New Numerical Methods for Quantum Field Theory and Critical Phenomena

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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. //