Anderson acceleration for contractive and noncontractive operators
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A one-step analysis of Anderson acceleration with general algorithmic depths is presented. The resulting residual bounds within both contractive and noncontractive settings clearly show 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. In the contractive setting, the 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. Several numerical tests illustrate the analysis primarily in the noncontractive setting, and demonstrate the use of the method on a nonlinear Helmholtz equation and the steady Navier-Stokes equations with high Reynolds number in three spatial dimensions.
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