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高效的数值模拟对于当代工程、科学和医疗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、等离子体流体动力学和引力波的数学模型建立在弯曲空间上，这些模型具有数值模拟必须遵守的几何特性，以获得准确、稳健和可靠的结果预测。该项目的两个主要推动应用是等离子体物理学和引力波。等离子体是高度电离的气体；它们出现在核聚变装置、太空探索推进系统以及星系形成过程中。引力波是爱因斯坦预言的时空中的涟漪，它们是由黑洞和中子星等大质量天体的碰撞产生的。针对此类问题构建数值方法使科学家能够设计更稳定、更高效的核聚变系统，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，研究人员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基础的优化算法的效率。研究生参与该研究。该项目结合了离散狄拉克力学和几何、变分积分器、对称空间和广义极分解之间的关系、非正则哈密顿系统的嵌入以及非变分方程及其伴随物的理论和计算工具转化为简并拉格朗日系统。这提供了一种系统方法，用于构造和分析简并非正则哈密顿系统的几何结构保持离散化、非变分方程及其伴随物以及对称空间上演化的问题。由此产生的方法对等离子体物理学（由非规范哈密顿系统描述）以及广义相对论（对称空间上的简并高阶规范场理论）都有影响。此外，伴随方程出现在许多重要的应用中，包括最优控制、最优设计、最优估计、不确定性量化和敏感性分析。更深入地理解任意微分方程组及其相关伴随方程的隐藏几何结构，以及尊重该几何结构的变分离散化，将对广泛依赖于求解的分析和数值技术产生深远的影响。伴随方程。研究生参与这项研究。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

期刊论文数量（15）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Geometric Methods for Adjoint Systems

伴随系统的几何方法

DOI：
10.1007/s00332-023-09999-7
发表时间：
2023-12
期刊：
Journal of Nonlinear Science
影响因子：
3
作者：
Tran, Brian Kha;Leok, Melvin
通讯作者：
Leok, Melvin

Practical perspectives on symplectic accelerated optimization

辛加速优化的实践观点

DOI：
10.1080/10556788.2023.2214837
发表时间：
2022-07-23
期刊：
Optimization Methods and Software
影响因子：
2.2
作者：
Valentin Duruisseaux;M. Leok
通讯作者：
M. Leok

A Variational Formulation of Accelerated Optimization on Riemannian Manifolds

黎曼流形加速优化的变分公式

DOI：
10.1137/21m1395648
发表时间：
2021-01-16
期刊：
SIAM J. Math. Data Sci.
影响因子：
0
作者：
Valentin Duruisseaux;M. Leok
通讯作者：
M. Leok

Simplifying Momentum-based Positive-definite Submanifold Optimization with Applications to Deep Learning

通过在深度学习中的应用简化基于动量的正定子流形优化

DOI：
发表时间：
2023-06
期刊：
Proceedings of Machine Learning Research
影响因子：
0
作者：
Lin, Wu;Duruisseaux, Valentin;Leok, Melvin;Nielsen, Frank;Khan, Mohammad Emtiyaz;Schmidt, Mark.
通讯作者：
Schmidt, Mark.

High-order symplectic Lie group methods on $ SO(n) $ using the polar decomposition

使用极分解的 $SO(n)$ 上的高阶辛李群方法