Approximation of Matrix Functions: Theory, Algorithms, and Software

这个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年里开发的一系列技术提供严格的数学基础，以执行从理论化学和分子物理到数据挖掘和统计等领域的计算机模拟。这些领域的一个共同主题是需要快速计算大型稀疏矩阵(如指数、平方根、对数及其组合)的函数的近似值。科学家使用物理直觉和试探法的组合已经取得了一些成功，但仍然缺乏严格的理由和分析，迫切需要。另一方面，到目前为止，计算数学家对这类问题的关注很少。本项目满足了这一需求。主要的概念工具是近些年发展起来的以衰减界的形式的局部化理论。另一个目标是构造更好的(即，更快、更准确的)算法来计算期望的近似。PI将为这一目标付出相当大的努力，特别是使用各种类型的多项式近似(插值法、切比雪夫法、费伯法等)。计算数学研究的最终目标是为科学家和工程师提供解决日益庞大和复杂的具有挑战性的科学和技术问题所需的算法和软件工具。美国科学技术的竞争力在很大程度上得益于(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。通过实现更快和更详细的计算机模拟，在解决PI所针对的问题方面取得的进展将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)将受到这项研究的影响，无论是通过直接参与，还是通过这项研究对埃默里大学计算数学和应用数学课程的PI教学产生的积极影响。

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.