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

点击翻译按钮获取中文摘要

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.