Divergence constraints are present in the governing equations of numerous physical phenomena, and they usually lead to a Poisson equation whose solution represents a bottleneck in many simulation codes. Algebraic Multigrid (AMG) is arguably the most powerful preconditioner for Poisson's equation, and its effectiveness results from the complementary roles played by the smoother, responsible for damping high-frequency error components, and the coarse-grid correction, which in turn reduces low-frequency modes. This work presents several strategies to make AMG more compute-intensive by leveraging reflection, translational and rotational symmetries. AMGR, our final proposal, does not require boundary conditions to be symmetric, therefore applying to a broad range of academic and industrial configurations. It is based on a multigrid reduction framework that introduces an aggressive coarsening to the multigrid hierarchy, reducing the memory footprint, setup and application costs of the top-level smoother. While preserving AMG's excellent convergence, AMGR allows replacing the standard sparse matrix-vector product with the more compute-intensive sparse matrix-matrix product, yielding significant accelerations. Numerical experiments on industrial CFD applications demonstrated up to 70% speed-ups when solving Poisson's equation with AMGR instead of AMG. Additionally, strong and weak scalability analyses revealed no significant degradation.
A multigrid-reduction-in-time solver with a new two-level convergence for unsteady fractional Laplacian problems
多重网格减少时间（MGRIT）技术已被证明是成功的，在实现更高的运行时加速利用时间上的并行性。本文的目标是开发和分析一个MGRIT算法，使用FCF松弛与时间相关的时间网格传播，寻求非定常分数拉普拉斯问题的有限元近似。文[L.陈河，巴西-地H. Nochetto，E. Ot{á}rola，A. J. Salgado，Math.Comp.85（2016）2583- 2607]被选择为空间解算器。由[B]驱动。S. Southworth，SIAM J. Matrix Anal. 40（2019）564- 608]，我们提供了一个新的时间特征值近似性质，然后推导出一个广义的两级收敛理论，该理论消除了[X. Q. Yue，S. Shu，X. W. Xu，W. P. Bu，K. J. Pan，Comput.数学应用78（2019）3471- 3484]。通过数值计算证实了理论预测的正确性，并证明了收敛上界的精确性。
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。
Multilevel convergence analysis of multigrid-reduction-in-time
University of Stuttgart（斯图加特大学）；University of Colorado at Boulder（科罗拉多大学博尔德分校）；King’s College London（伦敦国王学院）；Lawrence Livermore National Laboratory（劳伦斯利弗莫尔国家实验室）；University of New Mexico（新墨西哥大学）
A high-order Newton multigrid method with a simplified Jacobian for steady-state shallow water equations
School of Mathematics, Nanjing University of Aeronautics and Astronautics（南京航空航天大学数学学院）；Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles (NUAA), MIIT（航空飞行器数学建模与高性能计算重点实验室（工信部））
A parallel Newton multigrid framework for monolithic fluid-structure interactions
Technische Universität München（慕尼黑工业大学）；Otto-von-Guericke Universität Magdeburg（奥托·冯·古里克马格德堡大学）；Interdisciplinary Center for Scientific Computing, Heidelberg University（海德堡大学跨学科科学计算中心）
A multigrid method for the Helmholtz equation with optimized coarse grid corrections
University of Amsterdam（阿姆斯特丹大学）；Jagannath University（贾格纳特大学）；University of Dhaka（达卡大学）；Al Imam Mohammad Ibn Saud Islamic University（伊玛目穆罕默德·本·沙特伊斯兰大学）
