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

这个研究项目的目标是联合收割机的想法，从有限元和多重网格方法，以开发一个自适应代数多重网格算法框架的潜力，使一个可观的和广泛的影响计算量子色动力学（QCD）。本论文的主要研究目标是：（1）使目前用于求解Wilson-Dirac方程组的Galerkin自适应代数多重网格方法更加稳健和高效：（2）设计和分析求解畴壁费米子方程组的新的Petrov-Galerkin多重网格方法;（3）发现和分析格点场论与研究者为有限元方法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，包括基于贪婪算法的新型多重网格平滑器的设计和分析。（随机化）子空间校正方法，用于范围近似的随机化方法，用于解决非埃尔米特问题的自适应多重网格方法，和计算特征对和奇异值三元组的多级方法。这个项目的智力价值来自于它的潜力，独特的数学进步，并将这些进步纳入多级算法和软件的大规模QCD应用。 这些进展预计将大大减少晶格计算中出现的错误，从而使人们有可能使用模拟来测试QCD的完全非线性，并将实验数据与从头计算预测相结合。 该项目产生更广泛影响的潜力将通过将所提出的算法解决方案应用于粒子物理学基础研究以外的广泛问题来实现，例如石墨烯的晶格场理论，涉及麦克斯韦方程的模型，例如，磁流体力学，大规模图形应用，例如，马尔可夫链出现在各种随机模型中，以及具有随机系数的偏微分方程，例如，在地下水流的不确定性量化中。参与该项目的研究生将参与PI领导的跨学科研究，并有机会访问和与来自美国和欧洲的多个合作者合作，以便他们将接受高级数学算法和高端科学计算的理论和实践方面的高级培训。

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.