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:这项工作引入、分析并展示了一种高效且在理论上合理的过滤策略,以确保在每次迭代时都解决了最小平方的问题。过滤策略包括两个步骤:第一个步骤控制最小平方矩阵各列之间的长度差异,第二个步骤在该矩阵各列所覆盖的子空间之间的角上执行一个较低的界限。合并策略显示在每次迭代时控制最小平方矩阵的条件数。该方法显示对部分差分方程式的离散所产生的一系列问题是有效的。它对于最初偏差可能远离解决方案的问题特别有效,而且对于通过不同的淡化和淡化阶段取得进展的问题则特别有效。