对称不定线性系统的绝对值预处理
Absolute value preconditioning for symmetric indefinite linear systems
AI总结：
提出一种基于矩阵绝对值构造对称正定预条件子的新策略，并通过几何多重网格方法实现，数值实验证明其鲁棒性和低内存需求。
摘要：
我们提出了一种新的策略，用于为对称不定矩阵的线性系统构造对称正定（SPD）预条件子。该策略称为绝对值预处理，其动机是观察到：以矩阵绝对值的逆作为预条件子的预处理最小残差方法最多两步即可收敛到系统的精确解。然而，通常无法计算精确的矩阵绝对值或其精确逆。但我们基于所建议的方法提供了一个实用的SPD预条件子示例。在该示例中，我们考虑了一个带有平移离散负拉普拉斯算子的模型问题，并建议使用几何多重网格（MG）预条件子，其中矩阵绝对值的逆仅出现在粗网格上，而细网格上的操作基于拉普拉斯算子。我们的数值测试证明了新MG预条件子的实际有效性，它产生了一个具有极低内存需求的鲁棒迭代方案。
英文摘要：
We introduce a novel strategy for constructing symmetric positive definite (SPD) preconditioners for linear systems with symmetric indefinite matrices. The strategy, called absolute value preconditioning, is motivated by the observation that the preconditioned minimal residual method with the inverse of the absolute value of the matrix as a preconditioner converges to the exact solution of the system in at most two steps. Neither the exact absolute value of the matrix nor its exact inverse are computationally feasible to construct in general. However, we provide a practical example of an SPD preconditioner that is based on the suggested approach. In this example we consider a model problem with a shifted discrete negative Laplacian, and suggest a geometric multigrid (MG) preconditioner, where the inverse of the matrix absolute value appears only on the coarse grid, while operations on finer grids are based on the Laplacian. Our numerical tests demonstrate practical effectiveness of the new MG preconditioner, which leads to a robust iterative scheme with minimalist memory requirements.
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