We propose a path cover adaptive algebraic multigrid (PC-$α$AMG) method for solving linear systems of weighted graph Laplacians and can also be applied to discretized second order elliptic partial differential equations. The PC-$α$AMG is based on unsmoothed aggregation AMG (UA-AMG). To preserve the structure of smooth error down to the coarse levels, we approximate the level sets of the smooth error by first forming vertex-disjoint path cover with paths following the level sets. The aggregations are then formed by matching along the paths in the path cover. In such manner, we are able to build a multilevel structure at a low computational cost. The proposed PC-$α$AMG provides a mechanism to efficiently re-build the multilevel hierarchy during the iterations and leads to a fast nonlinear multilevel algorithm. Traditionally, UA-AMG requires more sophisticated cycling techniques, such as AMLI-cycle or K-cycle, but as our numerical results show, the PC-$α$AMG proposed here leads to nearly optimal standard V-cycle algorithm for solving linear systems with weighted graph Laplacians. Numerical experiments for some real world graph problems also demonstrate PC-$α$AMG's effectiveness and robustness, especially for ill-conditioned graphs.
Aspects of an adaptive finite element method for the fractional Laplacian: a priori and a posteriori error estimates, efficient implementation and multigrid solver
Brown University（布朗大学）；Oak Ridge National Laboratory（橡树岭国家实验室）
Fully Finite Element Adaptive Algebraic Multigrid Method for Time-Space Caputo-Riesz Fractional Diffusion Equations
School of Mathematics and Computational Science, Xiangtan University（湘潭大学数学与计算科学学院）；Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University（湘潭大学湖南省科学与工程计算重点实验室）
Solving the (Navier-)Stokes equations with space and time adaptivity using deal.II
Technical University of Berlin（柏林工业大学）；Charles University（查理大学）；Ruhr University Bochum（波鸿鲁尔大学）；Otto von Guericke University Magdeburg（马格德堡奥托·冯·格里克大学）；Weierstrass Institute for Applied Analysis and Stochastics（魏尔斯特拉斯应用分析与随机性研究所）
MGARD: A multigrid framework for high-performance, error-controlled data compression and refactoring
Oak Ridge National Laboratory（橡树岭国家实验室）；University of Alabama at Birmingham（阿拉巴马大学伯明翰分校）；University of Wisconsin Eau Claire（威斯康星大学欧克莱尔分校）；University of Kentucky（肯塔基大学）；University of Florida（佛罗里达大学）；Georgia State University（佐治亚州立大学）；New Jersey Institute of Technology（新泽西理工学院）
本文介绍 MGARD 多重网格框架，通过多重网格自适应降维实现浮点科学数据的高性能压缩与精确误差控制，并支持多架构统一 API 与 GPU 优化。
Kernel-learning parameter prediction and evaluation in algebraic multigrid method for several PDEs
Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education, School of Mathematics and Computational Science, Xiangtan University（湘潭大学数学与计算科学学院）
Goal-Oriented Adaptivity and Multilevel Preconditioning for the Poisson-Boltzmann Equation
TOBB University of Economics and Technology（TOBB经济与技术大学）；Louisiana State University（路易斯安那州立大学）；Sandia National Laboratories（桑迪亚国家实验室）；University of California San Diego（加州大学圣地亚哥分校）
Time-space Finite Element Adaptive AMG for Multi-term Time Fractional Advection Diffusion Equations
School of Mathematics and Computational Science, Xiangtan University（湘潭大学数学与计算科学学院）；Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University（湘潭大学湖南省计算与模拟科学重点实验室）
