Anderson加速的合同与非扩张不动点问题的非单调全局化框架
A nonmonotone globalization framework for Anderson acceleration for contractive and nonexpansive fixed point problems
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中文总结 AI 辅助
针对Anderson加速缺乏全局收敛保证的问题，提出非单调全局化框架，在保留局部加速的同时确保合同与非扩张映射的全局收敛，并给出参数选择规则，数值实验验证了鲁棒性与效率。
中文摘要 AI 辅助
Anderson加速（AA）是加速不动点迭代的有效技术，但通常缺乏全局收敛保证。我们提出了一种非单调全局化的Anderson加速框架，该框架保留了AA的局部加速特性，同时确保全局收敛。对于合同映射，无需预先知道收缩因子，我们证明了所提方法在有限次迭代后精确退化为纯AA，并且对于记忆大小$m\geq1$具有全局$r$-线性收敛性，对于$m=1$具有全局残差$q$-线性收敛性，其收敛因子不大于底层不动点映射的收缩因子。对于非扩张映射，我们证明了不动点残差全局收敛到零。进一步开发了一种与类别无关的参数选择规则，以确保在这两种设置下的收敛性质。数值实验证明了所提方法的鲁棒性和效率。
英文摘要
Anderson acceleration (AA) is an effective technique for accelerating fixed point iterations, but it generally lacks global convergence guarantees. We propose a nonmonotone globalized Anderson acceleration framework that retains the local acceleration of AA while ensuring global convergence. For contractive mappings, without requiring prior knowledge of the contraction factor, we prove that the proposed method reduces exactly to pure AA after finitely many iterations and enjoys global $r$-linear convergence for memory size $m\geq1$ and global residual $q$-linear convergence for $m=1$, with convergence factors no greater than the contraction factor of the underlying fixed point mapping. For nonexpansive mappings, we prove that the fixed point residuals converge globally to zero. A class-agnostic parameter selection rule is further developed to ensure these convergence properties in both settings. Numerical experiments demonstrate the robustness and efficiency of the proposed method.
发表机构
- School of Mathematics, Harbin Institute of Technology（哈尔滨工业大学数学学院）
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