New Numerical Methods for Quantum Field Theory and Critical Phenomena

该项目的目标是开发新的和更多的 有效数值算法及其在量子领域的应用 理论，临界现象的统计力学，以及 应用数学中的相关问题。 重点研究 三个主要应用领域：蒙特卡罗模拟 格点Spain模型和场论;大线性 系统的方程与无序系数;和蒙特 自回避随机游动的卡洛模拟。 校长 算法策略是采用“集体模式”更新，例如 作为多重网格、随机变量、嵌入和推广 它们的 其中一些算法的动机是技术 用于椭圆型偏微分方程的数值解 方程，但需要新的适应。 的方法 包括启发式的物理推理，严格的 数学分析和系统的数值测试。 一 主要项目是继续发展多- 网格蒙特卡罗方法，重点是格点规范理论。 第二个项目是开发多网格算法， 具有无序系数的线性系统;例如 背景中格点费米子和Faddeev-Popov传播子 规范场 此外，该集团计划继续 基于辅助的集体模式算法研究 变量（Swendsen-Wang及其推广），集体- 基于嵌入的模式算法（Wolff及其 一般化），以及用于自避免行走的算法。 有效的数值算法是必要的， 新的计算技术。 这项研究工作 将对计算数学和物理学产生影响。 //

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. //