Anderson acceleration (AA) is widely used for accelerating the convergence of an underlying fixed-point iteration $\bm{x}_{k+1} = \bm{q}( \bm{x}_{k} )$, $k = 0, 1, \ldots$, with $\bm{x}_k \in \mathbb{R}^n$, $\bm{q} \colon \mathbb{R}^n \to \mathbb{R}^n$. Despite AA's widespread use, relatively little is understood theoretically about the extent to which it may accelerate the underlying fixed-point iteration. To this end, we analyze a restarted variant of AA with a restart size of one, a method closely related to GMRES(1). We consider the case of $\bm{q}( \bm{x} ) = M \bm{x} + \bm{b}$ with matrix $M \in \mathbb{R}^{n \times n}$ either symmetric or skew-symmetric. For both classes of $M$ we compute the worst-case root-average asymptotic convergence factor of the AA method, partially relying on conjecture in the symmetric setting, proving that it is strictly smaller than that of the underlying fixed-point iteration. For symmetric $M$, we show that the AA residual iteration corresponds to a fixed-point iteration for solving an eigenvector-dependent nonlinear eigenvalue problem (NEPv), and we show how this can result in the convergence factor strongly depending on the initial iterate, which we quantify exactly in certain special cases. Conversely, for skew-symmetric $M$ we show that the AA residual iteration is closely related to a power iteration for $M$, and how this results in the convergence factor being independent of the initial iterate. Supporting numerical results are given, which also indicate the theory is applicable to the more general setting of nonlinear $\bm{q}$ with Jacobian at the fixed point that is symmetric or skew symmetric.

翻译：安德森加速（AA）被广泛用于加速基础定点迭代 $\bm{x}_{k+1} = \bm{q}( \bm{x}_{k} )$, $k = 0, 1, \ldots$（其中 $\bm{x}_k \in \mathbb{R}^n$, $\bm{q} \colon \mathbb{R}^n \to \mathbb{R}^n$）的收敛。尽管AA应用广泛，但理论上对其能在多大程度上加速基础定点迭代的理解相对有限。为此，我们分析了一种重启规模为一的AA变体，该方法与GMRES(1)密切相关。我们考虑 $\bm{q}( \bm{x} ) = M \bm{x} + \bm{b}$ 的情形，其中矩阵 $M \in \mathbb{R}^{n \times n}$ 为对称矩阵或斜对称矩阵。针对这两类$M$，我们计算了AA方法的最坏情况根平均渐近收敛因子（在对称矩阵情形下部分依赖于猜想），证明该因子严格小于基础定点迭代的收敛因子。对于对称矩阵$M$，我们证明了AA残差迭代对应于求解一个特征向量依赖的非线性特征值问题（NEPv）的定点迭代，并说明了这如何导致收敛因子强烈依赖于初始迭代点，我们在某些特殊情况下对此进行了精确量化。相反，对于斜对称矩阵$M$，我们证明了AA残差迭代与$M$的幂迭代密切相关，并说明了这如何使得收敛因子与初始迭代点无关。文中给出了支持性的数值结果，这些结果也表明该理论可推广至更一般的非线性$\bm{q}$情形，即其在定点处的雅可比矩阵为对称或斜对称矩阵。