Generally, discretization of partial differential equations (PDEs) creates a sequence of linear systems $A_k x_k = b_k, k = 0, 1, 2, ..., N$ with well-known and structured sparsity patterns. Preconditioners are often necessary to achieve fast convergence When solving these linear systems using iterative solvers. We can use preconditioner updates for closely related systems instead of computing a preconditioner for each system from scratch. One such preconditioner update is the sparse approximate map (SAM), which is based on the sparse approximate inverse preconditioner using a least squares approximation. A SAM then acts as a map from one matrix in the sequence to another nearby one for which we have an effective preconditioner. To efficiently compute an effective SAM update (i.e., one that facilitates fast convergence of the iterative solver), we seek to compute an optimal sparsity pattern. In this paper, we examine several sparsity patterns for computing the SAM update to characterize optimal or near-optimal sparsity patterns for linear systems arising from discretized PDEs.

翻译：一般而言，偏微分方程（PDE）的离散化会产生一系列线性系统 $A_k x_k = b_k, k = 0, 1, 2, ..., N$，这些系统具有已知且结构化的稀疏模式。在使用迭代求解器求解这些线性系统时，预条件子通常是实现快速收敛所必需的。对于紧密相关的系统，我们可以采用预条件子更新策略，而非为每个系统从头计算预条件子。其中一种预条件子更新方法是稀疏近似映射（SAM），它基于采用最小二乘近似的稀疏近似逆预条件子。SAM 可视为从一个序列矩阵到另一个邻近矩阵的映射，而后者我们已经拥有一个有效的预条件子。为了高效计算有效的 SAM 更新（即能促进迭代求解器快速收敛的更新），我们致力于寻找最优的稀疏模式。本文研究了用于计算 SAM 更新的几种稀疏模式，以刻画由离散化 PDE 产生的线性系统的最优或接近最优的稀疏模式特征。