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

TL;DR

This paper introduces nlKrylov, a unified framework for nonlinear Krylov subspace methods that generalizes classical linear solvers to nonlinear problems, providing convergence guarantees and demonstrating robustness through numerical experiments.

Contribution

It presents a novel unified framework for nonlinear Krylov methods, extending classical linear solvers to nonlinear and matrix-valued problems with theoretical and empirical validation.

Findings

Convergence results under relaxed assumptions.

Robustness demonstrated through extensive numerical experiments.

Efficient solution of nonlinear and matrix-valued rootfinding problems.

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.