几何缩放：一种针对具有间断系数的特定线性系统的简单预条件器
Geometric scaling: a simple preconditioner for certain linear systems with discontinuous coefficients
AI总结：
本文提出几何缩放（GS）预条件方法，通过按系数的Lp范数缩放方程，改善具有间断系数的线性系统的收敛性，并在对流扩散椭圆PDE导出的非对称系统上验证了其有效性。
摘要：
在偏微分方程（PDE）模拟涉及非均匀介质的物理现象时，经常出现系数差异很大的线性系统（“间断系数”）。解决此类问题的标准方法是使用区域分解技术，区域边界与不同介质之间的边界一致。当区域边界几何复杂或网格非结构化时，这种方法难以实现。本文研究了一种简单的预条件技术：将每个方程除以其系数的Lp范数来缩放方程。这种预条件称为几何缩放（GS）。长期以来，人们知道对角缩放有助于改善收敛性，但尚未有研究探讨该方法对间断系数的普遍有效性。GS在多个由对流扩散椭圆PDE导出的具有间断系数的非对称线性系统上进行了测试，这些PDE的对流项从小到中等。结果表明，无论是否使用ILUT预条件器，GS都改善了重启GMRES和Bi-CGSTAB的收敛性质。GS还显著减少了特征值在原点附近的集中，从而改善了特征值分布。
英文摘要：
Linear systems with large differences between coefficients ("discontinuous coefficients") arise in many cases in which partial differential equations(PDEs) model physical phenomena involving heterogeneous media. The standard approach to solving such problems is to use domain decomposition techniques, with domain boundaries conforming to the boundaries between the different media. This approach can be difficult to implement when the geometry of the domain boundaries is complicated or the grid is unstructured. This work examines the simple preconditioning technique of scaling the equations by dividing each equation by the Lp-norm of its coefficients. This preconditioning is called geometric scaling (GS). It has long been known that diagonal scaling can be useful in improving convergence, but there is no study on the general usefulness of this approach for discontinuous coefficients. GS was tested on several nonsymmetric linear systems with discontinuous coefficients derived from convection-diffusion elliptic PDEs with small to moderate convection terms. It is shown that GS improved the convergence properties of restarted GMRES and Bi-CGSTAB, with and without the ILUT preconditioner. GS was also shown to improve the distribution of the eigenvalues by reducing their concentration around the origin very significantly.
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