CAREER: Predictive Simulations of Complex Kinetic Systems

该项目旨在建立一个研究和教育的综合计划，重点是复杂动力学系统的预测模拟方面的进展。这样的系统是由大量随机运动的粒子组成的，用玻尔兹曼和相关的动力学方程来描述是最好的。在实际应用中，动力学系统中可能会出现许多不确定因素：初始条件和边界条件的测量不精确，对粒子之间基本相互作用机制的不完全了解，等等。了解这些不确定性的影响对于复杂动力学系统的模拟至关重要，并将使科学家和工程师获得更可靠的预测和更好的风险评估。由于动力学方程的多尺度、高维和正性等独特的挑战，现有的通用不确定性量化(UQ)算法可以直接应用的很少。为了弥补这一差距，本项目的研究目标是发展高效的随机和多尺度数值方法来求解类Boltzmann动力学方程。一个平行的教育目标是为各级学生创造创新机会，以改善科学、技术、工程和数学(STEM)教育，并促进对这些学科的职业兴趣，特别是在女学生中。具体地说，我们将追求四个研究和教育目标：1)开发多尺度动力学方程的随机渐近保持方法；2)为Boltzmann碰撞算子构建高性能的随机算法；3)为动力学系统设计物理保持的UQ算法；4)通过本科生STEM课堂创新为学生创造教育和外展活动；开发运动学理论的研究生课程；研究生和本科生辅导；以及为高中女生和中小学家庭数学/科学之夜组织课外数学研究计划。

This project aims to build an integrated program of research and education focused on advances in predictive simulations of complex kinetic systems. Such systems are comprised of a large number of particles in random motion and are best described by the Boltzmann and related kinetic equations. In practical applications, there are many sources of uncertainties that can arise in kinetic systems: imprecise measurements for initial and boundary conditions, incomplete knowledge of the fundamental interaction mechanism between particles, and so on. Understanding the impact of these uncertainties is critical to the simulations of the complex kinetic systems, and will allow scientists and engineers to obtain more reliable predictions and perform better risk assessment. Due to the unique challenges arising in kinetic equations, such as multiple scales, high dimensionality, and positivity, very few existing generic uncertainty quantification (UQ) algorithms can be applied directly. To bridge this gap, the research objective of this project is to develop highly efficient stochastic and multiscale numerical methods for Boltzmann-like kinetic equations. A parallel educational objective is to create innovative opportunities for students at all levels to improve science, technology, engineering, and mathematics (STEM) education and promote career interest in these disciplines, especially among female students. Specifically, we will pursue four research and educational aims: 1) develop stochastic asymptotic-preserving methods for multiscale kinetic equations; 2) construct high performance stochastic algorithms for the Boltzmann collision operator; 3) design physics-preserving UQ algorithms for kinetic systems; and 4) create education and outreach activities for students through undergraduate STEM classroom innovation; graduate curriculum development in kinetic theory; graduate and undergraduate mentoring; and organizing an after-school math research program for high school girls and family math/science nights at middle and elementary schools.