# 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…

## 142 Citations

### Anderson Acceleration for Nonsmooth Fixed Point Problems

- MathematicsSIAM J. Numer. Anal.
- 2022

. We give new convergence results of Anderson acceleration for the composite max ﬁxed point problem. We prove that Anderson(1) and EDIIS(1) are q-linear convergent with a smaller q-factor than…

### On the Asymptotic Linear Convergence Speed of Anderson Acceleration, Nesterov Acceleration, and Nonlinear GMRES

- Mathematics, Computer ScienceSIAM J. Sci. Comput.
- 2021

This work considers nonlinear convergence acceleration methods for fixed-point iteration of AA, nonlinear GMRES, and Nesterov-type acceleration, and determines coefficients that result in optimal asymptotic convergence factors, given knowledge of the spectrum of q'(x) at the fixed point.

### Anderson Acceleration as a Krylov Method with Application to Asymptotic Convergence Analysis

- Mathematics, Computer ScienceArXiv
- 2021

This work finds that the AA(m) residual polynomials observe a periodic memory effect where increasing powers of the error iteration matrix M act on the initial residual as the iteration number increases, and derives several further results based on these polynomial residual update formulas.

### Linear Asymptotic Convergence of Anderson Acceleration: Fixed-Point Analysis

- MathematicsSIAM Journal on Matrix Analysis and Applications
- 2022

The asymptotic convergence of AA(m), i.e., Anderson acceleration with window size m for accelerating fixed-point methods xk+1 = q(xk), xk ∈ Rn, is studied and it is shown that, despite the discontinuity of β(z), the iteration function Ψ(z) is Lipschitz continuous and directionally differentiable at z∗ for AA(1).

### A Proof That Anderson Acceleration Improves the Convergence Rate in Linearly Converging Fixed-Point Methods (But Not in Those Converging Quadratically)

- MathematicsSIAM J. Numer. Anal.
- 2020

This paper provides the first proof that Anderson acceleration (AA) improves the convergence rate of general fixed point iterations to first order by a factor of the gain at each step.

### A proof that Anderson acceleration increases the convergence rate in linearly converging fixed point methods (but not in quadratically converging ones)

- Mathematics
- 2018

This paper provides the first proof that Anderson acceleration (AA) increases the convergence rate of general fixed point iterations. AA has been used for decades to speed up nonlinear solvers in…

### Newton-Anderson at Singular Points

- MathematicsArXiv
- 2022

In this paper we develop convergence and acceleration theory for Anderson acceleration applied to Newton's method for nonlinear systems in which the Jacobian is singular at a solution. For these…

### Globally Convergent Type-I Anderson Acceleration for Nonsmooth Fixed-Point Iterations

- Mathematics, Computer ScienceSIAM J. Optim.
- 2020

This work proposes the first globally convergent variant of Anderson acceleration assuming only that the fixed-point iteration is non-expansive, and shows by extensive numerical experiments that many first order algorithms can be improved, especially in their terminal convergence, with the proposed algorithm.

### Anderson Acceleration of Proximal Gradient Methods

- Computer ScienceICML
- 2020

This work introduces novel methods for adapting Anderson acceleration to (non-smooth and constrained) proximal gradient algorithms and proposes a simple scheme for stabilization that combines the global worst-case guarantees of proximalgradient methods with the local adaptation and practical speed-up of Anderson acceleration.

### Convergence analysis of Anderson-type acceleration of Richardson's iteration

- MathematicsNumer. Linear Algebra Appl.
- 2019

It is established that sufficient conditions for convergence are established for Anderson extrapolation to accelerate the (stationary) Richardson iterative method for sparse linear systems, and an augmented version of this technique is proposed.

## References

SHOWING 1-10 OF 33 REFERENCES

### Anderson Acceleration for Fixed-Point Iterations

- MathematicsSIAM J. Numer. Anal.
- 2011

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.

### Globally Convergent Inexact Newton Methods

- MathematicsSIAM J. Optim.
- 1994

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

- Computer ScienceSIAM J. Sci. Comput.
- 2013

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

- Mathematics
- 2007

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

- Mathematics
- 1993

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…

### Iterative Procedures for Nonlinear Integral Equations

- MathematicsJACM
- 1965

A procedure is synthesized to offset some of the disadvantages of these t e c h n i q u e s in this context; however, the procedure is not restricted to this pt~rticular class of s y s t e m s of nonlinear equations.

### Numerical methods for unconstrained optimization and nonlinear equations

- MathematicsPrentice Hall series in computational mathematics
- 1983

Newton's Method for Nonlinear Equations and Unconstrained Minimization and methods for solving nonlinear least-squares problems with Special Structure.