Algebraic multigrid methods for solving the Dirac equation in Lattice Quantum Chromodynamics

本研究项目的目标是结合有限元和多网格方法的思想，以开发一种自适应代数多网格算法框架，该框架有可能对计算量子色动力学（QCD）产生可观而广泛的影响。本文的研究目标主要有三个：(1)提高目前用于求解Wilson-Dirac系统的Galerkin自适应代数多重网格方法的鲁棒性和效率；(2)设计并分析了求解畴壁费米子系统的新的Petrov-Galerkin多重网格方法；(3)发现并分析点阵场理论与研究人员为有限元法所发展的丰富理论之间的关系。本研究涵盖了代数多重网格方法的广泛基础研究，包括基于贪婪（随机）子空间校正方法的新型多重网格平滑器的设计和分析，范围逼近的随机方法，解决非厄米问题的自适应多重网格方法，以及计算特征对和奇异值三元组的多层次方法。这个项目的智力价值来自于它的潜力，可以使几个不同的数学进步，并将这些进步集成到多层次算法和大规模QCD应用程序的软件中。这些进步有望显著减少晶格计算中出现的误差，反过来，使使用模拟来测试QCD的全部非线性以及用从头开始预测来面对实验数据成为可能。该项目的潜力将产生更广泛的影响，通过将提出的算法解决方案应用于粒子物理学基础研究以外的领域的广泛问题，如石墨烯的晶格场理论，涉及麦克斯韦方程的模型，例如磁流体动力学，大规模图形应用，例如各种随机模型中出现的马尔可夫链，以及随机系数的偏微分方程。例如，在地下水流量的不确定性量化中就出现了这种情况。参与该项目的研究生将参与由PI领导的跨学科研究，并有机会与来自美国和欧洲的多位合作者进行访问和合作，从而使他们在先进数学算法和高端科学计算的理论和实践方面接受高级培训。

The goal of this research project is to combine ideas from the finite element and multigrid methodologies in order to develop an adaptive algebraic multigrid algorithmic framework with the potential to make an appreciable and broad impact on computational Quantum Chromodynamics (QCD). The proposed research is driven by three specific interrelated research goals: (1) To render the Galerkin adaptive algebraic multigrid methods currently being used to solve the Wilson-Dirac system more robust and more efficient; (2) To design and analyze new Petrov-Galerkin multigrid methods for solving the domain wall fermion system; (3) To discover and analyze relationships between lattice field theory and the rich theory that researchers have developed for the finite element method. This research encompasses a broad range of fundamental research in algebraic multigrid methods, including the design and analysis of new multigrid smoothers based on greedy (randomized) subspace correction methods, randomized methods for range approximation, adaptive multigrid methods for solving non-hermitian problems, and multilevel methods for computing eigenpairs and singular value triplets.The intellectual merit of this project derives from its potential to make several distinct mathematical advances and to integrate those advances into multilevel algorithms and software for large-scale QCD applications. These advances are expected to significantly reduce the errors that arise in lattice calculations and, in turn, to make it possible to use simulations to test the full non-linearities of QCD and confront experimental data with ab initio predictions. The project's potential to make a broader impact will be realized by applying the proposed algorithmic solutions to a wide range of problems in areas beyond the primary focus on fundamental investigations into particle physics, such as lattice field theories of graphene, models involving Maxwell's equations, e.g., magnetohydrodynamics, large-scale graph applications, e.g., Markov chains as arise in various Stochastic models, and partial differential equations with random coefficients, as arise, for example, in uncertainty quantification for groundwater flow. Graduate students involved in the project will engage in interdisciplinary research led by the PI and have opportunities to visit and work with multiple collaborators from the US and Europe, such that they will receive advanced training in both the theory and practice of advanced mathematical algorithms and high-end scientific computing.