Anderson Acceleration for Fixed-Point Iterations. 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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  12. Anderson, Donald G. M.: Comments on: “Anderson acceleration, mixing and extrapolation” (2019)
  13. Benzi, Michele; Beik, Fatemeh Panjeh Ali: Uzawa-type and augmented Lagrangian methods for double saddle point systems (2019)
  14. Both, Jakub Wiktor; Kumar, Kundan; Nordbotten, Jan Martin; Radu, Florin Adrian: Anderson accelerated fixed-stress splitting schemes for consolidation of unsaturated porous media (2019)
  15. Brezinski, Claude; Redivo-Zaglia, Michela: Extrapolation methods for the numerical solution of nonlinear Fredholm integral equations (2019)
  16. Brezinski, Claude; Redivo-Zaglia, Michela: The genesis and early developments of Aitken’s process, Shanks’ transformation, the (\varepsilon)-algorithm, and related fixed point methods (2019)
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  20. Jiang, Jiamin; Tchelepi, Hamdi A.: Nonlinear acceleration of sequential fully implicit (SFI) method for coupled flow and transport in porous media (2019)

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