Anderson acceleration (AA) is a technique for accelerating the convergence of an underlying fixed-point iteration. AA is widely used within computational science, with applications ranging from electronic structure calculation to the training of neural networks. Despite AA's widespread use, relatively little is understood about it theoretically. An important and unanswered question in this context is: To what extent can AA actually accelerate convergence of the underlying fixed-point iteration? While simple enough to state, this question appears rather difficult to answer. For example, it is unanswered even in the simplest (non-trivial) case where the underlying fixed-point iteration consists of applying a two-dimensional affine function. In this note we consider a restarted variant of AA applied to solve symmetric linear systems with restart window of size one. Several results are derived from the analytical solution of a nonlinear eigenvalue problem characterizing residual propagation of the AA iteration. This includes a complete characterization of the method to solve $2 \times 2$ linear systems, rigorously quantifying how the asymptotic convergence factor depends on the initial iterate, and quantifying by how much AA accelerates the underlying fixed-point iteration. We also prove that even if the underlying fixed-point iteration diverges, the associated AA iteration may still converge.

翻译：Anderson加速（AA）是一种加速基础不动点迭代收敛性的技术。AA广泛应用于计算科学领域，从电子结构计算到神经网络训练均有涉及。尽管AA应用广泛，但其理论理解相对有限，其中一个重要且未解决的问题是：AA能在多大程度上实际加速基础不动点迭代的收敛性？这个问题看似简单，实则相当棘手——即便在基础不动点迭代仅包含二维仿射函数这一最简（非平凡）情形下也尚未得到解答。本文研究窗口大小为1的重启型AA在求解对称线性系统中的应用。通过求解刻画AA迭代残差传播的非线性特征值问题的解析解，我们推导出若干理论结果。这包括：完整刻画求解$2 \times 2$线性系统的方法，严格量化渐近收敛因子对初始迭代的依赖性，以及量化AA对基础不动点迭代的加速程度。我们同时证明：即使基础不动点迭代发散，其对应的AA迭代仍可能收敛。