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首页> 外文期刊>SIAM Journal on Scientific Computing >PARALLEL SOLVER FOR SHIFTED SYSTEMS IN A HYBRID CPU-GPU FRAMEWORK
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PARALLEL SOLVER FOR SHIFTED SYSTEMS IN A HYBRID CPU-GPU FRAMEWORK

机译：混合CPU-GPU框架中的移位系统的并行求解器

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摘要

This paper proposes a combination of a hybrid CPU-GPU and a pure GPU software implementation of a direct algorithm for solving shifted linear systems (A - sigma I) X = B with a large number of complex shifts sigma and multiple right-hand sides. Such problems often appear, e.g., in control theory when evaluating the transfer function, or as a part of an algorithm performing interpolatory model reduction, as well as when computing pseudospectra and structured pseudospectra, or solving large linear systems of ordinary differential equations. The proposed algorithm first jointly reduces the general full n x n matrix A and the n x m full right-hand side matrix B to the controller Hessenberg canonical form that facilitates efficient solution: A is transformed to a so-called m-Hessenberg form, and B is made upper triangular. This is implemented as a blocked highly parallel CPU-GPU hybrid algorithm; individual blocks are reduced by the CPU, and the necessary updates of the rest of the matrix are split among the cores of the CPU and the GPU. To enhance parallelization, the reduction and the updates are overlapped. In the next phase, the reduced m-Hessenberg-triangular systems are solved entirely on the GPU, with shifts divided into batches. The benefits of such load distribution are demonstrated by numerical experiments. In particular, we show that our proposed implementation provides an excellent basis for efficient implementations of computational methods in systems and control theory, from evaluation of transfer function to the interpolatory model reduction.
机译：本文提出了一种混合CPU-GPU和纯GPU软件实现的直接算法的组合，用于求解偏移的线性系统（A - Sigma i）X = B，具有大量复杂的换档Σ和多个右手侧。这些问题通常出现，例如，在控制理论中，当评估传递函数时，或作为执行插值模型减少的算法的一部分，以及计算Pseudospectra和结构化伪谱时，或求解常微分方程的大线性系统。所提出的算法首先将通用的全NXN矩阵A和NXM全右手侧矩阵B联合减小到控制器Hessenberg规范形式，这有利于有效的解决方案：A被转换为所谓的M-Hessenberg形式，B是制造的上三角形。这实现为阻塞高度平行的CPU-GPU混合算法; CPU的单个块减少，并且矩阵的其余部分的必要更新被分割在CPU和GPU的核心之间。为了增强并行化，重叠减少和更新。在下阶段，整体上的M-Hessenberg-三角形系统完全求解在GPU上，换档分为批次。通过数值实验证明了这种载荷分布的益处。特别是，我们表明我们的拟议实施为系统和控制理论中的计算方法的有效实施提供了优异的基础，从转移函数降低了转移功能。

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