Anderson mixing (AM) is a classical method that can accelerate fixed-point iterations by exploring historical information. Despite the successful application of AM in scientific computing, the theoretical properties of AM are still under exploration. In this paper, we study the restarted version of the Type-I and Type-II AM methods, i.e., restarted AM. With a multi-step analysis, we give a unified convergence analysis for the two types of restarted AM and justify that the restarted Type-II AM can locally improve the convergence rate of the fixed-point iteration. Furthermore, we propose an adaptive mixing strategy by estimating the spectrum of the Jacobian matrix. If the Jacobian matrix is symmetric, we develop the short-term recurrence forms of restarted AM to reduce the memory cost. Finally, experimental results on various problems validate our theoretical findings.

翻译：安德森混合（Anderson mixing, AM）是一种通过利用历史信息加速不动点迭代的经典方法。尽管AM在科学计算中已成功应用，但其理论性质仍在探索之中。本文研究重启型Type-I和Type-II AM方法（即重启型AM）的收敛性。通过多步分析，我们为两类重启型AM建立了统一的收敛性分析框架，并证明重启型Type-II AM能局部提升不动点迭代的收敛速率。此外，我们提出一种通过估计雅可比矩阵谱的自适应混合策略。当雅可比矩阵对称时，我们推导出重启型AM的短递推形式以降低内存开销。最后，针对多种问题的实验结果验证了我们的理论发现。