Exploiting nested task-parallelism in the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mi mathvariant="bold-script">H</mml:mi></mml:math>-LU factorization

Carratalá-Sáez, Rocío; Christophersen, Sven; Aliaga, José I.; Beltrán, Vicenç; Börm, Steffen; Quintana–Ort́ı, Enrique S.

doi:10.1016/j.jocs.2019.02.004

Cited by 10 publications

(3 citation statements)

References 12 publications

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“…In particular, Gillman et al [19] propose an inversion scheme based on the Sherman-Morrison-Woodbury formula. However, to our knowledge, the algorithms presented here are the first of the literature to exploit an LU factorization for BLR 2 matrices, similarly to LU-based algorithms for other matrix formats, such as BLR (Algorithm 2.1, [5]), H [14,20], HSS [15], and H 2 [26] At each step k, we first compute the LU factorization of the diagonal block A kk = L kk U kk (line 8). Then, we perform the triangular solves…”

Section: Lu Factorization and Solution Algorithmsmentioning

confidence: 99%

Block Low-Rank Matrices with Shared Bases: Potential and Limitations of the BLR$^2$ Format

Ashcraft¹,

Buttari²,

Mary³

2021

SIAM J. Matrix Anal. Appl.

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We investigate a special class of data sparse rank-structured matrices that combine a flat block low-rank (BLR) partitioning with the use of shared (called nested in the hierarchical case) bases. This format is to H 2 matrices what BLR is to H matrices: we therefore call it the BLR 2 matrix format. We present algorithms for the construction and LU factorization of BLR 2 matrices, and perform their cost analysis-both asymptotically and for a fixed problem size. With weak admissibility, BLR 2 matrices reduce to block separable matrices (the flat version of HBS/HSS). Our analysis and numerical experiments reveal some limitations of BLR 2 matrices with weak admissibility, which we propose to overcome with two approaches: strong admissibility, and the use of multiple shared bases per row and column.

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Section: Lu Factorization and Solution Algorithmsmentioning

confidence: 99%

Block Low-Rank Matrices with Shared Bases: Potential and Limitations of the BLR$^2$ Format

Ashcraft¹,

Buttari²,

Mary³

2021

SIAM J. Matrix Anal. Appl.

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“…A weak dependency from a parent task to a sub-task does not require the parent task to wait for the sub-task to nish as would be needed in OmpSs (or OpenMP) thereby avoiding unnecessary task synchronisation. is extension would permit the implemention of nested functions with ne grained dependencies as the Hmatrix arithmetic makes use of and was used in [10] to fully implement task-based H-matrix arithmetic. e presented numerical results demonstrate that the technique has some potential but needs further optimizations to be e cient for a wide range of H-matrix structures.…”

Section: Introductionmentioning

confidence: 99%

Semi-Automatic Task Graph Construction for $\mathcal{H}$-Matrix Arithmetic

Börm¹,

Christophersen²,

Kriemann³

2019

Preprint

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A new method to construct task graphs for Hmatrix arithmetic is introduced, which uses the information associated with all tasks of the standard recursive H-matrix algorithms, e.g., the block index set of the matrix blocks involved in the computation. Task re nement, i.e., the replacement of tasks by sub-computations, is then used to proceed in the H-matrix hierarchy until the matrix blocks containing the actual matrix data are reached. is process is a natural extension of the classical, recursive way in which H-matrix arithmetic is de ned and thereby simpli es the e cient usage of many-core systems. Examples for standard and accumulator based H-arithmetic are shown for model problems with di erent block structures.

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“…For example, when designing linear algebra solvers based on low-rank approximation algorithms, it is almost impossible to predict the right DAG to ensure good numerical accuracy. [8][9][10][11][12] In general, most modern task-based runtime systems suffer from a lack of dynamism in task-graph generation. Some programming models, such as, 7,13,14 support just-in-time DAG submission, but the generation either follows static rules or requires a significant amount of programming effort.…”

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confidence: 99%

Programming heterogeneous architectures using hierarchical tasks

Faverge

Furmento

Guermouche

et al. 2023

Concurrency and Computation

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SummaryTask‐based systems have become popular due to their ability to utilize the computational power of complex heterogeneous systems. A typical programming model used is the Sequential Task Flow (STF) model, which unfortunately only supports static task graphs. This can result in submission overhead and a static task graph that is not well‐suited for execution on heterogeneous systems. A common approach is to find a balance between the granularity needed for accelerator devices and the granularity required by CPU cores to achieve optimal performance. To address these issues, we have extended the STF model in the StarPU runtime system by introducing the concept of hierarchical tasks. This allows for a more dynamic task graph and, when combined with an automatic data manager, it is possible to adjust granularity at runtime to best match the targeted computing resource. That data manager makes it possible to switch between various data layout without programmer input and allows us to enforce the correctness of the DAG as hierarchical tasks alter it during runtime. Additionally, submission overhead is reduced by using large‐grain hierarchical tasks, as the submission process can now be done in parallel. We have shown that the hierarchical task model is correct and have conducted an early evaluation on shared memory heterogeneous systems using the Chameleon dense linear algebra library.

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Exploiting nested task-parallelism in the H-LU factorization

Cited by 10 publications

References 12 publications

Block Low-Rank Matrices with Shared Bases: Potential and Limitations of the BLR$^2$ Format

Block Low-Rank Matrices with Shared Bases: Potential and Limitations of the BLR$^2$ Format

Semi-Automatic Task Graph Construction for $\mathcal{H}$-Matrix Arithmetic

Programming heterogeneous architectures using hierarchical tasks

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