Modeling 1D Distributed-Memory Dense Kernels for an Asynchronous Multifrontal Sparse Solver

Amestoy, Patrick; L’Excellent, Jean-Yves; Rouet, François-Henry; Sid-Lakhdar, Wissam M.

doi:10.1007/978-3-319-17353-5_14

Cited by 2 publications

(3 citation statements)

References 14 publications

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“…Amestoy, L’Excellent and Sid-Lakhdar (2014 a ) further notice that, although greatly improving the communication performance, this broadcast scheme breaks the fundamental properties on which they were relying to ensure deadlock-free factorizations. They then propose adaptations of deadlock prevention and avoidance algorithms to the context of asynchronous distributed-memory environments.…”

Section: Multifrontal Methodsmentioning

confidence: 99%

“…Amestoy, L’Excellent, Rouet and Sid-Lakhdar (2014 b ) propose improvements to the one-dimensional asynchronous distributed-memory dense kernels algorithms to improve the scalability of their multifrontal solver. They notice that, in a left-looking approach, the master process produces factorized panels faster at the beginning than at the end of its factorization, thus resulting in improved scheduling between master and slaves.…”

Section: Multifrontal Methodsmentioning

confidence: 99%

See 1 more Smart Citation

A survey of direct methods for sparse linear systems

Davis¹,

Rajamanickam²,

Sid-Lakhdar³

2016

Acta Numerica

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174

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Wilkinson defined a sparse matrix as one with enough zeros that it pays to take advantage of them.1 This informal yet practical definition captures the essence of the goal of direct methods for solving sparse matrix problems. They exploit the sparsity of a matrix to solve problems economically: much faster and using far less memory than if all the entries of a matrix were stored and took part in explicit computations. These methods form the backbone of a wide range of problems in computational science. A glimpse of the breadth of applications relying on sparse solvers can be seen in the origins of matrices in published matrix benchmark collections (Duff and Reid 1979a, Duff, Grimes and Lewis 1989a, Davis and Hu 2011). The goal of this survey article is to impart a working knowledge of the underlying theory and practice of sparse direct methods for solving linear systems and least-squares problems, and to provide an overview of the algorithms, data structures, and software available to solve these problems, so that the reader can both understand the methods and know how best to use them.

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Section: Multifrontal Methodsmentioning

confidence: 99%

Section: Multifrontal Methodsmentioning

confidence: 99%

A survey of direct methods for sparse linear systems

Davis¹,

Rajamanickam²,

Sid-Lakhdar³

2016

Acta Numerica

Self Cite

174

View full text Add to dashboard Cite

show abstract

“…As shown in Figure 8(a), more rows must then be associated to the processors that appear first in the front. Note that we also want to adjust relative size of q and r to balance the workload between the master and each worker by splitting nodes of the separator tree (Amestoy et al, 2001(Amestoy et al, , 2014.…”

Section: Differences Between the Factorization And The Solve Algorithmsmentioning

confidence: 99%

Efficient use of sparsity by direct solvers applied to 3D controlled-source EM problems

et al. 2019

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Controlled-source electromagnetic (CSEM) surveying becomes a widespread method for oil and gaz exploration, which requires fast and efficient software for inverting large-scale EM datasets. In this context, one often needs to solve sparse systems of linear equations with a large number of sparse right-hand sides, each corresponding to a given transmitter position. Sparse direct solvers are very attractive for these problems, especially when combined with low-rank approximations which significantly reduce the complexity and the cost of the factorization. In the case of thousands of right-hand sides, the time spent in the sparse triangular solve tends to dominate the total simulation time and here we propose several approaches to reduce it. A significant reduction is demonstrated for marine CSEM application by utilizing the sparsity of the right-hand sides (RHS) and of the solutions that results from the geometry of the problem. Large gains are achieved by restricting computations at the forward substitution stage to exploit the fact that the RHS matrix might have empty rows (vertical sparsity) and/or empty blocks of columns within a non-empty row (horizontal sparsity). We also adapt the parallel algorithms that were designed for the factorization to solve-oriented algorithms and describe performance optimizations particularly relevant for the very large numbers of right-hand sides of the CSEM application. We show that both the operation count and the elapsed time for the solution phase can be significantly reduced. The total time of CSEM simulation can be divided by approximately a factor of 3 on all the matrices from our set (from 3 to 30 million unknowns, and from 4 to 12 thousands RHSs). Exploitation efficace du creux dans les solveurs directs sur des problèmes d'électromagnétisme 3D à source contrôléeRésumé : L'électromagnétisme à source controlée (CSEM) est une méthode de plus en plus utilisée pour l'exploration du gaz et du pétrole. L'inversion des données électromagnétiques nécessite souvent la résolution de systèmes d'équations linéaires avec un grand nombre de seconds membres creux, chacun correspondant à la position d'une source/d'un émetteur dans l'application. Les solveurs creux directs sont très attrayants pour ce type de problèmes, surtout lorsqu'ils sont combinés avec des approximations de rang faible qui réduisent la complexité et le coût de la factorisation. Comme il peut y avoir des milliers de seconds membres, le coût de la phase de résolution devient alors prédominant. Dans cet article, nous montrons qu'il est possible d'exploiter la géométrie du problème et la structure creuse qui apparaît à la fois dans les seconds membres et dans la matrice des solutions et que cela peut avoir un impact important sur les performances des phases de résolutions triangulaires. Pour cela, nous organisons les calculs de manière à minimiser le nombre d'opérations, tout en traitant les seconds membres par blocs qui tiennent en mémoire et qui visent à conserver au mieux le parallélisme lors de la phae de résolution....

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Modeling 1D Distributed-Memory Dense Kernels for an Asynchronous Multifrontal Sparse Solver

Cited by 2 publications

References 14 publications

A survey of direct methods for sparse linear systems

A survey of direct methods for sparse linear systems

Efficient use of sparsity by direct solvers applied to 3D controlled-source EM problems

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