广义Dryja--Smith--Widlund (GDSW) 预处理器是一种两层重叠Schwarz区域分解 (DD) 预处理器，它将经典的一级重叠Schwarz预处理器与能量最小化粗空间相耦合。当用于加速Krylov子空间迭代方法的收敛速度时，GDSW预处理器为求解由各种偏微分方程离散化产生的稀疏线性系统提供了鲁棒性和可扩展性。在本文中，我们提出了FROSch (Fast and Robust Schwarz)，这是一个区域分解求解器包，为CPU和GPU集群实现了GDSW型预处理器。为了提高GPU上的求解器性能，我们使用了一种新颖的分解方法在每个GPU上运行多个MPI进程，降低了求解器的计算和存储成本，并潜在地提高了收敛速度。这使我们能够在使用GPU时获得与仅使用CPU相比具有竞争力或更快的性能。我们在配备NVIDIA V100 GPU的Summit超级计算机上演示了FROSch的性能，其中我们使用NVIDIA Multi-Process Service (MPS) 来实现我们的分解策略。该求解器具有多种算法和实现选择，这为GPU实现带来了机遇和挑战。我们使用不同的求解器选项进行了彻底的实验研究，包括在GPU上对局部重叠子域问题的精确或不精确求解。我们还讨论了使用不完全LU分解和稀疏三角求解的迭代变体作为近似局部求解器的效果，以及使用较低精度计算整个FROSch预处理器的效果。总体而言，使用GPU使求解时间减少了约 $2\times$ 倍，而数值建立时间的GPU加速则取决于求解器选项和局部矩阵大小。
英文摘要：
The generalized Dryja--Smith--Widlund (GDSW) preconditioner is a two-level overlapping Schwarz domain decomposition (DD) preconditioner that couples a classical one-level overlapping Schwarz preconditioner with an energy-minimizing coarse space. When used to accelerate the convergence rate of Krylov subspace iterative methods, the GDSW preconditioner provides robustness and scalability for the solution of sparse linear systems arising from the discretization of a wide range of partial different equations. In this paper, we present FROSch (Fast and Robust Schwarz), a domain decomposition solver package which implements GDSW-type preconditioners for both CPU and GPU clusters. To improve the solver performance on GPUs, we use a novel decomposition to run multiple MPI processes on each GPU, reducing both solver's computational and storage costs and potentially improving the convergence rate. This allowed us to obtain competitive or faster performance using GPUs compared to using CPUs alone. We demonstrate the performance of FROSch on the Summit supercomputer with NVIDIA V100 GPUs, where we used NVIDIA Multi-Process Service (MPS) to implement our decomposition strategy.
The solver has a wide variety of algorithmic and implementation choices, which poses both opportunities and challenges for its GPU implementation. We conduct a thorough experimental study with different solver options including the exact or inexact solution of the local overlapping subdomain problems on a GPU. We also discuss the effect of using the iterative variant of the incomplete LU factorization and sparse-triangular solve as the approximate local solver, and using lower precision for computing the whole FROSch preconditioner. Overall, the solve time was reduced by factors of about $2\times$ using GPUs, while the GPU acceleration of the numerical setup time depend on the solver options and the local matrix sizes.
Overlapping Schwarz Preconditioners for Pose-Graph SLAM in Robotics
Faculty of Mathematics and Computer Science, Technische Universität Bergakademie Freiberg（弗莱贝格矿业工业大学数学与计算机科学学院）；Universitätsrechenzentrum (URZ), Technische Universität Bergakademie Freiberg（弗莱贝格矿业大学计算中心）
Additive Schwarz preconditioner for the general finite volume element discretization of symmetric elliptic problems
University of Warsaw（华沙大学）；Bergen University College（卑尔根大学学院）；University of Bergen（卑尔根大学）；University of South Bohemia（南波希米亚大学）；Institute of Information Theory and Automation of the ASCR（捷克科学院信息论与自动化研究所）
Monolithic and Block Overlapping Schwarz Preconditioners for the Incompressible Navier-Stokes Equations
Delft Institute of Applied Mathematics, Delft University of Technology（代尔夫特理工大学应用数学研究所）；Department of Mathematics and Computer Science, University of Cologne（科隆大学数学与计算机科学系）；Center for Data and Simulation Science, University of Cologne（科隆大学数据与模拟科学中心）
A Class of Iterative Solvers for the Helmholtz Equation: Factorizations, Sweeping Preconditioners, Source Transfer, Single Layer Potentials, Polarized Traces, and Optimized Schwarz Methods
University of Geneva（日内瓦大学）；Zhejiang Ocean University（浙江海洋大学）
Neutron transport criticality calculations using a parallel monolithic multilevel Schwarz preconditioner together with a nonlinear diffusion acceleration method
Block Schwarz methods and preconditioning strategies using Generalized locally Toeplitz tools - part I: analysis of the preconditioners and numerical validation
Two-level hybrid Schwarz preconditioners with preasymptotic piecewise-polynomial coarse spaces for the high-frequency Helmholtz equation
University College London（伦敦大学学院）；University of Bath（巴斯大学）；Sorbonne Université, Université Paris Cité, CNRS, INRIA, Laboratoire Jacques-Louis Lions（索邦大学，巴黎西岱大学，法国国家科学研究中心，法国国家信息与自动化研究所，雅克-路易·利翁斯实验室）
A nonoverlapping spectral additive Schwarz method for interior penalty discontinuous Galerkin discretizations of Anisotropic Elliptic Problems
Massachusetts Institute of Technology（麻省理工学院）；University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；State University of New York Polytechnic Institute（纽约州立理工大学）；Worcester Polytechnic Institute（伍斯特理工学院）
Nonlinear Preconditioning: How to use a Nonlinear Schwarz Method to Precondition Newton's Method
University of Strathclyde（斯特拉斯克莱德大学）；Université de Genève（日内瓦大学）；Université Nice Sophia Antipolis（尼斯索菲亚安蒂波利斯大学）；Inria（法国国家数字科学研究所）；Hong Kong Baptist University（香港浸会大学）
Discrete Edge Multiscale Space based Hybrid Schwarz Preconditioner for High-Frequency Helmholtz Problems
Eastern Institute for Advanced Study, Eastern Institute of Technology（宁波东方理工大学高级研究院）；School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳）理工学院）；Department of Mathematics, The University of Hong Kong（香港大学数学系）；Department of Mathematics, Purdue University（普渡大学数学系）
An algebraic multiscale framework for two-level overlapping Schwarz preconditioners
Delft Institute of Applied Mathematics, Delft University of Technology（代尔夫特应用数学研究所，代尔夫特理工大学）；Department of Geoscience and Engineering, Delft University of Technology（地球科学与工程学院，代尔夫特理工大学）
Additive average Schwarz with adaptive coarse spaces: scalable algorithms for multiscale problems
Faculty of Mathematics, Univeristy of Warsaw（华沙大学数学系）；Department of Computing, Mathematics and Physics, Western Norway University of Applied Sciences（西挪威应用科学大学计算、数学与物理系）
