In this report, we present a versatile and efficient preconditioned Anderson acceleration (PAA) method for fixed-point iterations. The proposed framework offers flexibility in balancing convergence rates (linear, super-linear, or quadratic) and computational costs related to the Jacobian matrix. Our approach recovers various fixed-point iteration techniques, including Picard, Newton, and quasi-Newton iterations. The PAA method can be interpreted as employing Anderson acceleration (AA) as its own preconditioner or as an accelerator for quasi-Newton methods when their convergence is insufficient. Adaptable to a wide range of problems with differing degrees of nonlinearity and complexity, the method achieves improved convergence rates and robustness by incorporating suitable preconditioners. We test multiple preconditioning strategies on various problems and investigate a delayed update strategy for preconditioners to further reduce the computational costs.

翻译：本报告提出了一种通用且高效的预条件安德森加速（PAA）方法，用于处理不动点迭代问题。所提出的框架在收敛速度（线性、超线性或二次收敛）以及与雅可比矩阵相关的计算成本之间实现了灵活平衡。本方法涵盖了多种不动点迭代技术，包括Picard、牛顿和拟牛顿迭代。PAA方法既可将安德森加速（AA）作为其自身的预条件子使用，也可在拟牛顿方法收敛不足时作为其加速器。该方法适用于非线性程度和复杂度各异的广泛问题，通过引入合适的预条件子，实现了更优的收敛速度和鲁棒性。我们在多种问题上测试了多种预条件策略，并研究了预条件子的延迟更新策略以进一步降低计算成本。