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nlKrylov: A Unified Framework for Nonlinear GCR-type Krylov Subspace Methods

Tom Werner, Ning Wan, Agnieszka Miedlar

TL;DR

A unified framework for nonlinear Krylov subspace methods (nlKrylov) to solve systems of nonlinear equations via nested algorithmic structures is introduced and rigorous convergence results are presented.

Abstract

In this paper, we introduce a unified framework for nonlinear Krylov subspace methods (nlKrylov) to solve systems of nonlinear equations. Building on classical GCR-like/type linear Krylov solvers such as GMRESR, we generalize these approaches to nonlinear problems via nested algorithmic structures. We present rigorous convergence results for problems, relying on relaxed assumptions that avoid the need for exact line searches. The framework is further extended to matrix-valued rootfinding problems using global nonlinear Krylov approaches. Extensive numerical experiments validate the theoretical insights and demonstrate the robustness and efficiency of our proposed algorithms.

nlKrylov: A Unified Framework for Nonlinear GCR-type Krylov Subspace Methods

TL;DR

A unified framework for nonlinear Krylov subspace methods (nlKrylov) to solve systems of nonlinear equations via nested algorithmic structures is introduced and rigorous convergence results are presented.

Abstract

In this paper, we introduce a unified framework for nonlinear Krylov subspace methods (nlKrylov) to solve systems of nonlinear equations. Building on classical GCR-like/type linear Krylov solvers such as GMRESR, we generalize these approaches to nonlinear problems via nested algorithmic structures. We present rigorous convergence results for problems, relying on relaxed assumptions that avoid the need for exact line searches. The framework is further extended to matrix-valued rootfinding problems using global nonlinear Krylov approaches. Extensive numerical experiments validate the theoretical insights and demonstrate the robustness and efficiency of our proposed algorithms.

Paper Structure

This paper contains 38 sections, 3 theorems, 138 equations, 12 figures, 1 table, 16 algorithms.

Table of Contents

  1. Introduction
  2. Contributions and Outline
  3. Notation
  4. Background
  5. Krylov subspace methods for linear equations
  6. Inexact Newton, Quasi-Newton and Newton-Krylov Methods
  7. The Nonlinear Truncated Generalized Conjugate Residual (nlTGCR)
  8. From Linear to Nonlinear Krylov methods
  9. The general nlKrylov framework
  10. nlGMRESR
  11. nlGCRO
  12. nlLGMRES
  13. Connection of nlKrylov methods to existing nonlinear methods
  14. Local updating and connection to quasi-Newton methods
  15. Connection to nonlinear Orthomin
  16. ...and 23 more sections

Key Result

Theorem 5.1

Let $f(x)$ be continuously differentiable in a neighborhood $B_1(x^*)$ of a solution $x^*$ to $f(x)=0$, with $J_f(x^*)$ nonsingular. Suppose that ass:upperbound holds. Then there exists a neighborhood $B_0(x^*)$ of $x^*$ such that, for any $x_0\in B_0(x^*)$, the nlKrylov iterates $\{x_j\}_j$ converg

Figures (12)

  • Figure 1: Convergence map for nlGCR(2) (left) and nlGMRESR(2,2) (right) on $\Omega=[-0.1,0.1]^2$
  • Figure 1: Convergence results for Lennard-Jones problem.
  • Figure 1: Convergence results for PDDE-NEP
  • Figure 2: Convergence trajectories of nlGCR(2) and nlGMRESR(2,2) for starting vectors $x^{(1)}_0=[0.1,1]^T$ (left) and $x^{(2)}_0=[0.001,0.05]^T$ (right).
  • Figure 2: Theoretical and observed convergence bounds for nlGCR (left) and nlGMRESR (right) applied to the Lennard-Jones-problem.
  • ...and 7 more figures

Theorems & Definitions (9)

  • Theorem 5.1
  • Proof 1
  • Remark 5.2
  • Lemma 5.3
  • Remark 5.4
  • Theorem 5.5
  • Proof 2
  • Remark 6.1
  • Remark 8.1