Geometric Numerical Integration of Plasma Physics and General Relativity

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中的涟漪，它们来自黑洞和中子星等大质量天体的碰撞。 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。

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.