Approximation of Matrix Functions: Theory, Algorithms, and Software

矩阵函数的逼近：理论、算法和软件

• 批准号: 0810862
• 负责人: Michele Benzi
• 金额: $ 22.95万
• 依托单位: Emory University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-08-01 至 2011-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0810862&HistoricalAwards=false
• 关键词: Approximation; Matrix; Functions; Theory; Algorithms

This project has two main goals. One goal is to provide rigorous mathematical foundations for a wide array of techniques developed by physicists, chemists, and other scientists over the last 10-15 years to perform computer simulations in fields ranging from theoretical chemistry and molecular physics to data mining and statistics. A common theme in these areas is the need to quickly compute approximations of functions of large and sparse matrices (such as the exponential, the square root, the logarithm, and combinations thereof). Scientists have had some success using a combination of physical intuition and heuristics, but rigorous justifications and analysis are still lacking and are sorely needed. On the other hand, computational mathematicians have until now devoted scant attention to these types of problems. The present project addresses this need. The main conceptual tool is a theory of localization, in the form of decay bounds, that the PI has been developing in recent years.Another goal is to construct better (i.e., faster and more accurate) algorithms to compute the desired approximations. The PI will devote considerable effort to this objective, in particular using various types of polynomial approximations (interpolation, Chebyshev, Faber, etc). The resulting software will be distributed to interested parties.The ultimate goal of research in computational mathematics is to provide scientists and engineers the algorithmic and software tools needed for the solution of challenging scientific and technical problems of increasing size and complexity. The competitiveness of American science and technology greatly benefits from (and to a large extent depends on) the creation of innovative computational methods and software and from the continuous improvement of existing techniques. Progress in the solution of the problems targeted by the PI will have a positive impact on science and engineering by enabling faster and more detailed computer simulations. In addition, several graduate students (and possibly a few undergraduates) will be impacted by this research either through direct involvement, or through the positive effects this research will have on the PI's teaching of computational and applied mathematics courses at Emory University.

该项目有两个主要目标。 一个目标是为物理学家，化学家和其他科学家在过去10-15年中开发的各种技术提供严格的数学基础，以在从理论化学和分子物理到数据挖掘和统计学的领域进行计算机模拟。 这些领域的一个共同主题是需要快速计算大型稀疏矩阵的函数的近似值（例如指数，平方根，对数及其组合）。 科学家们结合物理直觉和物理学已经取得了一些成功，但仍然缺乏严格的论证和分析，而且非常需要。 另一方面，计算数学家到目前为止很少关注这类问题。 本项目就是为了满足这一需要。 主要的概念工具是PI近年来一直在发展的以衰变边界形式存在的局部化理论。另一个目标是构建更好的（即，更快和更精确）的算法来计算期望的近似。 PI将为此目标投入大量精力，特别是使用各种类型的多项式近似（插值，Chebyshev，Faber等）。 计算数学研究的最终目标是为科学家和工程师提供所需的算法和软件工具，以解决规模和复杂性不断增加的具有挑战性的科学和技术问题。 美国科学技术的竞争力极大地得益于（并在很大程度上取决于）创新计算方法和软件的创造，以及现有技术的不断改进。 PI所针对的问题的解决进展将通过实现更快和更详细的计算机模拟对科学和工程产生积极影响。 此外，一些研究生（可能还有一些本科生）将通过直接参与这项研究或通过这项研究将对埃默里大学计算和应用数学课程的PI教学产生积极影响而受到影响。