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.

翻译：我们提出了一种针对线性问题 $A x=y$ 的高效预处理子。该方法能保证对所有形如 $A = L + V$ 的增生系统（其中 $L$ 是 $A$ 的近似，且系统经过缩放使得偏差满足 $\lVert V \rVert<1$），内存高效的不动点迭代单调收敛。与常见分裂预处理子不同，我们的方法不局限于任何特定的分裂方式。因此，可选取解析解存在的近似问题来高效评估预处理子。我们证明了唯一具有该特性的预处理子形式为 $(L+I)(I - V)^{-1}$。这种独特形式还允许从预处理系统中消除前向问题，通常可使每次迭代所需时间减半。我们针对波动问题、扩散问题以及受电弓时滞微分方程展示了该方法并进行评估。通过后者，我们展示了该方法如何推广至一般（不必为增生）线性系统。