基于数据驱动的调度松弛Jacobi格式快速收敛算法
A data driven heuristic for rapid convergence of Scheduled Relaxation Jacobi schemes
摘要：
调度松弛雅可比（SRJ）方法是求解椭圆型偏微分方程（PDE）的一种可行的高性能线性求解器。该方法通过在算法的每个循环中应用一系列精心选择的$M$超松弛因子和欠松弛因子来有效地衰减解的误差，从而大大改善了标准Jacobi迭代的收敛性。在以前的工作中，最佳SRJ计划（套松弛因子）已被推导出加速收敛的椭圆偏微分方程的特定离散。在这项工作中，我们开发了一个家庭的SRJ计划，可以适用于解决椭圆型偏微分方程，无论具体的离散化。为了实现良好的收敛性，我们训练一种算法，根据将这些方案应用于一维泊松方程所收集的收敛数据，在线性求解过程的每个循环中选择该族中的哪种方案。自动选择启发式，是基于这一有限的数据被发现提供了良好的收敛范围广泛的问题。
英文摘要：
The Scheduled Relaxation Jacobi (SRJ) method is a viable candidate as a high performance linear solver for elliptic partial differential equations (PDEs). The method greatly improves the convergence of the standard Jacobi iteration by applying a sequence of $M$ well-chosen overrelaxation and underrelaxation factors in each cycle of the algorithm to effectively attenuate the solution error. In previous work, optimal SRJ schemes (sets of relaxation factors) have been derived to accelerate convergence for specific discretizations of elliptic PDEs. In this work, we develop a family of SRJ schemes which can be applied to solve elliptic PDEs regardless of the specific discretization employed. To achieve favorable convergence, we train an algorithm to select which scheme in this family to apply at each cycle of the linear solve process, based on convergence data collected from applying these schemes to the one-dimensional Poisson equation. The automatic selection heuristic that is developed based on this limited data is found to provide good convergence for a wide range of problems.
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