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Title: One-step convergence of inexact Anderson acceleration for contractive and non-contractive mappings
Authors:
Award ID(s):
1819097
Publication Date:
NSF-PAR ID:
10344797
Journal Name:
ETNA - Electronic Transactions on Numerical Analysis
Volume:
55
Page Range or eLocation-ID:
285 to 309
ISSN:
1068-9613
Sponsoring Org:
National Science Foundation
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  2. Abstract A one-step analysis of Anderson acceleration with general algorithmic depths is presented. The resulting residual bounds within both contractive and noncontractive settings reveal the balance between the contributions from the higher and lower order terms, which are both dependent on the success of the optimization problem solved at each step of the algorithm. The new residual bounds show the additional terms introduced by the extrapolation produce terms that are of a higher order than was previously understood. In the contractive setting these bounds sharpen previous convergence and acceleration results. The bounds rely on sufficient linear independence of the differences between consecutive residuals, rather than assumptions on the boundedness of the optimization coefficients, allowing the introduction of a theoretically sound safeguarding strategy. Several numerical tests illustrate the analysis primarily in the noncontractive setting, and demonstrate the use of the method, the safeguarding strategy and theory-based guidance on dynamic selection of the algorithmic depth, on a p-Laplace equation, a nonlinear Helmholtz equation and the steady Navier–Stokes equations with high Reynolds number in three spatial dimensions.