Journal
SIAM JOURNAL ON NUMERICAL ANALYSIS
Volume 49, Issue 4, Pages 1715-1735Publisher
SIAM PUBLICATIONS
DOI: 10.1137/10078356X
Keywords
acceleration methods; fixed-point iterations; generalized minimal residual method; Arnoldi (full orthogonalization) method; iterative methods; expectation-maximization algorithm; mixture densities; alternating least-squares; nonnegative matrix factorization; domain decomposition
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Funding
- U.S. National Science Foundation [DMS 0915183]
- U.S. Department of Energy [DE-SC0004880]
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This paper concerns an acceleration method for fixed-point iterations that originated in work of D. G. Anderson [J. Assoc. Comput. Mach., 12 (1965), pp. 547-560], which we accordingly call Anderson acceleration here. This method has enjoyed considerable success and wide usage in electronic structure computations, where it is known as Anderson mixing; however, it seems to have been untried or underexploited in many other important applications. Moreover, while other acceleration methods have been extensively studied by the mathematics and numerical analysis communities, this method has received relatively little attention from these communities over the years. A recent paper by H. Fang and Y. Saad [Numer. Linear Algebra Appl., 16 (2009), pp. 197-221] has clarified a remarkable relationship of Anderson acceleration to quasi-Newton (secant updating) methods and extended it to define a broader Anderson family of acceleration methods. In this paper, our goals are to shed additional light on Anderson acceleration and to draw further attention to its usefulness as a general tool. We first show that, on linear problems, Anderson acceleration without truncation is essentially equivalent in a certain sense to the generalized minimal residual (GMRES) method. We also show that the Type 1 variant in the Fang-Saad Anderson family is similarly essentially equivalent to the Arnoldi (full orthogonalization) method. We then discuss practical considerations for implementing Anderson acceleration and illustrate its performance through numerical experiments involving a variety of applications.
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