Comments on "Anderson Acceleration, Mixing and Extrapolation"

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Comments on "Anderson Acceleration, Mixing and Extrapolation"

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Title: Comments on "Anderson Acceleration, Mixing and Extrapolation"
Author: Anderson, Donald G.M.
Citation: Anderson, Donald G.M. 2017. Comments on "Anderson Acceleration, Mixing and Extrapolation." Working paper.
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Abstract: In 1962, during the course of my doctoral dissertation research, I devised a technique for accelerating the convergence of the Picard iteration associated with a fixed point problem, which I called the Extrapolation Algorithm. More recently, versions of this method have been labelled as Anderson Acceleration in the applied mathematics community and Anderson Mixing in the computational quantum mechanics community. It would never have occurred to me to use the term Anderson Extrapolation, thence the quotation marks in the title.

I continued to work with the Extrapolation Algorithm, off and on, over the better part of two decades, but until recently have not had occasion to do so since. Only minimal records of this earlier work survive, so what follows is based on recollection and reconstruction, plus some new ideas. I am inclined to retire the Extrapolation language in favor of the Acceleration language, provided the purview of the latter is broadened to include the version that I shall outline hereafter. I shall also argue that that purview should be narrowed to focus on fixed point rather than root finding problems. Mixing is a term of art in the computational quantum mechanics literature, with broader connotations, so this terminology seems likely to predominate in that community. A number of methods equivalent or related to versions of the Extrapolation Algorithm are now extant. Some of the existing literature will be reviewed from conceptual and implementation perspectives.
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