Anderson Acceleration (AA) is a popular algorithm designed to enhance the convergence of fixed-point iterations. In this paper, we introduce a variant of AA based on a Truncated Gram-Schmidt process (AATGS) which has a few advantages over the classical AA. In particular, an attractive feature of AATGS is that its iterates obey a three-term recurrence in the situation when it is applied to solving symmetric linear problems and this can lead to a considerable reduction of memory and computational costs. We analyze the convergence of AATGS in both full-depth and limited-depth scenarios and establish its equivalence to the classical AA in the linear case. We also report on the effectiveness of AATGS through a set of numerical experiments, ranging from solving nonlinear partial differential equations to tackling nonlinear optimization problems. In particular, the performance of the method is compared with that of the classical AA algorithms.

翻译：安德森加速（AA）是一种旨在提高不动点迭代收敛速度的流行算法。本文提出了一种基于截断格拉姆-施密特过程（AATGS）的AA变体，相较于经典AA具有若干优势。特别地，AATGS的一个显著特征是：当应用于对称线性问题求解时，其迭代过程满足三项递推关系，这可以显著降低内存和计算成本。我们分析了AATGS在全深度和有限深度场景下的收敛性，并在线性情形下证明了其与经典AA的等价性。通过求解非线性偏微分方程到处理非线性优化问题等一系列数值实验，我们验证了AATGS的有效性，并系统比较了该方法与经典AA算法的性能表现。