English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

This paper provides the first proof that Anderson acceleration (AA) improvesthe convergence rate of general fixed point iterations. AA has been used fordecades to speed up nonlinear solvers in many applications, however a rigorousmathematical justification of the improved convergence rate has remainedlacking. The key ideas of the analysis presented here are relating thedifference of consecutive iterates to residuals based on performing theinner-optimization in a Hilbert space setting, and explicitly defining the gainin the optimization stage to be the ratio of improvement over a step of theunaccelerated fixed point iteration. The main result we prove is that AAimproves the convergence rate of a fixed point iteration to first order by afactor of the gain at each step. In addition to improving the convergence rate,our results indicate that AA increases the radius of convergence. Lastly, ourestimate shows that while the linear convergence rate is improved, additionalquadratic terms arise in the estimate, which shows why AA does not typicallyimprove convergence in quadratically converging fixed point iterations. Resultsof several numerical tests are given which illustrate the theory.

A proof that Anderson acceleration improves the convergence rate in  linearly converging fixed point methods (but not in those converging  quadratically)