Distributed $${{\mathcal H}^2}$$ -matrices for non-local operators

Börm, Steffen; Bendoraityte, Joana

doi:10.1007/s00791-008-0095-z

Cited by 12 publications

(3 citation statements)

References 17 publications

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“…For the 2D tests, the problem size is 2 20 . With P = 8 GPUs, the local problem size is only p N = 2 17 and results in an efficiency reduction to near 50%. On 32 GPUs, the local problem size is p N = 2 15 and the limit of strong scalability is essentially reached: there is very little local work to do and the whole operation takes a few ms, spent mostly in communications.…”

Section: Strong Scalabilitymentioning

confidence: 99%

“…A high quality software for distributed-memory environments that targets large scale problems is STRUMPACK [5]. Distributed memory algorithms for constructing and operating on these matrices were proposed in [17,37,50]. Dynamic run-time systems to manage the scheduling of the operations of the irregular hierarchical matrix structure in a more convenient fashion through explicit task graphs are presented in [7,18].…”

Section: Introductionmentioning

confidence: 99%

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H2Opus: A distributed-memory multi-GPU software package for non-local operators

Zampini¹,

Boukaram²,

Turkiyyah³

et al. 2021

Preprint

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Hierarchical H 2 -matrices are asymptotically optimal representations for the discretizations of non-local operators such as those arising in integral equations or from kernel functions. Their O(N ) complexity in both memory and operator application makes them particularly suited for largescale problems. As a result, there is a need for software that provides support for distributed operations on these matrices to allow large-scale problems to be represented. In this paper, we present high-performance, distributed-memory GPU-accelerated algorithms and implementations for matrix-vector multiplication and matrix recompression of hierarchical matrices in the H 2 format.The algorithms are a new module of H2Opus, a performance-oriented package that supports a broad variety of H 2 matrix operations on CPUs and GPUs. Performance in the distributed GPU setting is achieved by marshaling the tree data of the hierarchical matrix representation to allow batched kernels to be executed on the individual GPUs. MPI is used for inter-process communication. We optimize the communication data volume and hide much of the communication cost with local compute phases of the algorithms. Results show near-ideal scalability up to 1024 NVIDIA V100 GPUs on Summit, with performance exceeding 2.3 Tflop/s/GPU for the matrix-vector multiplication, and 670 Gflops/s/GPU for matrix compression, which involves batched QR and SVD operations.We illustrate the flexibility and efficiency of the library by solving a 2D variable diffusivity integral fractional diffusion problem with an algebraic multigridpreconditioned Krylov solver and demonstrate scalability up to 16M degrees of freedom problems on 64 GPUs.

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Section: Strong Scalabilitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

H2Opus: A distributed-memory multi-GPU software package for non-local operators

Zampini¹,

Boukaram²,

Turkiyyah³

et al. 2021

Preprint

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“…For H 2 -matrices, which are special variants of H-matrices, a parallel algorithm on distributed memory clusters was proposed in Ref. [17]. The algorithm was implemented using flat-MPI approach, and the parallel scalability was also evaluated on a small cluster system with 16 single-CPU machines.…”

Section: ) To O(n) ∼ O(n Log N)mentioning

confidence: 99%

Parallel Hierarchical Matrices with Adaptive Cross Approximation on Symmetric Multiprocessing Clusters

Ida

Iwashita

Mifune

et al. 2014

Journal of Information Processing

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Abstract:We discuss a scheme for hierarchical matrices with adaptive cross approximation on symmetric multiprocessing clusters. We propose a set of parallel algorithms that are applicable to hierarchical matrices. The proposed algorithms are implemented using the flat-MPI and hybrid MPI+OpenMP programming models. The performance of these implementations is evaluated using an electric field analysis computed on two symmetric multiprocessing cluster systems. Although the flat-MPI version gives better parallel scalability when constructing hierarchical matrices, the speed-up reaches a limit in the hierarchical matrix-vector multiplication. We succeeded in developing a hybrid MPI+OpenMP version to improve the parallel scalability. In numerical experiments, the hybrid version exhibits a better parallel speed-up for the hierarchical matrix-vector multiplication up to 256 cores.

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