Anderson Acceleration for Fixed-Point Iteration

本研究将以d.g. Anderson于1965年提出的一种方法为基础，重点研究不动点迭代的加速方法。该方法作为加速电子结构计算中自洽场迭代的一种手段，在计算物理、化学和材料界得到了广泛的应用，并取得了相当大的成功。然而，在许多其他重要的应用中，它尚未被尝试或未被充分利用，而在这些应用中，它似乎同样取得了成功。此外，尽管有许多重要的数学问题没有得到解答，但数学家和数值分析师对它的关注相对较少。本研究的目标是分析该方法的收敛性，探索其在广泛的重要应用中的有效性，并最终开发具有改进的全局收敛性和稳定性的扩展。要研究的方法目前是加速材料科学中使用的计算的重要方法，例如，在纳米技术应用中确定“设计”材料的性质。在实践中，这种方法通常非常有效，但偶尔也会失败。这项研究的第一个目标是对解释这种行为的方法进行理论理解，并指出更持久成功的方法。该方法在本质上是通用的，可以潜在地更广泛地应用。本研究的第二个目标是确定该方法在各种重要应用中的有效性，从统计估计到复杂物理现象的计算建模。如果成功的话，这项研究可以为具有挑战性的任务带来更快的计算方法，例如从极大的数据集中提取信息，模拟流体流动与化学反应。

This research will focus on acceleration methods for fixed-point iteration that are based on a method introduced by D. G. Anderson in 1965. This method has been used widely and with considerable success within the computational physics, chemistry, and materials communities as a means of accelerating the self-consistent field iteration used in electronic structure computations. However, it has been untried or underexploited in many other important applications in which it seems likely to be equally successful. Moreover, it has received relatively little attention from mathematicians and numerical analysts, despite there being many significant unanswered mathematical questions. The goals of this research are to analyze the convergence of the method, to explore its effectiveness across a broad range of important applications, and ultimately to develop extensions with improved global convergence and stability properties.The method to be investigated is at present an important method for accelerating computations used in materials science, for example, to determine properties of "designer" materials in nanotechnology applications. In practice, the method is usually very effective but occasionally fails. The first goal of this research is to develop a theoretical understanding of the method that explains this behavior and points the way to methods that are more consistently successful. The method is general in nature and can potentially be much more broadly applied. The second goal of this research is to determine the effectiveness of the method in a variety of important applications, ranging from statistical estimation to computational modeling of complex physical phenomena. If successful, this research can result in much faster computational methods for challenging tasks such as extracting information from extremely large data sets and simulating fluid flow coupled with chemical reactions.