Geometric Numerical Integration of Plasma Physics and General Relativity

等离子体物理与广义相对论的几何数值积分

基本信息

批准号：
1813635
负责人：
Melvin Leok
金额：
$ 23.76万
依托单位：
University of California-San Diego
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-09-01 至 2023-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1813635&HistoricalAwards=false
关键词：
Geometric Numerical Integration Plasma Physics

项目摘要

The accurate and efficient numerical simulation of complex mathematical models is critical to the design and analysis of contemporary engineering, scientific, and medical systems. Mathematical models of drones, computer vision and graphics, medical imaging, fluid dynamics of plasmas, and gravitational waves are posed on curved spaces, which possess geometric properties that have to be respected by the numerical simulations in order to obtain accurate, robust, and reliable predictions. The two main motivating applications for this project are to plasma physics and gravitational waves. Plasmas are highly ionized gases; theyarise in nuclear fusion devices, propulsion systems for space exploration, and during the formation of galaxies. Gravitational waves are ripples in spacetime that were predicted by Einstein, and they arise from the collision of massive astrophysical bodies like black holes and neutron stars. The construction of numerical methods for such problems enables scientists to design more stable and efficient nuclear fusion systems, and to more accurately determine the astrophysical events that correspond to gravitational waves that are detected. In addition, the investigator develops optimization and sensitivity analysis techniques that improve the efficiency of optimization algorithms that underlie deep learning and other machine learning techniques in data science. Graduate students participate in the research.The project combines theoretical and computational tools arising from discrete Dirac mechanics and geometry, variational integrators, the relationship between symmetric spaces and the generalized polar decomposition, and embeddings of noncanonical Hamiltonian systems as well as nonvariational equations and their adjoints into degenerate Lagrangian systems. This provides a systematic method for constructing and analyzing geometric structure-preserving discretizations of degenerate noncanonical Hamiltonian systems, nonvariational equations and their adjoints, and problems that evolve on symmetric spaces. The resulting methods have implications for plasma physics, which is described by noncanonical Hamiltonian systems, as well as general relativity, which is a degenerate higher-order gauge field theory on a symmetric space. In addition, adjoint equations arise in many important applications, including optimal control, optimal design, optimal estimation, uncertainty quantification, and sensitivity analysis. A deeper understanding of the hidden geometric structure underlying an arbitrary system of differential equations and their associated adjoint equations, and variational discretizations that respect that geometric structure, would have profound implications on the broad range of analytical and numerical techniques that rely critically on the solution of adjoint equations. Graduate students participate in the research.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.

复杂数学模型的准确有效的数值模拟对于当代工程，科学和医学系统的设计和分析至关重要。无人机，计算机视觉和图形，医学成像，等离子体的流体动力学和重力波的数学模型在弯曲空间上构成，这些空间具有数值模拟必须尊重的几何特性，以获得准确，稳健和可靠的预测。该项目的两个主要激励应用是血浆物理和引力波。等离子体是高度离子化的气体；它们在核融合设备，用于太空勘探的推进系统以及星系形成期间。引力波是由爱因斯坦预测的时空中的涟漪，它们是由于黑洞和中子星（如黑洞和中子星）的巨大天体物理体的碰撞而产生的。用于此类问题的数值方法的构建使科学家能够设计更稳定，有效的核融合系统，并更准确地确定与检测到的引力波相对应的天体物理事件。此外，研究者还开发了优化和敏感性分析技术，以提高数据科学中深度学习和其他机器学习技术的优化算法的效率。研究生参与了研究。该项目结合了由离散的狄拉克力学和几何形状，变异集成商，对称空间与广义极性分解之间的关系以及非规范性汉密尔顿系统的嵌入以及非变化方程以及非变性的嵌套以及它们的邻接中的嵌入，并将其嵌入到退化的lagrangian lagrangian系统中。这提供了一种系统的方法，用于构建和分析堕落的非适用哈密顿系统，非不同方程式及其伴随的几何结构的离散，以及在对称空间上进化的问题。所得的方法对血浆物理学具有影响，该血浆物理学由非规范性的哈密顿系统描述，以及一般相对论，这是对称空间上的退化高阶仪表理论。此外，伴随方程在许多重要的应用中都会出现，包括最佳控制，最佳设计，最佳估计，不确定性定量和灵敏度分析。对差异方程式的任意系统及其相关的伴随方程的隐藏几何结构的更深入的了解，以及尊重几何结构的变异离散化，将对广泛的分析和数值技术产生深远的影响，这些分析和数值技术依赖于批判性地依赖于相邻方程的解决方案。研究生参加了研究。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响的评论标准来评估值得支持的。