众核处理器上的$\mathcal{H}$矩阵的数学模式
Algorithmic patterns for $\mathcal{H}$-matrices on many-core processors
摘要：
在这项工作中，我们考虑重新制定的分层（$\mathcal{H}$）矩阵算法的众核处理器的图形处理单元（GPU）上的模型实现。$\mathcal{H}$矩阵近似特定的稠密矩阵，例如，从离散积分方程或核岭回归，导致在密集矩阵-向量乘积中的对数线性时间复杂度。$\mathcal{H}$矩阵运算在众核处理器上的并行化是困难的，这是由于底层算法的复杂性。虽然众核硬件以前的算法进步集中在加速现有的$\mathcal{H}$矩阵CPU实现众核处理器，我们在这里的目标是完全依赖于处理器类型。作为主要贡献，我们介绍了必要的并行算法模式，允许映射到众核硬件的完整的$\mathcal{H}$矩阵建设和快速矩阵向量积。在这里，关键的成分是空间填充曲线，并行树遍历和线性代数运算的并行化。由此产生的模型GPU实现hmglib是，据作者所知，第一个完全基于GPU的开源$\mathcal{H}$矩阵库。我们结束这项工作的深入性能分析和比较性能研究对标准的$\mathcal{H}$矩阵库，突出了深刻的加速我们的众核并行方法。
英文摘要：
In this work, we consider the reformulation of hierarchical ($\mathcal{H}$) matrix algorithms for many-core processors with a model implementation on graphics processing units (GPUs). $\mathcal{H}$ matrices approximate specific dense matrices, e.g., from discretized integral equations or kernel ridge regression, leading to log-linear time complexity in dense matrix-vector products. The parallelization of $\mathcal{H}$ matrix operations on many-core processors is difficult due to the complex nature of the underlying algorithms. While previous algorithmic advances for many-core hardware focused on accelerating existing $\mathcal{H}$ matrix CPU implementations by many-core processors, we here aim at totally relying on that processor type. As main contribution, we introduce the necessary parallel algorithmic patterns allowing to map the full $\mathcal{H}$ matrix construction and the fast matrix-vector product to many-core hardware. Here, crucial ingredients are space filling curves, parallel tree traversal and batching of linear algebra operations. The resulting model GPU implementation hmglib is the, to the best of the authors knowledge, first entirely GPU-based Open Source $\mathcal{H}$ matrix library of this kind. We conclude this work by an in-depth performance analysis and a comparative performance study against a standard $\mathcal{H}$ matrix library, highlighting profound speedups of our many-core parallel approach.
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