非线性广义最小残差法（NGMRES）在压缩和非压缩迭代中的收敛性分析及加速证明

中文摘要 AI 辅助

本文首次对应用于求解一般非线性系统的压缩和非压缩不动点迭代（FPI）的非线性广义最小残差法（NGMRES）进行了收敛性分析和加速证明。主要结果表明，在压缩和非压缩情况下，优化问题的比率增益是加速（或实现）收敛的机制。分析还揭示了与优化问题相关的第二个重要量，它直接预测每次迭代的线性收敛速率，并证明其至多为1，因此只有高阶项导致NGMRES不收敛。给出了几个具有挑战性的非线性测试问题的数值结果，说明了该理论，展示了加速如何改善收敛，表明预测线性收敛速率的量非常准确，可用于自适应选择NGMRES深度，展示了重启如何改善非压缩迭代中的收敛，展示了NGMRES如何自然适用于寻找多解偏微分方程的不同解，以及NGMRES应用于超线性FPI时如何比安德森加速表现更好。

英文摘要

This paper gives the first convergence analysis and proof of acceleration for nonlinear GMRES (NGMRES) applied to contractive and noncontractive fixed point iterations (FPIs) for solving general nonlinear systems. Our main results are that in both the contractive and noncontractive cases, the ratio gain of the optimization problem is the mechanism responsible for accelerating (or enabling) convergence. Our analysis also reveals a second important quantity related to the optimization problem, which directly predicts the linear convergence rate at each iteration and proves it is at most 1; hence only higher order terms are responsible for NGMRES non-convergence. Numerical results for several challenging nonlinear test problems are given that illustrate the theory, show how the acceleration improves convergence, show that the quantity predicting the linear convergence rate is remarkably accurate and moreover can be useful for adaptively choosing NGMRES depth, show how restarts can improve convergence in noncontractive iterations, show how NGMRES is naturally suited for finding distinct solutions of a multi-solution PDE, and that NGMRES can perform better than Anderson acceleration when applied to superlinear FPIs.

发表机构

• University of Houston（休斯顿大学）
• Clemson University（克莱姆森大学）
• University of West Florida（西佛罗里达大学）

机构由 AI 辅助整理，请以论文原文为准。