physics, semiconductors, and biology; they often introduce severe numerical challenges due to the stiffness that comes from the small scales. Optimal transport plays a fundamental role in image registration, video restoration, urban transport, kinetic theory and many others. However, numerical methods for it have not reached their full capacity to meet the most demanding practical applications. This project aims at both advancing the multiscale computational methods - particularly the asymptotic preserving (AP) schemes - in new prospects for kinetic equations including multi-stage and fractional asymptotic limit, and developing fast parallelizable algorithms for optimal transport via advanced optimization technique. Specifically, the following topics will be investigated in this project: (1) theoretically study the AP schemes for semiconductor Boltzmann equation with two-scale collisions at a deeper depth and generalize them to implicit/high order schemes and to capture the hierarchy of macroscopic models; (2) extend the AP scheme for kinetic equation with fractional diffusion limit to a broader scope including anisotropic scattering, degenerate collision, and Levy-Fokker-Planck interaction (applications to nonclassical photon transport in clouds will be addressed); (3) develop efficient algorithms for optimal transport problems and conduct convergence analysis and apply it to practical problems especially for human crowd dynamics in panic situations. With increasing interest in multiscale kinetic equations and optimal transport, the computational methods developed here will impact beyond the particular applications in this proposal. The dynamics of electron transport in semiconductor devices are one of the main concerns in physics and engineering; the developed methods from this proposal will be equally applicable in a broader context such as gas discharges and multi-group radiative transfer. Nonclassical transport that leads to a fractional diffusion has attracted much attention in plasma physics and economy; it has now been applied in climate science to model the photon transport in clouds as well as in criminology to model the hotspots in residential burglaries. Optimal transport has become a useful tool in image processing, urban transport, computer vision and etc; the development of fast parallelizable algorithms will substantially advance these areas and the application in modeling human crowds is crucial for better preparation of safe mass events.