Preconditioners are generally essential for fast convergence in the iterative solution of linear systems of equations. However, the computation of a good preconditioner can be expensive. So, while solving a sequence of many linear systems, it is advantageous to recycle preconditioners, that is, update a previous preconditioner and reuse the updated version. In this paper, we introduce a simple and effective method for doing this. Although our approach can be used for matrices changing slowly in any way, we focus on the important case of sequences of the type $(s_k\textbf{E}(\textbf{p}) + \textbf{A}(\textbf{p}))\textbf{x}_k = \textbf{b}_k$, where the right hand side may or may not change. More general changes in matrices will be discussed in a future paper. We update preconditioners by defining a map from a new matrix to a previous matrix, for example the first matrix in the sequence, and combine the preconditioner for this previous matrix with the map to define the new preconditioner. This approach has several advantages. The update is entirely independent from the original preconditioner, so it can be applied to any preconditioner. The possibly high cost of an initial preconditioner can be amortized over many linear solves. The cost of updating the preconditioner is more or less constant and independent of the original preconditioner. There is flexibility in balancing the quality of the map with the computational cost. In the numerical experiments section we demonstrate good results for several applications, in particular when using an algebraic multigrid preconditioner.

翻译：预设条件通常对于直线方程系统的迭接式解决方案的快速趋同至关重要。 但是, 计算一个好的前提条件可能是昂贵的。 所以, 在解决许多线性系统的序列时, 回收预设条件, 也就是说, 更新先前的前提条件并重新使用更新版本。 在本文中, 我们引入一个简单而有效的方法来做到这一点。 尽管我们的方法可以用来以任何方式缓慢变化矩阵, 但是我们关注美元( s_ k\ textb{E} (\ textbf{p}) 类型序列的重要案例。 我们侧重于 $( textbf{p}) +\ textbf{A} (\ textbf{f{p})\ textb{x{x{ k =\ textbf{b} k =\\\\\ k$, 右手边可能改变或不改变。 在未来的文件中, 将会讨论更笼统的矩阵变化。 尽管我们的方法可以用新的矩阵定义地图到以前的矩阵, 例如序列的第一个矩阵, 我们的预设条件与新的预设条件合并起来。 在初始前置前提中, 这个方法可以有多种不同的前置前置前置前置前置前置前置前置前置前置前置前提。 。 。 更新是整个前置前置前置前置前置的 。 。 。 可能具有一种最独立的前置前置前置前置前置前置前置前置前置的优点是成本 。 。 。 。