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Properties of Nonlinear GMRES Applied to the Preconditioned Richardson Iteration

Yunhui He

Abstract

In this work, we propose new variants of Anderson acceleration and nonlinear GMRES for general fixed-point iterations, based on modified least-squares problems associated with the methods. To solve the underlying linear systems, we apply these new approaches to accelerate the preconditioned Richardson iteration. We establish connections between the proposed variants and both left- and right-preconditioned GMRES. In particular, we show that full NGMRES applied to the preconditioned Richardson iteration is equivalent to right-preconditioned GMRES, while full NGMRES equipped with the new least-squares formulation is equivalent to left-preconditioned GMRES. Furthermore, under certain conditions on the preconditioned coefficient matrix, an equivalence between windowed NGMRES with any depth and preconditioned GMRES. These theoretical results deepen our understanding of NGMRES for solving linear systems and clarify its relationship to classical preconditioned GMRES. Finally, we establish conditions for monotonicity of the various variants. Numerical results are presented to validate our theoretical findings.

Properties of Nonlinear GMRES Applied to the Preconditioned Richardson Iteration

Abstract

In this work, we propose new variants of Anderson acceleration and nonlinear GMRES for general fixed-point iterations, based on modified least-squares problems associated with the methods. To solve the underlying linear systems, we apply these new approaches to accelerate the preconditioned Richardson iteration. We establish connections between the proposed variants and both left- and right-preconditioned GMRES. In particular, we show that full NGMRES applied to the preconditioned Richardson iteration is equivalent to right-preconditioned GMRES, while full NGMRES equipped with the new least-squares formulation is equivalent to left-preconditioned GMRES. Furthermore, under certain conditions on the preconditioned coefficient matrix, an equivalence between windowed NGMRES with any depth and preconditioned GMRES. These theoretical results deepen our understanding of NGMRES for solving linear systems and clarify its relationship to classical preconditioned GMRES. Finally, we establish conditions for monotonicity of the various variants. Numerical results are presented to validate our theoretical findings.

Paper Structure

This paper contains 9 sections, 10 theorems, 73 equations, 5 figures, 1 table, 5 algorithms.

Table of Contents

  1. Introduction
  2. AA and NGMRES on preconditioned Richardson iteration
  3. AA on preconditioned Richardson iteration
  4. NGMRES on preconditioned Richardson iteration
  5. New Variants Using Modified Least-Squares Problem
  6. Numerical Results
  7. Symmetric case
  8. Nonsymmetric case
  9. Conclusion

Key Result

Theorem 2.1

Consider AA($m$) presented in Algorithm alg:AA to accelerate preconditioned Richardson iteration, eq:PR-FP. Let $B=I-PA$ and $r_{k+1}=x_{k+1}-q(x_{k+1})$. Assume that $B$ is invertible. Then, the least-squares problem defined by eq:min-AA can be rewritten as where In addition, for $k\geq 0$ and for $i, j =0, 1, \cdots, m_k$, we have the following orthogonal property: Furthermore, if $\|B\|<1$,

Figures (5)

  • Figure 1: Example \ref{['ex:Laplace']} with $N=16$. Convergence history (residual norms) for the right-preconditioned GMRES, full NGRMES and full AA. The graph on the right shows the first 10 of the 40 iterations.
  • Figure 2: Example \ref{['ex:Laplace']} with $N=16$. Convergence history (preconditioned residual norms) for left-preconditioned GMRES, full NGRMESr, full NGMRES, and full AA. The graph on the right shows the first 10 of the 40 iterations.
  • Figure 3: Example \ref{['ex:Laplace']} with $N=16$. Convergence history (preconditioned residual norms) for full NGRMESr, full NGMRES, full AA, full AAr and full AAg. The graph on the right shows the first 10 of the 40 iterations.
  • Figure 4: Example \ref{['ex:Laplace']} with $N=64$. Convergence history (preconditioned residual norms) for windowed NGMRES-type and AA-type methods using $m=1$ (left), $m=5$ (middle) and $m=15$ (right).
  • Figure 5: Example \ref{['ex:conv-diff']} with $N=64$. Convergence history (preconditioned residual norms) for windowed NGMRES-type and AA-type methods using $m=1$ (left), $m=5$ (middle) and $m=15$ (right).

Theorems & Definitions (23)

  • Theorem 2.1
  • proof
  • Remark 2.1
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Example 2.1
  • Theorem 2.4
  • Lemma 2.1
  • ...and 13 more