This work introduces, analyzes and demonstrates an efficient and theoretically sound filtering strategy to ensure the condition of the least-squares problem solved at each iteration of Anderson acceleration. The filtering strategy consists of two steps: the first controls the length disparity between columns of the least-squares matrix, and the second enforces a lower bound on the angles between subspaces spanned by the columns of that matrix. The combined strategy is shown to control the condition number of the least-squares matrix at each iteration. The method is shown to be effective on a range of problems based on discretizations of partial differential equations. It is shown particularly effective for problems where the initial iterate may lie far from the solution, and which progress through distinct preasymptotic and asymptotic phases.

翻译：本文提出、分析并验证了一种高效且理论完备的滤波策略，用于确保Anderson加速每次迭代中求解的最小二乘问题的条件数可控。该滤波策略包含两个步骤：第一步控制最小二乘矩阵各列之间的长度差异，第二步则对该矩阵各列张成子空间之间的夹角施加下界约束。研究表明，该组合策略能有效控制每次迭代中最小二乘矩阵的条件数。该方法在基于偏微分方程离散化的多类问题上展现出良好效果，尤其适用于初始迭代可能远离解、且需要经历前渐近与渐近两个不同阶段的问题。