Preparing sparse solvers for exascale computing

Anzt, Hartwig; Boman, Erik G.; Falgout, Robert D.; Ghysels, Pieter; Heroux, Michael A.; Li, Xiaoye; McInnes, Lois Curfman; Mills, Richard Tran; Rajamanickam, Sivasankaran; Rupp, Karl; Barry, Smith; Yamazaki, Ichitaro; Yang, Ulrike Meier

doi:10.1098/rsta.2019.0053

Cited by 17 publications

(15 citation statements)

References 54 publications

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“…While the main components of PDE solvers involve sparse linear algebra that are also memorybound due to their low arithmetic intensity [28], optimal design and implementation of solvers for good performance on GPUs [95] remains a challenge that is being currently addressed [96,97]. In particular PETSc's algebraic multigrd (GAMG) can execute the solve phase entirely on the GPUs and there is ongoing development on performing the setup phase on the GPUs as well Fig.…”

Section: Scaling Resultsmentioning

confidence: 99%

Comparison of distributed memory algorithms for X-ray wave propagation in inhomogeneous media

Ali¹,

Du²,

Adams³

et al. 2020

Opt. Express

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Calculations of X-ray wave propagation in large objects are needed for modeling diffractive X-ray optics and for optimization-based approaches to image reconstruction for objects that extend beyond the depth of focus. We describe three methods for calculating wave propagation with large arrays on parallel computing systems with distributed memory: (1) a full-array Fresnel multislice approach, (2) a tiling-based short-distance Fresnel multislice approach, and (3) a finite difference approach. We find that the first approach suffers from internode communication delays when the transverse array size becomes large, while the second and third approaches have similar scaling to large array size problems (with the second approach offering about three times the compute speed).

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Section: Scaling Resultsmentioning

confidence: 99%

Comparison of distributed memory algorithms for X-ray wave propagation in inhomogeneous media

Ali¹,

Du²,

Adams³

et al. 2020

Opt. Express

Self Cite

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“…In this section we analyze performance results obtained by using the smoothers discussed in Table 1 on CPU cores of Piz 3 Refer to https://www.top500.org/lists/top500/2020/11/. Daint.…”

Section: Performance Of the Smoothersmentioning

confidence: 99%

“…where A ∈ R n×n is a very large and sparse matrix and x, b ∈ R n are vectors, is the most time consuming computational kernel in many areas of computational science and engineering, including more recent fields such as data and network analysis. Efficient and scalable methods and software libraries for solving such systems on the current generation of pre-exascale parallel computers are a key technology for high-resolution and high-fidelity simulations and analysis [3].…”

mentioning

confidence: 99%

“…Since the introduction of AMG methods, substantial efforts have been put in the design and implementation of efficient packages for their use as parallel preconditioners. For a detailed description of the most important software projects aimed at preparing parallel linear solvers and preconditioners for the exascale challenge we refer the reader to [3]. For comparison purposes we consider here the Hypre library [24]; Hypre is very well known for its BoomerAMG multigrid preconditioner package [29], which includes many de facto standard AMG preconditioners, and is also interfaced by more comprehensive software frameworks, such as PETSc [1] and Trilinos [30].…”

mentioning

confidence: 99%

“…3 , with homogeneous Dirichlet boundary conditions, K = 1 and unitary right-hand side. The discretization of this problem is obtained by the usual 7-points finite difference stencil.…”

mentioning

confidence: 99%

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AMG Preconditioners for Linear Solvers towards Extreme Scale

D'Ambra¹,

Durastante²,

Filippone³

2021

SIAM J. Sci. Comput.

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Linear solvers for large and sparse systems are a key element of scientific applications, and their efficient implementation is necessary to harness the computational power of current computers. Algebraic MultiGrid (AMG) preconditioners are a popular ingredient of such linear solvers; this is the motivation for the present work where we examine some recent developments in a package of AMG preconditioners to improve efficiency, scalability and robustness on extreme scale problems. The main novelty is the design and implementation of a parallel coarsening algorithm based on aggregation of unknowns employing weighted graph matching techniques; this is a completely automated procedure, requiring no information from the user, and applicable to general symmetric positive definite (s.p.d.) matrices. The new coarsening algorithm improves in terms of numerical scalability at low operator complexity over decoupled aggregation algorithms available in previous releases of the package. The preconditioners package is built on the parallel software framework PSBLAS, which has also been updated to progress towards exascale. We present weak scalability results on one of the most powerful supercomputer in Europe for linear systems with sizes up to O(10 10 ) unknowns.

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Adaptively restarted block Krylov subspace methods with low-synchronization skeletons

Lund

2022

Numer Algor

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With the recent realization of exascale performance by Oak Ridge National Laboratory’s Frontier supercomputer, reducing communication in kernels like QR factorization has become even more imperative. Low-synchronization Gram-Schmidt methods, first introduced in Świrydowicz et al. (Numer. Lin. Alg. Appl. 28(2):e2343, 2020), have been shown to improve the scalability of the Arnoldi method in high-performance distributed computing. Block versions of low-synchronization Gram-Schmidt show further potential for speeding up algorithms, as column-batching allows for maximizing cache usage with matrix-matrix operations. In this work, low-synchronization block Gram-Schmidt variants from Carson et al. (Linear Algebra Appl. 638:150–195, 2022) are transformed into block Arnoldi variants for use in block full orthogonalization methods (BFOM) and block generalized minimal residual methods (BGMRES). An adaptive restarting heuristic is developed to handle instabilities that arise with the increasing condition number of the Krylov basis. The performance, accuracy, and stability of these methods are assessed via a flexible benchmarking tool written in MATLAB. The modularity of the tool additionally permits generalized block inner products, like the global inner product.

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Preparing sparse solvers for exascale computing

Cited by 17 publications

References 54 publications

Comparison of distributed memory algorithms for X-ray wave propagation in inhomogeneous media

Comparison of distributed memory algorithms for X-ray wave propagation in inhomogeneous media

AMG Preconditioners for Linear Solvers towards Extreme Scale

Adaptively restarted block Krylov subspace methods with low-synchronization skeletons

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