论文标题
并行无基质多项式预处理，并应用于离散裂缝网络中的流量模拟
Parallel Matrix-free polynomial preconditioners with application to flow simulations in discrete fracture networks
论文作者
论文摘要
我们为大型和稀疏的对称正定线性系统的共轭梯度方法开发了一种稳健的无基质，避免了平行的，高度的多项式预处理。我们讨论旨在避免频谱极端矩阵特征值的缩放参数的选择。我们使用此预处理框架来解决$ 3 \ times 3 $块系统，该系统在大型离散裂缝网络中的流体流量模拟中产生。我们将多项式预处理应用于与该系统相关的合适SCHUR，由于其大小和密度无法明确计算。数值结果证实了所提出的预处理的出色特性，高于非常高的多项式程度。并行实现通过利用减少的标量产品和全球通信的优势来实现令人满意的可伸缩性。
We develop a robust matrix-free, communication avoiding parallel, high-degree polynomial preconditioner for the Conjugate Gradient method for large and sparse symmetric positive definite linear systems. We discuss the selection of a scaling parameter aimed at avoiding unwanted clustering of eigenvalues of the preconditioned matrices at the extrema of the spectrum. We use this preconditioned framework to solve a $3 \times 3$ block system arising in the simulation of fluid flow in large-size discrete fractured networks. We apply our polynomial preconditioner to a suitable Schur complement related with this system, which can not be explicitly computed because of its size and density. Numerical results confirm the excellent properties of the proposed preconditioner up to very high polynomial degrees. The parallel implementation achieves satisfactory scalability by taking advantage from the reduced number of scalar products and hence of global communications.
