论文标题
slablu：椭圆pdes的两级稀疏直接求解器
SlabLU: A Two-Level Sparse Direct Solver for Elliptic PDEs
论文作者
论文摘要
本文描述了由椭圆形PDE在二维结构域上离散化的线性系统的稀疏直接求解器。求解器旨在降低沟通成本并在GPU上表现良好；它使用了两级框架，比基于层次嵌套的解剖订单的传统多额外方案更容易实现和优化。该方案将域分解为薄的子域或“平板”。在每个平板中，执行局部分解，以利用局部域的几何形状。然后，通过块状三核减少系数矩阵的LU分解来获得全局分解。求解器具有复杂性$ O（n^{5/3}）$用于分解步骤，一旦分解完成，每个求解的$ o（n^{7/6}）$。 描述的求解器与一系列不同的局部离散范围兼容，数值实验证明了其针对矩形和弯曲几何形状定期离散的性能。当与非常高的收敛多域光谱搭配方案结合使用时，该技术变得特别有效。通过这种离散化，大小$1000λ\ times1000λ$（$ n = 100 \ mbox {m} $）上的helmholtz问题在15分钟内解决了与GPU加速度的高功率台式机上的6个正确数字的解决方案。
The paper describes a sparse direct solver for the linear systems that arise from the discretization of an elliptic PDE on a two dimensional domain. The solver is designed to reduce communication costs and perform well on GPUs; it uses a two-level framework, which is easier to implement and optimize than traditional multi-frontal schemes based on hierarchical nested dissection orderings. The scheme decomposes the domain into thin subdomains, or "slabs". Within each slab, a local factorization is executed that exploits the geometry of the local domain. A global factorization is then obtained through the LU factorization of a block-tridiagonal reduced coefficient matrix. The solver has complexity $O(N^{5/3})$ for the factorization step, and $O(N^{7/6})$ for each solve once the factorization is completed. The solver described is compatible with a range of different local discretizations, and numerical experiments demonstrate its performance for regular discretizations of rectangular and curved geometries. The technique becomes particularly efficient when combined with very high-order convergent multi-domain spectral collocation schemes. With this discretization, a Helmholtz problem on a domain of size $1000 λ\times 1000 λ$ (for which $N=100 \mbox{M}$) is solved in 15 minutes to 6 correct digits on a high-powered desktop with GPU acceleration.
