Multiscale computational methods in kinetic theory and optimal transport

多尺度动力学方程出现在不同的应用，如稀薄气体动力学，等离子体物理，半导体和生物学；由于来自小尺度的刚度，它们经常引入严峻的数值挑战。最优交通在图像配准、视频复原、城市交通、动力学理论等领域发挥着重要作用。然而，数值方法还没有达到满足最苛刻的实际应用的全部能力。本项目旨在推进多尺度计算方法，特别是渐近保持（AP）方案，为包括多阶段和分数阶渐近极限在内的动力学方程提供新的前景，并通过先进的优化技术开发快速并行化的最优传输算法。具体而言，本项目将研究以下主题：(1)从理论上深入研究具有两尺度碰撞的半导体玻尔兹曼方程的AP格式，并将其推广到隐式/高阶格式，并捕获宏观模型的层次结构；(2)将具有分数扩散极限的动力学方程的AP格式扩展到更广泛的范围，包括各向异性散射、简并碰撞和Levy-Fokker-Planck相互作用（应用于云中的非经典光子输运）；(3)开发最优交通问题的高效算法，并进行收敛分析，并将其应用于实际问题，特别是恐慌情况下的人群动态问题。随着对多尺度动力学方程和最优输运的兴趣的增加，这里开发的计算方法将影响超出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关注的主要问题之一；从这个建议发展的方法将同样适用于更广泛的背景下，如气体排放和多群辐射转移。引起分数扩散的非经典输运引起了等离子体物理学和经济学的广泛关注。现在，它已被应用于气候科学，以模拟云中的光子传输，以及犯罪学，以模拟住宅盗窃的热点。优化交通已成为图像处理、城市交通、计算机视觉等领域的有用工具；快速并行算法的发展将大大推动这些领域的发展，在人群建模中的应用对于更好地准备安全的大规模活动至关重要。

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.