Convergence Analysis for Anderson Acceleration

@article{Toth2015ConvergenceAF,
  title={Convergence Analysis for Anderson Acceleration},
  author={Alexander Raymond Toth and C. T. Kelley},
  journal={SIAM J. Numer. Anal.},
  year={2015},
  volume={53},
  pages={805-819}
}
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… 

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References

SHOWING 1-10 OF 33 REFERENCES

Anderson Acceleration for Fixed-Point Iterations

It is shown that, on linear problems, Anderson acceleration without truncation is “essentially equivalent” in a certain sense to the generalized minimal residual (GMRES) method and the Type 1 variant in the Fang-Saad Anderson family is similarly essentially equivalent to the Arnoldi (full orthogonalization) method.

Nonlinear Krylov and moving nodes in the method of lines

Globally Convergent Inexact Newton Methods

The primary goal is to introduce and analyze new inexact Newton methods, but consideration is also given to “globally convergence” features designed to improve convergence from arbitrary starting points.

Elliptic Preconditioner for Accelerating the Self-Consistent Field Iteration in Kohn-Sham Density Functional Theory

The elliptic preconditioner is shown to be more effective in accelerating the convergence of a fixed point iteration than the existing approaches for large inhomogeneous systems at low temperature.

Inexact Newton Methods for Singular Problems

In this paper we describe the eeects of an inexact implementation of Newton's method on the behavior of the iteration for certain nonlinear equations in Banach space for which the Fr echet derivative

Inexact newton methods for singular problems

In this paper we describe the effects of an inexact implementation of Newton's method on the behavior of the iteration for certain nonlinear equations in Banach space for which the Frechet derivative

An analysis for the DIIS acceleration method used in quantum chemistry calculations

This work features an analysis for the acceleration technique DIIS that is standardly used in most of the important quantum chemistry codes, e.g. in DFT and Hartree–Fock calculations and in the Coupled Cluster method and shows that for the general nonlinear case, DIIS corresponds to a projected quasi-Newton/secant method.

A Reflective Newton Method for Minimizing a Quadratic Function Subject to Bounds on Some of the Variables

A new algorithm, a reflective Newton method, for the minimization of a quadratic function of many variables subject to upper and lower bounds on some of the variables, which appears to have significant practical potential for large-scale problems.