Multiscale computational methods in kinetic theory and optimal transport

动力学理论和最优输运中的多尺度计算方法

• 批准号: 1620135
• 负责人: Li Wang
• 金额: $ 20万
• 依托单位: SUNY at Buffalo
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2018-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1620135&HistoricalAwards=false
• 关键词: Multiscale; computational; methods; kinetic; theory

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.

多尺度动力学方程广泛应用于稀薄气体动力学、等离子体物理、半导体和生物等领域；由于小尺度的刚性，它们常常带来严重的数值挑战。最优交通在图像配准、视频恢复、城市交通、运动理论等诸多领域都具有重要的应用价值。然而，针对这一问题的数值方法还不能完全满足实际应用的要求。该项目的目的是在包括多阶段和分数渐近极限在内的动力学方程的新前景中推进多尺度计算方法，特别是渐近保持(AP)格式，并通过先进的优化技术开发快速可并行化的最优运输算法。具体地说，本项目将研究以下内容：(1)从理论上研究具有双尺度碰撞的半导体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)将具有分数扩散限制的动力学方程的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(将研究非经典光子在云中的传输)；(3)开发求解最优传输问题的高效算法，并进行收敛分析，并将其应用于实际问题，特别是在恐慌情况下的人群动力学问题。随着人们对多尺度动力学方程和最优输运的兴趣与日俱增，这里开发的计算方法的影响将超出本提案中的特定应用。半导体器件中电子传输的动力学是物理和工程中的主要问题之一；从这一提议开发的方法将同样适用于更广泛的背景，如气体放电和多基团辐射传输。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极大的关注；它现在已经被应用于气候科学来模拟云中的光子输运，以及在犯罪学中用来模拟住宅入室盗窃的热点。最优交通已经成为图像处理、城市交通、计算机视觉等领域的有用工具；快速可并行化算法的发展将极大地推动这些领域的发展，而在人群建模中的应用对于更好地准备安全的群体性事件至关重要。