Geometric Numerical Integration of Plasma Physics and General Relativity

复杂数学模型的精确和高效的数值模拟对当代工程、科学和医疗系统的设计和分析至关重要。无人机、计算机视觉和图形学、医学成像、等离子体流体动力学和引力波的数学模型被放置在弯曲空间上，这些空间具有几何特性，为了获得准确、稳健和可靠的预测，数值模拟必须尊重这些几何特性。这个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；它们出现在核聚变装置、太空探索的推进系统以及星系的形成过程中。引力波是爱因斯坦预言的时空涟漪，它们是由黑洞和中子星等大质量天体碰撞产生的。为这些问题建立数值方法，使科学家能够设计出更稳定、更高效的核聚变系统，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，研究者还开发了优化和敏感性分析技术，以提高优化算法的效率，这些算法是数据科学中深度学习和其他机器学习技术的基础。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。这为构造和分析退化非正则哈密顿系统、非变分方程及其伴随、以及在对称空间上演化的问题的几何结构保持离散化提供了一个系统的方法。由此产生的方法对等离子体物理学和广义相对论都有启示，后者是由非正则哈密顿系统描述的，广义相对论是对称空间上的简并高阶规范场理论。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。对任意微分方程及其相关伴随方程系统的隐藏几何结构的更深入理解，以及尊重该几何结构的变分离散化，将对广泛的分析和数值技术产生深远的影响，这些技术严重依赖于伴随方程的解。研究生参与研究。该奖项反映了美国国家科学基金会的法定使命，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。

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.