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Convergence Analysis for Anderson Acceleration

2015; Society for Industrial and Applied Mathematics; Volume: 53; Issue: 2 Linguagem: Inglês

10.1137/130919398

1095-7170

Alex Toth, C. T. Kelley,

Advanced Numerical Methods in Computational Mathematics

Anderson($m$) is a method for acceleration of fixed point iteration which stores m+1 prior evaluations of the fixed point map and computes the new iteration as a linear combination of those evaluations. Anderson(0) is fixed point iteration. In this paper we show that Anderson($m$) is locally r-linearly convergent if the fixed point map is a contraction and the coefficients in the linear combination remain bounded. Without assumptions on the coefficients, we prove q-linear convergence of Anderson(1) and, in the case of linear problems, Anderson($m$). We observe that the optimization problem for the coefficients can be formulated and solved in nonstandard ways and report on numerical experiments which illustrate the ideas.