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:
  • 金额:
    $ 23.76万
  • 依托单位:
  • 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

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.
复杂数学模型的精确高效的数值模拟对于现代工程、科学和医疗系统的设计和分析至关重要。 无人机、计算机视觉和图形学、医学成像、等离子体流体动力学和引力波的数学模型都是在弯曲的空间上建立的,这些空间具有数值模拟必须尊重的几何特性,以便获得准确、稳健和可靠的预测。 该项目的两个主要激励应用是等离子体物理和引力波。 等离子体是高度电离的气体;它们在核聚变装置、太空探索的推进系统以及星系的形成过程中出现。 引力波是爱因斯坦预测的时空涟漪,它们产生于黑洞和中子星等大质量天体物理体的碰撞。 这些问题的数值方法的构建使科学家能够设计更稳定和有效的核聚变系统,并更准确地确定与探测到的引力波相对应的天体物理事件。 此外,研究人员开发优化和敏感性分析技术,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到退化拉格朗日系统中的理论和计算工具。 这提供了一个系统的方法,用于构造和分析退化非正则哈密顿系统,非变分方程及其伴随的几何结构保持离散化,并在对称空间上发展的问题。 由此产生的方法有影响等离子体物理,这是描述非正则哈密顿系统,以及广义相对论,这是一个退化的高阶规范场理论的对称空间。 此外,伴随方程在许多重要的应用中出现,包括最优控制、最优设计、最优估计、不确定性量化和灵敏度分析。 更深入地了解隐藏的几何结构下的任意系统的微分方程及其相关的伴随方程,变分离散尊重的几何结构,将有深远的影响广泛的分析和数值技术,严重依赖于伴随方程的解决方案。 该奖项反映了NSF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。

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
  • 期刊:
  • 影响因子:
    3
  • 作者:
    Tran, Brian Kha;Leok, Melvin
  • 通讯作者:
    Leok, Melvin
Variational Discretizations of Gauge Field Theories Using Group-Equivariant Interpolation
Practical perspectives on symplectic accelerated optimization
  • DOI:
    10.1080/10556788.2023.2214837
  • 发表时间:
    2023-06-06
  • 期刊:
  • 影响因子:
    2.2
  • 作者:
    Duruisseaux,Valentin;Leok,Melvin
  • 通讯作者:
    Leok,Melvin
A Variational Formulation of Accelerated Optimization on Riemannian Manifolds
  • DOI:
    10.1137/21m1395648
  • 发表时间:
    2021-01
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Valentin Duruisseaux;M. Leok
  • 通讯作者:
    Valentin Duruisseaux;M. Leok
Variational Symplectic Accelerated Optimization on Lie Groups
李群上的变分辛加速优化
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Melvin Leok其他文献

Safe Stabilizing Control for Polygonal Robots in Dynamic Elliptical Environments
动态椭圆环境中多边形机器人的安全稳定控制
  • DOI:
  • 发表时间:
    2023
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Kehan Long;Khoa Tran;Melvin Leok;Nikolay Atanasov
  • 通讯作者:
    Nikolay Atanasov
On Properties of Adjoint Systems for Evolutionary PDEs
演化偏微分方程伴随系统的性质
  • DOI:
  • 发表时间:
    2024
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Brian K. Tran;Benjamin Southworth;Melvin Leok
  • 通讯作者:
    Melvin Leok
A Type II Hamiltonian Variational Principle and Adjoint Systems for Lie Groups

Melvin Leok的其他文献

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{{ truncateString('Melvin Leok', 18)}}的其他基金

Hierarchical Geometric Accelerated Optimization, Collision-based Constraint Satisfaction, and Sensitivity Analysis for VLSI Chip Design
VLSI 芯片设计的分层几何加速优化、基于碰撞的约束满足和灵敏度分析
  • 批准号:
    2307801
  • 财政年份:
    2023
  • 资助金额:
    $ 23.76万
  • 项目类别:
    Standard Grant
Geometric Numerical Discretizations of Gauge Field Theories and Interconnected Systems
规范场论和互连系统的几何数值离散
  • 批准号:
    1411792
  • 财政年份:
    2014
  • 资助金额:
    $ 23.76万
  • 项目类别:
    Standard Grant
Collaborative Research: Ergodic Trajectories in Discrete Mechanics
协作研究:离散力学中的遍历轨迹
  • 批准号:
    1334759
  • 财政年份:
    2013
  • 资助金额:
    $ 23.76万
  • 项目类别:
    Standard Grant
Collaborative Research: Computational Geometric Uncertainty Propagation for Hamiltonian Systems on a Lie Group
合作研究:李群上哈密顿系统的计算几何不确定性传播
  • 批准号:
    1029445
  • 财政年份:
    2010
  • 资助金额:
    $ 23.76万
  • 项目类别:
    Standard Grant
CAREER: Computational Geometric Mechanics: Foundations, Computation, and Applications
职业:计算几何力学:基础、计算和应用
  • 批准号:
    1010687
  • 财政年份:
    2009
  • 资助金额:
    $ 23.76万
  • 项目类别:
    Continuing Grant
LTB: Generalized Variational Integrators for Large-Scale Scientific Computation
LTB:用于大规模科学计算的广义变分积分器
  • 批准号:
    1001521
  • 财政年份:
    2009
  • 资助金额:
    $ 23.76万
  • 项目类别:
    Standard Grant
CAREER: Computational Geometric Mechanics: Foundations, Computation, and Applications
职业:计算几何力学:基础、计算和应用
  • 批准号:
    0747659
  • 财政年份:
    2008
  • 资助金额:
    $ 23.76万
  • 项目类别:
    Continuing Grant
LTB: Generalized Variational Integrators for Large-Scale Scientific Computation
LTB:用于大规模科学计算的广义变分积分器
  • 批准号:
    0714223
  • 财政年份:
    2007
  • 资助金额:
    $ 23.76万
  • 项目类别:
    Standard Grant
Computational Geometric Mechanics and its Applications to Geometric Control Theory
计算几何力学及其在几何控制理论中的应用
  • 批准号:
    0726263
  • 财政年份:
    2007
  • 资助金额:
    $ 23.76万
  • 项目类别:
    Standard Grant
Computational Geometric Mechanics and its Applications to Geometric Control Theory
计算几何力学及其在几何控制理论中的应用
  • 批准号:
    0504747
  • 财政年份:
    2005
  • 资助金额:
    $ 23.76万
  • 项目类别:
    Standard Grant

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