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On the Convergence of CROP-Anderson Acceleration Method

Ning Wan, Agnieszka Międlar

TL;DR

This paper aims to delve into the intricate connections between the classical Anderson Acceleration method and the CROP algorithm, and contribute valuable insights that can enhance the understanding and application of acceleration methods in practical computations, as well as the developments of new and more efficient acceleration schemes.

Abstract

Anderson Acceleration is a well-established method that allows to speed up or encourage convergence of fixed-point iterations. It has been successfully used in a variety of applications, in particular within the Self-Consistent Field (SCF) iteration method for quantum chemistry and physics computations. In recent years, the Conjugate Residual with OPtimal trial vectors (CROP) algorithm was introduced and shown to have a better performance than the classical Anderson Acceleration with less storage needed. This paper aims to delve into the intricate connections between the classical Anderson Acceleration method and the CROP algorithm. Our objectives include a comprehensive study of their convergence properties, explaining the underlying relationships, and substantiating our findings through some numerical examples. Through this exploration, we contribute valuable insights that can enhance the understanding and application of acceleration methods in practical computations, as well as the developments of new and more efficient acceleration schemes.

On the Convergence of CROP-Anderson Acceleration Method

TL;DR

This paper aims to delve into the intricate connections between the classical Anderson Acceleration method and the CROP algorithm, and contribute valuable insights that can enhance the understanding and application of acceleration methods in practical computations, as well as the developments of new and more efficient acceleration schemes.

Abstract

Anderson Acceleration is a well-established method that allows to speed up or encourage convergence of fixed-point iterations. It has been successfully used in a variety of applications, in particular within the Self-Consistent Field (SCF) iteration method for quantum chemistry and physics computations. In recent years, the Conjugate Residual with OPtimal trial vectors (CROP) algorithm was introduced and shown to have a better performance than the classical Anderson Acceleration with less storage needed. This paper aims to delve into the intricate connections between the classical Anderson Acceleration method and the CROP algorithm. Our objectives include a comprehensive study of their convergence properties, explaining the underlying relationships, and substantiating our findings through some numerical examples. Through this exploration, we contribute valuable insights that can enhance the understanding and application of acceleration methods in practical computations, as well as the developments of new and more efficient acceleration schemes.
Paper Structure (20 sections, 10 theorems, 73 equations, 6 figures, 2 tables, 3 algorithms)

This paper contains 20 sections, 10 theorems, 73 equations, 6 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Contributions and Outline
  3. Previous work
  4. Background
  5. Anderson Acceleration
  6. CROP Algorithm
  7. The Least-Squares Problem
  8. The damping parameter $\beta$
  9. The truncation parameter $m^{(k)}$
  10. Anderson Acceleration vs CROP Method
  11. Broader View of Anderson Acceleration Method and CROP Algorithm
  12. Connection with Krylov methods
  13. Connection with Multisecant Methods
  14. Convergence Theory for CROP Algorithm
  15. Convergence of CROP Algorithm for Linear Problems
  16. ...and 5 more sections

Key Result

Theorem 3.1

\newlabelthm:AequivC0 Let us consider applying Anderson Acceleration method ($\beta_k=1$) and CROP algorithm to the nonlinear problem $f(x) = 0$ with initial values $x_A^{(0)} = x_C^{(0)}$ and no truncation ($m^{(k)}=k$). Then, for $k=0,1,\ldots$ until convergence ($f_C^{(0)} \neq0$) with $\bar{x}^{(k)}_A$, $x^{(k+1)}_A$ defined as in eq:WAvg and eq:Anderson, and $\widetilde{x}^{(k+1)}_{C}$, $x^

Figures (6)

  • Figure 1: Anderson Acceleration in Anderson notation.
  • Figure 1: A linear problem in Example \ref{['exp:Ex1']} with (a) a tridiagonal and (b) a seven-diagonal matrix $A$.
  • Figure 2: CROP method in CROP notation.
  • Figure 2: Convergence for a nonlinear problem in Example \ref{['exp:Ex4']}.
  • Figure 3: A small nonlinear problem in Example \ref{['exp:Ex5']} with (a) control residuals and (b) real residuals.
  • ...and 1 more figures

Theorems & Definitions (28)

  • Remark 1.1
  • Theorem 3.1
  • Proof 1
  • Remark 3.2
  • Corollary 3.3
  • Lemma 4.1
  • Proof 2
  • Lemma 4.2
  • Proof 3
  • Theorem 4.3
  • ...and 18 more