Optimization-Based Moment Models for Multiscale Kinetic Equations

本提案的重点是设计，分析和实现基于优化的力矩模型，用于求解具有扩散极限的碰撞动力学方程。在计算动力学理论领域，矩量法是一种有效但高度复杂的工具，用于离散动力学方程和捕获多尺度现象。他们将多粒子系统的动力学描述简化为动力学分布速度平均值的一组偏微分方程。基于优化的力矩模型是通过求解一个物理驱动的凸优化问题推导出来的，该问题强制执行潜在动力学方程的重要属性。这项工作的驱动应用将是核心坍缩超新星模拟中的中微子传输，其中完全解析动力学模拟的计算要求远远超出了最大的超级计算机的能力。为复杂系统设计可处理的简化模型是及时而重要的。事实上，对于范围广泛的物理、生物和社会科学问题，需要从难以处理的大量微观数据中提取相关宏观特征的方法。在这种情况下，矩模型的设计和实现是理解多粒子系统行为的重要组成部分，对于直接的数值模拟是不可能的。重要的例子包括稀薄气体、辐射和带电粒子输运——所有这些都在涉及国家利益的先进能源和技术应用中发挥着关键作用。因此，除其他事项外，该项目将为学生和博士后提供有益和相关的研究经验。

The focus of this proposal is the design, analysis, and implementation of optimization-based moment models for solving collisional kinetic equations with diffusive limits. In the field of computational kinetic theory, moment methods are an effective, yet highly complex tool for discretizing kinetic equations and capturing multiscale phenomena. They reduce the kinetic description of a many-particle system to a set of partial differential equations for velocity averages of the kinetic distribution. Optimization-based moment models are derived via the solution of a physically motivated, convex optimization problem that enforces important properties of the underlying kinetic equation. The driving application for this effort will be neutrino transport in core-collapse supernovae simulations, where the computational requirements for a fully resolved kinetic simulation are well beyond the capabilities of the largest supercomputers.The design of tractable reduced models for complex systems is both timely and important. Indeed, for a broad range of physical, biological, and social science problems, methodologies are needed to extract relevant macroscopic features from intractable amounts of microscopic data. In this context, the design and implementation of moment models is an important part of understanding the behavior of many-particle systems, for which direct numerical simulation is not possible. Important examples include rarefied gases, radiation, and charged-particle transport---all of which play critical roles in advanced energy and technology applications of national interest. Thus, among other things, the project will provide a rewarding and relevant research experience to students and postdocs.