Book ChapterDOI
Iterative Sparse Triangular Solves for Preconditioning
Hartwig Anzt,Edmond Chow,Jack Dongarra +2 more
- pp 650-661
TLDR
This work proposes using an iterative approach for solving sparse triangular systems when an approximation is suitable, and demonstrates the performance gains that this approach can have on GPUs in the context of solving sparse linear systems with a preconditioned Krylov subspace method.Abstract:
Sparse triangular solvers are typically parallelized using level-scheduling techniques, but parallel efficiency is poor on high-throughput architectures like GPUs. We propose using an iterative approach for solving sparse triangular systems when an approximation is suitable. This approach will not work for all problems, but can be successful for sparse triangular matrices arising from incomplete factorizations, where an approximate solution is acceptable. We demonstrate the performance gains that this approach can have on GPUs in the context of solving sparse linear systems with a preconditioned Krylov subspace method. We also illustrate the effect of using asynchronous iterations.read more
Citations
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Proceedings ArticleDOI
10M-core scalable fully-implicit solver for nonhydrostatic atmospheric dynamics
Chao Yang,Wei Xue,Haohuan Fu,Hongtao You,Xinliang Wang,Yulong Ao,Fangfang Liu,Lin Gan,Ping Xu,Lanning Wang,Guangwen Yang,Weimin Zheng +11 more
TL;DR: An ultra-scalable fully-implicit solver is developed for stiff time-dependent problems arising from the hyperbolic conservation laws in nonhydrostatic atmospheric dynamics and a highly efficient hybrid domain-decomposed multigrid preconditioner is proposed that can greatly accelerate the convergence rate at the extreme scale.
Book ChapterDOI
A Synchronization-Free Algorithm for Parallel Sparse Triangular Solves
TL;DR: This paper proposes a novel approach for SpTRSV in which the ordering between components is naturally enforced within the solution stage, and is an order of magnitude faster for the preprocessing stage than existing methods.
Journal ArticleDOI
Incomplete Sparse Approximate Inverses for Parallel Preconditioning
Hartwig Anzt,Hartwig Anzt,Thomas Huckle,Jürgen Bräckle,Jack Dongarra,Jack Dongarra,Jack Dongarra +6 more
TL;DR: A new preconditioning method that can be seen as a generalization of block-Jacobi methods, or as a simplification of the sparse approximate inverse (SAI) preconditionsers is proposed.
Journal ArticleDOI
Fast synchronization‐free algorithms for parallel sparse triangular solves with multiple right‐hand sides
TL;DR: Novel approaches for SpTRSV and SpTRSM in which the ordering between components is naturally enforced within the solution stage are proposed, so the cost for preprocessing can be greatly reduced, and the synchronizations between sets are completely eliminated.
Journal ArticleDOI
Using Jacobi iterations and blocking for solving sparse triangular systems in incomplete factorization preconditioning
Edmond Chow,Hartwig Anzt,Hartwig Anzt,Jennifer A. Scott,Jack Dongarra,Jack Dongarra,Jack Dongarra +6 more
TL;DR: It is shown that by using block Jacobi iterations, the range of problems for which such an approach can be effective is extended, and it is essential for the blocking to be cognizant of the matrix structure.
References
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Book
Iterative Methods for Sparse Linear Systems
TL;DR: This chapter discusses methods related to the normal equations of linear algebra, and some of the techniques used in this chapter were derived from previous chapters of this book.
Journal ArticleDOI
The university of Florida sparse matrix collection
Timothy A. Davis,Yifan Hu +1 more
TL;DR: The University of Florida Sparse Matrix Collection, a large and actively growing set of sparse matrices that arise in real applications, is described and a new multilevel coarsening scheme is proposed to facilitate this task.
Journal ArticleDOI
A flexible inner-outer preconditioned GMRES algorithm
TL;DR: A variant of the GMRES algorithm is presented that allows changes in the preconditioning at every step, resulting in a result of the flexibility of the new variant that any iterative method can be used as a preconditionser.
Journal ArticleDOI
The effect of ordering on preconditioned conjugate gradients
Iain S. Duff,Gérard Meurant +1 more
TL;DR: It is shown empirically that there can be a significant difference in the number of iterations required by the conjugate gradient method and reasons for this marked difference in performance are suggested.
Journal ArticleDOI
A comparative study of sparse approximate inverse preconditioners
Michele Benzi,Miroslav Tůma +1 more
TL;DR: A number of recently proposed preconditioning techniques based on sparse approximate inverses are considered, and an experimental comparison performed on one processor of a Cray C98 vector computer using sparse matrices from a variety of applications is presented.