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A theoretical justification that Anderson acceleration improves linear convergence rates.

Banff International Research Station for Mathematical Innovation and Discovery

The extrapolation method known as Anderson acceleration has been used for decades to speed the convergence of nonlinear solvers in many applications. A mathematical justification of the improved convergence rate however has remained elusive. Here, we provide theory to establish the improved convergence rate. The key ideas of the analysis are relating the difference of consecutive iterates to residuals based on performing the inner-optimization in a Hilbert space setting, and explicitly defining the gain in the optimization stage to be the ratio of improvement over a step of the unaccelerated fixed point iteration. The main result we prove is this method of acceleration improves the convergence rate of a fixed point iteration to first order by a factor of the gain at each step as the method converges.

Mathematics; Numerical analysis; Partial differential equations; Women and early careers in math

video/mp4

Author affiliation: University of Florida

2020-09-12

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/75990

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/