论文标题
非对称线性系统的定点迭代溶液的通用矩阵拆分预处理
A universal matrix-free split preconditioner for the fixed-point iterative solution of non-symmetric linear systems
论文作者
论文摘要
我们提出了线性问题的有效预处理$ a x = y $。它保证了表格$ a = l + v $的所有增值系统的存储效率定点迭代的单调收敛，其中$ l $是$ a $的近似值，并且对系统进行了缩放，以使差异与$ \ lvert v \ rvert v \ rvert <1 $。与普通的拆分预处理相反，我们的方法不仅限于任何特定的分裂。因此，可以选择近似问题，以便可以使用分析解决方案有效地评估预处理。我们证明，此属性的唯一预处理具有$（l+i）（i -v）^{ - 1} $的表单。此外，这种独特的形式允许从预处理系统中消除正向问题，通常会使迭代所需的时间减半。我们证明并评估了我们的波浪问题，扩散问题和pantograph延迟微分方程的方法。在后者的情况下，我们显示该方法如何扩展到一般，不一定是增强的线性系统。
We present an efficient preconditioner for linear problems $A x=y$. It guarantees monotonic convergence of the memory-efficient fixed-point iteration for all accretive systems of the form $A = L + V$, where $L$ is an approximation of $A$, and the system is scaled so that the discrepancy is bounded with $\lVert V \rVert<1$. In contrast to common splitting preconditioners, our approach is not restricted to any particular splitting. Therefore, the approximate problem can be chosen so that an analytic solution is available to efficiently evaluate the preconditioner. We prove that the only preconditioner with this property has the form $(L+I)(I - V)^{-1}$. This unique form moreover permits the elimination of the forward problem from the preconditioned system, often halving the time required per iteration. We demonstrate and evaluate our approach for wave problems, diffusion problems, and pantograph delay differential equations. With the latter we show how the method extends to general, not necessarily accretive, linear systems.
