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Performance and Scalability of Hierarchical Hybrid Multigrid Solvers for Stokes Systems

Björn Gmeiner, Ulrich Rüde, Holger Stengel, Christian Waluga, Barbara Wohlmuth
2015 SIAM Journal on Scientific Computing  
of hierarchical hybrid grids.  ...  The design of our fast multigrid solver is guided by an innovative performance analysis for the computational kernels in combination with a quantification of the communication overhead.  ...  We are grateful to the Jülich Supercomputing Center and the Leibniz Rechenzentrum for providing computational resources.  ... 
doi:10.1137/130941353 fatcat:zd5m3ab7fnat5chvfrfpzwmouq

An extreme-scale implicit solver for complex PDEs

Johann Rudi, Omar Ghattas, A. Cristiano I. Malossi, Tobin Isaac, Georg Stadler, Michael Gurnis, Peter W. J. Staar, Yves Ineichen, Costas Bekas, Alessandro Curioni
2015 Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis on - SC '15  
Here we present a new implicit solver that exhibits optimal algorithmic performance and is capable of extreme scaling for hard PDE problems, such as mantle convection.  ...  accuracy, hybrid spectral/geometric/algebraic multigrid, and novel Schur-complement preconditioning.  ...  This hybrid multigrid setup sits at the core of our nonlinear solver and thus a careful design of the intergrid transfer operators was critical for efficiency and performance.  ... 
doi:10.1145/2807591.2807675 dblp:conf/sc/RudiMISGSIBCG15 fatcat:vqy3ko3x35g4rbxsdbaxrkzine

Textbook efficiency: massively parallel matrix-free multigrid for the Stokes system [article]

Nils Kohl, Ulrich Rüde
2020 arXiv   pre-print
We employ textbook multigrid efficiency (TME), as introduced by Achi Brandt, to construct an asymptotically optimal monolithic multigrid solver for the Stokes system.  ...  The geometric multigrid solver builds upon the concept of hierarchical hybrid grids (HHG), which is extended to higher-order finite-element discretizations, and a corresponding matrix-free implementation  ...  Acknowledgements The authors gratefully acknowledge the Gauss Centre for Supercomputing e.V.  ... 
arXiv:2010.13513v1 fatcat:5efjox5ka5dkznt2dyxljcq6ua

High Resolution Aerospace Applications Using the NASA Columbia Supercomputer

Dimitri J. Mavriplis, Michael J. Aftosmis, Marsha Berger
2007 The international journal of high performance computing applications  
These packages include both a high-fidelity, unstructured, Reynolds-averaged Navier-Stokes solver, and a fully-automated inviscid flow package for cut-cell Cartesian grids.  ...  Both packages are industrial-level codes designed for complex geometry and incorporate customized multigrid solution algorithms.  ...  Since the system is not cache-coherent across all 4 of these nodes and the solver module does not have a hybrid OpenMP+MPI build mode, performance was evaluated using MPI only.  ... 
doi:10.1177/1094342006074872 fatcat:cxzxcdmlgjgetarekd2j57bttu

Fast multipole preconditioners for sparse matrices arising from elliptic equations

Huda Ibeid, Rio Yokota, Jennifer Pestana, David Keyes
2017 Computing and Visualization in Science  
However, the convergence of these hierarchical solvers can be fragile with respect to coefficient distribution in the second-order term, and, if present, with respect to the first-order and zeroth-order  ...  It is capable of algebraic convergence rates down to the truncation error of the discretized PDE comparable to those of multigrid methods, and it offers potentially superior multicore and distributed memory  ...  We thank Dave Hewett, David May, Andy Wathen, and Ulrike Yang for helpful discussions and comments and are indebted to Nathan Collier for his help with the PetIGA framework.  ... 
doi:10.1007/s00791-017-0287-5 fatcat:vmacwxjeu5gzpg3euyzfp2rd6a

A quantitative performance analysis for Stokes solvers at the extreme scale [article]

Björn Gmeiner and Markus Huber and Lorenz John and Ulrich Rüde and Barbara Wohlmuth
2015 arXiv   pre-print
Three parallel iterative solvers for the Stokes system, discretized by low order tetrahedral elements, are compared with respect to their numerical efficiency and their scalability running on up to 786  ...  Brandt's notion of "textbook multigrid efficiency" is employed to study the algorithmic performance of iterative solvers.  ...  The last three authors gratefully acknowledge the hospitality and support of the Institute for Mathematical Sciences of the National University of Singapore where part of this work was performed.  ... 
arXiv:1511.02134v1 fatcat:ilorgjuqqjg3ddruso7shcvjnu

Fast Multipole Preconditioners for Sparse Matrices Arising from Elliptic Equations [article]

Huda Ibeid, Rio Yokota, Jennifer Pestana, David Keyes
2016 arXiv   pre-print
Compared with multigrid methods, it is capable of comparable algebraic convergence rates down to the truncation error of the discretized PDE, and it offers potentially superior multicore and distributed  ...  Among optimal hierarchical algorithms for the computational solution of elliptic problems, the Fast Multipole Method (FMM) stands out for its adaptability to emerging architectures, having high arithmetic  ...  We thank Dave Hewett, David May, Andy Wathen, and Ulrike Yang for helpful discussions and comments and are indebted to Nathan Collier for his help with the PetIGA framework.  ... 
arXiv:1308.3339v4 fatcat:skniridmdvfolbfpbwhga3wnl4

A Scalable and Modular Software Architecture for Finite Elements on Hierarchical Hybrid Grids [article]

Nils Kohl, Dominik Thönnes, Daniel Drzisga, Dominik Bartuschat, and Ulrich Rüde
2018 arXiv   pre-print
Example scenarios with coupled systems of PDEs show the applicability of the concepts to performing geophysical simulations.  ...  Combining an unstructured topology with structured grid refinement facilitates high geometric adaptability and matrix-free multigrid implementations with excellent performance.  ...  Acknowledgements This work was partly supported by the German Research Foundation through the Priority Programme 1648 "Software for Exascale Computing" (SPPEXA) and by grant WO671/11-1.  ... 
arXiv:1805.10167v1 fatcat:uxexq24lgjdgxp66mtvkyir5zy

TerraNeo—Mantle Convection Beyond a Trillion Degrees of Freedom [chapter]

Simon Bauer, Hans-Peter Bunge, Daniel Drzisga, Siavash Ghelichkhan, Markus Huber, Nils Kohl, Marcus Mohr, Ulrich Rüde, Dominik Thönnes, Barbara Wohlmuth
2020 Lecture Notes in Computational Science and Engineering  
This contribution reports on the TerraNeo project which delivered novel matrix-free geometric multigrid solvers for the Stokes system that forms the core of mantle convection models.  ...  In TerraNeo the hierarchical hybrid grids paradigm was employed to demonstrate that scalability can be achieved when solving the Stokes system with more than ten trillion (1.1 · 10 13 ) degrees of freedom  ...  Stokes Solvers and Performance Multigrid Approaches for the Stokes System In this subsection we briefly review different classes of solvers for Stokes type systems involving a multigrid component.  ... 
doi:10.1007/978-3-030-47956-5_19 fatcat:4jhueyv5qrhfdkmpfxxv5lsmqa

GPU acceleration of an unmodified parallel finite element Navier-Stokes solver

Dominik Goddeke, Sven H.M. Buijssen, Hilmar Wobker, Stefan Turek
2009 2009 International Conference on High Performance Computing & Simulation  
In this paper we explore the limitations of our approach by accelerating a Navier-Stokes solver.  ...  We have previously suggested a minimally invasive approach to include hardware accelerators into an existing large-scale parallel finite element PDE solver toolkit, and implemented it into our software  ...  ACKNOWLEDGEMENTS The authors would like to thank Christian Becker and the co-developers of FEAST.  ... 
doi:10.1109/hpcsim.2009.5191718 dblp:conf/ieeehpcs/GoddekeBWT09 fatcat:mm4tch3fjnc3vgmh7hxkdnrq6u

FEAST-realization of hardware-oriented numerics for HPC simulations with finite elements

Stefan Turek, Dominik Göddeke, Christian Becker, Sven H. M. Buijssen, Hilmar Wobker
2010 Concurrency and Computation  
FEAST (Finite Element Analysis & Solutions Tools) is a Finite Element based solver toolkit for the simulation of PDE problems on parallel HPC systems which implements the concept of 'hardware-oriented  ...  We demonstrate good performance and weak and strong scalability for the prototypical Poisson problem and more challenging applications from solid mechanics and fluid dynamics.  ...  Thanks to NVIDIA for donating hardware that was used in the development of Feast's GPU backend.  ... 
doi:10.1002/cpe.1584 fatcat:6omy4woanrbn7dqigs3q4ql5hq

Chaotic multigrid methods for the solution of elliptic equations

J. Hawkes, G. Vaz, A.B. Phillips, C.M. Klaij, S.J. Cox, S.R. Turnock
2019 Computer Physics Communications  
The chaotic-cycle multigrid shows good scalability and numerical performance compared to classical V-, W-and F-cycles.  ...  On 2048 cores the chaotic-cycle multigrid solver performs up to 7.7× faster than Flexible-GMRES and 13.3× faster than classical V-cycle multigrid.  ...  Acknowledgements Thousands of simulations have been performed to obtain the results presented herein, and thousands more during development and preliminary investigations.  ... 
doi:10.1016/j.cpc.2018.10.031 fatcat:nr7gsm4dirbmdlyzrl6ffvewe4

Extreme-scale Multigrid Components within PETSc [article]

Dave A. May, Patrick Sanan, Karl Rupp, Matthew G. Knepley, Barry F. Smith
2016 arXiv   pre-print
Multilevel preconditioners represent a family of scalable techniques for solving discrete PDEs of this type and thus are the method of choice for high-resolution simulations.  ...  The scalability and time-to-solution of massively parallel multilevel preconditioners can be adversely effected by using a coarse-level solver with sub-optimal algorithmic complexity.  ...  , and perform publicly and display publicly, by or on behalf of the Government.  ... 
arXiv:1604.07163v1 fatcat:gpnszozdfzhepjv7xrkqmdvj7y

A Multigrid Preconditioner for Spatially Adaptive High-order Meshless Method on Fluid-solid Interaction Problems [article]

Zisheng Ye, Xiaozhe Hu, Wenxiao Pan
2022 arXiv   pre-print
For constructing the interpolation and restriction operators - the key ingredients of the multigrid preconditioner, we utilize the geometric information of hierarchical sets of GMLS nodes generated in  ...  Through numerical examples with the inclusion of different numbers and shapes of solid bodies, we demonstrate the performance and assess the scalability of the designed preconditioner.  ...  The linear system resulting from the adaptive GMLS discretization is solved by the proposed scalable linear solver with multigrid preconditioner.  ... 
arXiv:2203.15908v1 fatcat:xuvjyzgzl5d5zpgbo6skgeiiha

Parallel geometric-algebraic multigrid on unstructured forests of octrees

Hari Sundar, George Biros, Carsten Burstedde, Johann Rudi, Omar Ghattas, Georg Stadler
2012 2012 International Conference for High Performance Computing, Networking, Storage and Analysis  
We use geometric multigrid (GMG) for each of the octrees and algebraic multigrid (AMG) as the coarse grid solver.  ...  We present weak and strong scaling results for the 3D variablecoefficient Poisson problem that demonstrate high parallel scalability.  ...  ACKNOWLEDGMENT The authors would like to thank Tobin Isaac for useful discussions and for providing the mesh for the Antarctic ice sheet. Support for this work was provided by: the U.S.  ... 
doi:10.1109/sc.2012.91 dblp:conf/sc/SundarBBRGS12 fatcat:o67eaqhs2vcbblgfr2ulonkp7m
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