Mixed-Precision GPU-Multigrid Solvers with Strong Smoothers

2010 ◽  
pp. 131-147 ◽  
Author(s):  
Dominik Goddeke ◽  
Robert Strzodka
Keyword(s):  
Multigrid Solvers ◽  
Mixed Precision

scholarly journals Discretization-Error-Accurate Mixed-Precision Multigrid Solvers

2021 ◽  
pp. S420-S447 ◽  
Author(s):  
Rasmus Tamstorf ◽  
Joseph Benzaken ◽  
Stephen F. McCormick
Keyword(s):  
Discretization Error ◽  
Multigrid Solvers ◽  
Mixed Precision

Development of O(Nm2) preconditioned multigrid solvers for Euler and Navier-Stokes equations

AIAA Journal ◽  
2000 ◽  
Vol 38 ◽  
pp. 717-720
Author(s):  
Jack R. Edwards ◽  
James L. Thomas
Keyword(s):  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Multigrid Solvers

Adaptive Precision Block-Jacobi for High Performance Preconditioning in the Ginkgo Linear Algebra Software

2021 ◽  
Vol 47 (2) ◽  
pp. 1-28
Author(s):  
Goran Flegar ◽  
Hartwig Anzt ◽  
Terry Cojean ◽  
Enrique S. Quintana-Ortí
Keyword(s):  
Linear Algebra ◽  
Graphics Processing Units ◽  
High Performance ◽  
Numerical Algorithms ◽  
Mixed Precision ◽  
Before And After ◽  
Memory Accesses ◽  
Specialized Hardware ◽  
The Individual ◽  
Graphics Processing

The use of mixed precision in numerical algorithms is a promising strategy for accelerating scientific applications. In particular, the adoption of specialized hardware and data formats for low-precision arithmetic in high-end GPUs (graphics processing units) has motivated numerous efforts aiming at carefully reducing the working precision in order to speed up the computations. For algorithms whose performance is bound by the memory bandwidth, the idea of compressing its data before (and after) memory accesses has received considerable attention. One idea is to store an approximate operator–like a preconditioner–in lower than working precision hopefully without impacting the algorithm output. We realize the first high-performance implementation of an adaptive precision block-Jacobi preconditioner which selects the precision format used to store the preconditioner data on-the-fly, taking into account the numerical properties of the individual preconditioner blocks. We implement the adaptive block-Jacobi preconditioner as production-ready functionality in the Ginkgo linear algebra library, considering not only the precision formats that are part of the IEEE standard, but also customized formats which optimize the length of the exponent and significand to the characteristics of the preconditioner blocks. Experiments run on a state-of-the-art GPU accelerator show that our implementation offers attractive runtime savings.

Rigorous floating-point mixed-precision tuning

2017 ◽  
Author(s):  
Wei-Fan Chiang ◽  
Mark Baranowski ◽  
Ian Briggs ◽  
Alexey Solovyev ◽  
Ganesh Gopalakrishnan ◽  
...  
Keyword(s):  
Floating Point ◽  
Mixed Precision

scholarly journals GRAM

2021 ◽  
Vol 18 (2) ◽  
pp. 1-24
Author(s):  
Nhut-Minh Ho ◽  
Himeshi De silva ◽  
Weng-Fai Wong
Keyword(s):  
Performance Improvement ◽  
Trade Off ◽  
Accuracy Requirement ◽  
Output Error ◽  
Fine Grain ◽  
Mixed Precision ◽  
And Performance ◽  
Effective Use

This article presents GRAM (<underline>G</underline>PU-based <underline>R</underline>untime <underline>A</underline>daption for <underline>M</underline>ixed-precision) a framework for the effective use of mixed precision arithmetic for CUDA programs. Our method provides a fine-grain tradeoff between output error and performance. It can create many variants that satisfy different accuracy requirements by assigning different groups of threads to different precision levels adaptively at runtime . To widen the range of applications that can benefit from its approximation, GRAM comes with an optional half-precision approximate math library. Using GRAM, we can trade off precision for any performance improvement of up to 540%, depending on the application and accuracy requirement.

Increased space-parallelism via time-simultaneous Newton-multigrid methods for nonstationary nonlinear PDE problems

2021 ◽  
Vol 35 (3) ◽  
pp. 211-225
Author(s):  
Jonas Dünnebacke ◽  
Stefan Turek ◽  
Christoph Lohmann ◽  
Andriy Sokolov ◽  
Peter Zajac
Keyword(s):  
Krylov Subspace ◽  
Solution Procedure ◽  
Grid Size ◽  
Waveform Relaxation ◽  
Test Problems ◽  
Subspace Method ◽  
Multigrid Algorithm ◽  
Multigrid Solvers ◽  
Large Systems ◽  
Linear Pdes

We discuss how “parallel-in-space & simultaneous-in-time” Newton-multigrid approaches can be designed which improve the scaling behavior of the spatial parallelism by reducing the latency costs. The idea is to solve many time steps at once and therefore solving fewer but larger systems. These large systems are reordered and interpreted as a space-only problem leading to multigrid algorithm with semi-coarsening in space and line smoothing in time direction. The smoother is further improved by embedding it as a preconditioner in a Krylov subspace method. As a prototypical application, we concentrate on scalar partial differential equations (PDEs) with up to many thousands of time steps which are discretized in time, resp., space by finite difference, resp., finite element methods. For linear PDEs, the resulting method is closely related to multigrid waveform relaxation and its theoretical framework. In our parabolic test problems the numerical behavior of this multigrid approach is robust w.r.t. the spatial and temporal grid size and the number of simultaneously treated time steps. Moreover, we illustrate how corresponding time-simultaneous fixed-point and Newton-type solvers can be derived for nonlinear nonstationary problems that require the described solution of linearized problems in each outer nonlinear step. As the main result, we are able to generate much larger problem sizes to be treated by a large number of cores so that the combination of the robustly scaling multigrid solvers together with a larger degree of parallelism allows a faster solution procedure for nonstationary problems.

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