High-performance Cholesky factorization for GPU-only execution

Abstract

We present our performance analysis, algorithm designs, and the optimizations needed for the development of high-performance GPU-only algorithms, and in particular, for the dense Cholesky factorization. In contrast to currently promoted designs that solve parallelism challenges on multicore architectures by representing algorithms as Directed Acyclic Graphs (DAGs), where nodes are tasks of fine granularity and edges are the dependencies between the tasks, our designs explicitly target manycore architectures like GPUs and feature coarse granularity tasks (that can be hierarchically split into fine grain data-parallel subtasks). Furthermore, in contrast to hybrid algorithms that schedule difficult to parallelize tasks on CPUs, we develop highly-efficient code for entirely GPU execution. GPU-only codes remove the expensive CPU-to-GPU communications and the tuning challenges related to slow CPU and/or low CPU-to-GPU bandwidth. We show that on latest GPUs, like the P100, this becomes so important that the GPU-only code even outperforms the hybrid MAGMA algorithms when the CPU tasks and communications can not be entirely overlapped with GPU computations. We achieve up to 4, 300 GFlop/s in double precision on a P100 GPU, which is about 7-8× faster than high-end multicore CPUs, e.g., two 10-cores Intel Xeon E5-2650 v3 Haswell CPUs, where MKL runs up to about 500-600 Gflop/s. The new algorithm also outperforms significantly the GPU-only implementation currently available in the NVIDIA cuSOLVER library. CCS CONCEPTS •General and reference →Design; Performance; •Theory of computation →Algorithm design techniques; •Computing methodologies →Linear algebra algorithms; Optimization

Simt-X

“…Hybrid CPU-GPU algorithms, for instance, incur memory transfer and synchronization overhead. For smaller batch sizes, GPU-only implementations have been shown to offer better overall energy efficiency on mixed parallel/serial algorithms [Haidar et al 2017]. Likewise, the latency of GPU tasks and data communication has been shown as an important factor affecting hybrid performance [Wong and Aamodt 2009].…”

“…Tino et al platforms has a cost, both in programmability and efficiency. For instance, the energy efficiency benefits of heterogeneous systems are negated due to the communication and synchronization overhead incurred by hybrid algorithms [Haidar et al 2017]. Likewise, the serial performance provided by GPUs demonstrates an impact on overall system performance [Wong and Aamodt 2009].…”

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“…Still, the evolution of those algorithms dedicated entirely to GPUs made it possible to envisage efficient implementations. In fact, Haidar et al [8] shows that in modern GPU architectures, GPU-only codes can achieve higher performance than the hybrid algorithms when the difficult-to-parallelize CPU tasks and communications cannot be overlapped entirely by the GPU computations, a typical advantage observed in hybrid implementations. With that in mind, we opt to use a parallel GPU-only LU factorization in our implementation of the scattering algorithm.…”

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