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的进展。这种系统由大量随机运动的粒子组成，最好用玻尔兹曼方程和相关的动力学方程来描述。在实际应用中，动力学系统中可能出现许多不确定性：初始和边界条件的不精确测量，粒子之间基本相互作用机制的不完整知识，等等。理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能够获得更可靠的预测并进行更好的风险评估。由于动力学方程中出现的独特挑战，如多尺度，高维和正性，很少有现有的通用不确定性量化（UQ）算法可以直接应用。为了弥补这一差距，本项目的研究目标是开发高效的随机和多尺度数值方法的玻尔兹曼动力学方程。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、工程和数学教育，并促进对这些学科的职业兴趣，特别是在女生中。具体来说，我们将追求四个研究和教育目标：1）为多尺度动力学方程开发随机渐近保持方法; 2）为玻尔兹曼碰撞算子构建高性能随机算法; 3）为动力学系统设计物理保持UQ算法; 4）通过本科STEM课堂创新为学生创建教育和推广活动;动力学理论研究生课程开发;研究生和本科生指导;为高中女生组织课后数学研究方案，并在中小学组织家庭数学/科学之夜。

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.