New Numerical Methods for Quantum Field Theory and Critical Phenomena

量子场论和临界现象的新数值方法

• 批准号: 9200719
• 负责人: Jonathan Goodman
• 金额: $ 32.45万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-07-01 至 1995-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9200719&HistoricalAwards=false
• 关键词: New; Numerical; Methods; Quantum; Field

The objective of this project is to develop new and more efficient numerical algorithms with applications to quantum field theory, the statistical mechanics of critical phenomena, and related problems in applied mathematics. The research focuses on three principal application areas: Monte Carlo simulation of lattice spain models and field theories; solution of large linear systems of equations with disordered coefficients; and Monte Carlo simulation of self-avoiding random walk. The principal algorithmic strategy is to employ "collective-mode" updates, such as multi-grid, auxiliary-variable, embedding and generalizations thereof. Some of these algorithms are motivated by techniques used in the numerical solution of elliptic partial differential equations, but novel adaptations are required. The methodology involves a combination of heuristic physical reasoning, rigorous mathematical analysis, and systematic numerical testing. One principal project is to continue the development of the multi- grid Monte Carlo method, with emphasis on lattice gauge theories. A second project is to develop multi-grid algorithms for large linear systems with disordered coefficients; examples are the lattice fermion and Faddeev-Popov propagators in a background gauge field. In addition, the group plans to continue its investigations of collective-mode algorithms based on auxiliary variables (Swendsen-Wang and its generalizations), collective- mode algorithms based on embeddings (Wolff and its generalizations), and algorithms for the self-avoiding walk. Efficient numerical algorithms are necessary to exploit adequately new computational technologies. This research effort will have impact both on computational mathematics and physics. //

该项目的目标是开发新的、更高效的数值算法，应用于量子场论、临界现象的统计力学以及应用数学中的相关问题。 该研究集中在三个主要应用领域：格子西班牙模型和场论的蒙特卡罗模拟；具有无序系数的大型线性方程组的解；以及自回避随机游走的蒙特卡罗模拟。 主要算法策略是采用“集体模式”更新，例如多重网格、辅助变量、嵌入及其泛化。 其中一些算法是受到椭圆偏微分方程数值求解中使用的技术的启发，但需要新颖的适应。 该方法论结合了启发式物理推理、严格的数学分析和系统的数值测试。 其中一个主要项目是继续开发多网格蒙特卡罗方法，重点是格子规范理论。 第二个项目是为具有无序系数的大型线性系统开发多重网格算法；例如背景规范场中的晶格费米子和 Faddeev-Popov 传播子。 此外，该小组计划继续研究基于辅助变量的集体模式算法（Swendsen-Wang 及其泛化）、基于嵌入的集体模式算法（Wolff 及其泛化）以及自回避行走算法。 为了充分利用新的计算技术，高效的数值算法是必要的。 这项研究工作将对计算数学和物理学产生影响。 //