Numerical Methods and Analysis for Multiscale Kinetic Equations with Uncertainties

动力学模型为物理系统的微观和宏观描述提供了桥梁；它们具有广泛的应用，包括航天，核工程，等离子体物理和半导体器件建模。这些模式往往涉及多个时间和空间尺度，这给数值模拟带来了极大的困难。此外，由于动力学模型是由近似产生的，因此所采用的方程和所使用的数据（初始条件和边界值）存在固有的不确定性。本项目旨在发展有效的数值方法，并对多尺度和不确定动力学方程进行分析。所研究的问题涉及现代科学和工程计算中的基本问题-多尺度建模和模拟以及不确定性量化。部分研究成果有望为应用数学和科学计算的研究生课程提供优秀的补充，从而有助于培养下一代现代应用数学和科学计算的研究人员。本项目旨在发展和分析具有不确定性的多尺度动力学方程的数值方法。这项工作解决了多个时间和空间尺度的数值挑战，以及碰撞核、散射系数、初始和边界数据、强迫和源项等固有模型的不确定性。研究人员计划通过几种计算和分析工具来解决这些数值挑战：处理多尺度的渐近保持方案；随机不确定性的多项式混沌展开和随机伽辽金（及其他非侵入式）方法；并利用低矫顽力理论研究这些方法的规律性、稳定性、灵敏度和长期行为。亚矫顽力分析为研究数学物理中一类重要的非线性偏微分方程提供了新的数值分析工具。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Kinetic models provide the bridges between microscopic and macroscopic descriptions for physical systems; they have a wide range of applications, including astronautics, nuclear engineering, plasma physics, and semiconductor device modeling. These models often involve multiple time and spatial scales, which pose tremendous difficulties in numerical simulations. Moreover, since kinetic models arise from approximations, there are intrinsic uncertainties in the equations employed and the data (initial conditions and boundary values) used. This project aims to develop efficient numerical methods and to conduct analysis for multiscale and uncertain kinetic equations. The questions under study concern fundamental issues in scientific and engineering computation in the modern age -- multiscale modeling and simulation and uncertainty quantification. Some of the research results are expected to provide excellent additions for graduate courses in applied mathematics and scientific computing, thus contributing to training of the future generation of researchers in modern applied mathematics and scientific computing.The project aims to develop and analyze numerical methods for multiscale kinetic equations with uncertainties. The work addresses the numerical challenges of multiple time and spatial scales as well as intrinsic model uncertainties in collision kernels, scattering coefficients, initial and boundary data, forcing and source terms, etc. The investigator plans to tackle these numerical challenges via several computational and analytical tools: asymptotic-preserving schemes to deal with multiple scales; polynomial chaos expansion and stochastic Galerkin (and other non-intrusive) methods for the random uncertainties; and hypocoercivity theory to study the regularity, stability, sensitivity, and long-time behavior of these methods. The hypocoercivity analysis provides new numerical analysis tools to study a wide class of physically important nonlinear partial differential equations in mathematical physics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.