在混合CPU-GPU框架中求解移位系统的并行求解器
Parallel solver for shifted systems in a hybrid CPU-GPU framework
AI总结：
本文提出一种混合CPU-GPU和纯GPU的直接算法，用于高效求解具有大量复数移位σ和多个右端矩阵的线性系统。该方法通过将矩阵转化为控制器赫森堡标准型，实现高效的并行计算，适用于控制理论、模型降阶和伪谱计算等场景。
摘要：
本文提出了一种结合混合CPU-GPU和纯GPU软件实现的直接算法，用于求解具有大量复数移位σ和多个右端矩阵的线性系统(A - σI)X = B。此类问题常出现在控制理论中评估传递函数或插值模型降阶算法中，以及计算伪谱和结构伪谱或求解大规模常微分方程组时。所提算法首先将一般n×n矩阵A和n×m右端矩阵B联合转化为控制器赫森堡标准型，使求解更高效：A转化为所谓的m-赫森堡形式，B则变为上三角矩阵。该过程实现为高度并行的混合CPU-GPU阻塞算法；每个块由CPU处理，其余矩阵的更新则分配给CPU和GPU的核心。为增强并行化，减少和更新过程被重叠。下一阶段，减少的m-赫森堡-三角系统完全在GPU上求解，移位被分成批次处理。通过数值实验展示了这种负载分配的优势。特别地，我们展示了所提实现为系统和控制理论中计算方法提供了良好的基础，从传递函数评估到插值模型降阶。
英文摘要：
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 $(A - σI)X = B$ with large number of complex shifts $σ$ 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 $n\times n$ matrix $A$ and the $n\times m$ 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 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
