Computational Riemannian Geometry: High-Order Methods, Analysis, and Structure Preservation

计算黎曼几何:高阶方法、分析和结构保持

基本信息

  • 批准号:
    2012427
  • 负责人:
  • 金额:
    $ 13.53万
  • 依托单位:
  • 依托单位国家:
    美国
  • 项目类别:
    Standard Grant
  • 财政年份:
    2020
  • 资助国家:
    美国
  • 起止时间:
    2020-06-15 至 2024-05-31
  • 项目状态:
    已结题

项目摘要

Riemannian geometry plays a fundamental role in mathematical physics and geometric analysis, and computational methods for Riemannian geometry have surprisingly many practical applications. Equations that govern time-varying Riemannian metrics, for example, are prototypes for curvature-driven flows that arise in science and engineering like surface tension-driven flow. Such equations also underly several algorithms that are used widely in computer graphics, machine vision, and medical imaging. Examples include algorithms for surface parameterization, texture mapping, and surface registration. Computational Riemannian geometry also plays an essential role in gravitational wave astronomy, where accurate numerical simulations of Einstein’s equations are needed to make inferences about gravitational wave signals. Despite its importance, Riemannian geometry is, in certain respects, underserved by traditional tools of numerical analysis, which are tailored toward problems posed in Euclidean space. This project centers on developing novel computational methods for Riemannian geometry. The computational methods will be made freely available in a public repository, and graduate students will participate in their development. The main goal of this project is to design and analyze high-order methods for three families of problems in computational Riemannian geometry: (1) the numerical solution of intrinsic geometric flows with finite element methods, (2) intrinsic curvature approximation with finite elements, and (3) the efficient computation of interpolants, geodesics, Riemannian means, and the Riemannian exponential map on matrix manifolds. These three problems are tightly intertwined. The vast majority of geometric flows in Riemannian geometry are curvature-driven flows, so their discretization with finite elements goes hand in hand with the construction of finite element approximations of the Riemann curvature tensor and its contractions. In turn, tensor field and frame field discretizations play an important role in curvature approximation, underscoring the need for efficient algorithms for computations on matrix manifolds. This project aims to design numerical methods for the aforementioned problems that are high-order, provably convergent, and structure-preserving. This project will support one graduate student each year.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.
黎曼几何在数学物理和几何分析中起着基础性的作用,黎曼几何的计算方法有着惊人的实际应用。 例如,支配时变黎曼度量的方程是曲率驱动流的原型,这些流出现在科学和工程中,如表面张力驱动流。 这些方程也是计算机图形学、机器视觉和医学成像中广泛使用的几种算法的基础。 例子包括用于表面参数化、纹理映射和表面配准的算法。 计算黎曼几何在引力波天文学中也起着至关重要的作用,需要对爱因斯坦方程进行精确的数值模拟来推断引力波信号。 尽管它的重要性,黎曼几何是,在某些方面,不足的传统工具的数值分析,这是针对问题提出的欧几里得空间。 这个项目的中心是发展黎曼几何的新的计算方法。 计算方法将在公共存储库中免费提供,研究生将参与其开发。 这个项目的主要目标是设计和分析计算黎曼几何中三类问题的高阶方法:(1)用有限元方法数值求解内在几何流,(2)用有限元方法逼近内在曲率,(3)插值,测地线,黎曼平均值和矩阵流形上的黎曼指数映射的有效计算。这三个问题紧密地交织在一起。黎曼几何中的绝大多数几何流都是曲率驱动流,因此它们的有限元离散化与黎曼曲率张量及其压缩的有限元近似的构造密切相关。反过来,张量场和框架场离散化在曲率近似中起着重要作用,强调了对矩阵流形计算的有效算法的需求。该项目旨在为上述问题设计高阶,可证明收敛和结构保持的数值方法。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(8)
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 Variational Finite Element Discretizations for Fluids
流体的几何变分有限元离散
  • DOI:
    10.1016/j.ifacol.2021.11.047
  • 发表时间:
    2021
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Gay-Balmaz, François;Gawlik, Evan
  • 通讯作者:
    Gawlik, Evan
Local finite element approximation of Sobolev differential forms
Sobolev 微分形式的局部有限元近似
Approximating the p th root by composite rational functions
用复合有理函数逼近 p 次方根
  • DOI:
    10.1016/j.jat.2021.105577
  • 发表时间:
    2021
  • 期刊:
  • 影响因子:
    0.9
  • 作者:
    Gawlik, Evan S.;Nakatsukasa, Yuji
  • 通讯作者:
    Nakatsukasa, Yuji
Rational Minimax Iterations for Computing the Matrix pth Root
用于计算矩阵 p 根的有理极小极大迭代
  • DOI:
    10.1007/s00365-020-09504-3
  • 发表时间:
    2021
  • 期刊:
  • 影响因子:
    2.7
  • 作者:
    Gawlik, Evan S.
  • 通讯作者:
    Gawlik, Evan S.
Finite Element Approximation of the Levi-Civita Connection and Its Curvature in Two Dimensions
  • DOI:
    10.1007/s10208-022-09597-1
  • 发表时间:
    2021-11
  • 期刊:
  • 影响因子:
    3
  • 作者:
    Yakov Berchenko-Kogan;Evan S. Gawlik
  • 通讯作者:
    Yakov Berchenko-Kogan;Evan S. Gawlik
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Evan Gawlik其他文献

Evan Gawlik的其他文献

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{{ truncateString('Evan Gawlik', 18)}}的其他基金

PostDoctoral Research Fellowship
博士后研究奖学金
  • 批准号:
    1703719
  • 财政年份:
    2017
  • 资助金额:
    $ 13.53万
  • 项目类别:
    Fellowship Award

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Stochastic processes in sub-Riemannian geometry
亚黎曼几何中的随机过程
  • 批准号:
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  • 财政年份:
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Differential Equations in Complex Riemannian Geometry
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Volume-collapsed manifolds in Riemannian geometry and geometric inference
黎曼几何中的体积塌陷流形和几何推理
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Geodesics in noncommutative Riemannian geometry
非交换黎曼几何中的测地线
  • 批准号:
    571975-2022
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    2022
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CAREER: Rethinking Dynamic Wireless Networks through the Lens of Riemannian Geometry
职业:通过黎曼几何的视角重新思考动态无线网络
  • 批准号:
    2144297
  • 财政年份:
    2022
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    Continuing Grant
Minimal submanifolds in Riemannian geometry
黎曼几何中的最小子流形
  • 批准号:
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Minimal Filling Estimates in Riemannian Geometry
黎曼几何中的最小填充估计
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  • 财政年份:
    2022
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    $ 13.53万
  • 项目类别:
    Postgraduate Scholarships - Doctoral
Geodesics in noncommutative Riemannian geometry
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  • 批准号:
    561794-2021
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New methods for variational problems in Riemannian geometry
黎曼几何中变分问题的新方法
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  • 财政年份:
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Minimal submanifolds in Riemannian geometry
黎曼几何中的最小子流形
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    RGPIN-2020-04225
  • 财政年份:
    2021
  • 资助金额:
    $ 13.53万
  • 项目类别:
    Discovery Grants Program - Individual
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