Anderson Acceleration for Fixed-Point Iteration

Worcester Polytechnic Institute, Worcester MA

Investigators

Abstract

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.