Mixed Precision Symmetric Eigensolvers: Proof of Concept

混合精度对称特征求解器：概念证明

• 批准号: EP/W018101/1
• 负责人: Francoise Tisseur
• 金额: $ 9.12万
• 依托单位: University of Manchester
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2022
• 资助国家: 英国
• 起止时间: 2022 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FW018101%2F1
• 关键词: Mixed; Precision; Symmetric; Eigensolvers; Proof

The numerical solution of algebraic eigenvalue problems is a key technology underpinning many areas of computational science and engineering, including acoustics, aeronautics, control theory, fluid mechanics, population modelling, quantum physics, robotics, and structural engineering. In all these areas, the need for fast and numerically reliable solution of eigenvalue problems arises. The problems can be large so that time to solution can be unacceptably long.Modern hardware (multicore processors and accelerators such as graphics processing units (GPUs)) increasingly supports half precision floating-point arithmetics. These low precisions provide new opportunities to considerably accelerate linear algebra computations.The research in this "small grant" proposal is a proof of concept for the next generation of efficient and numerically stable eigensolvers that exploit the different arithmetic precisions of modern hardware while maintaining numerical stability. Our investigation concentrates on the symmetric eigenvalue problem, for which eigenvalues are real with a full set of orthonormal eigenvectors, but any advances will have direct impact on future algorithms for nonsymmetric eigenproblems, generalized eigenproblems, and the singular value decomposition.The algorithms will be developed as prototypes in MATLAB, using simulated half precision. Their numerical stability will be analyzed as well as their efficiency in terms of arithmetic costs and communications costs so as to determine which one(s) should be fully implemented in state of the art numerical linear algebra libraries such as the freely available matrix algebra on GPU and multicore architectures (MAGMA) library.

代数特征值问题的数值求解是支撑计算科学和工程许多领域的关键技术，包括声学、航空、控制理论、流体力学、总体建模、量子物理、机器人和结构工程。在所有这些领域，都需要快速且数值可靠地解决特征值问题。这些问题可能很大，以至于解决问题的时间可能长得令人无法接受。现代硬件（多核处理器和加速器，例如图形处理单元 (GPU)）越来越多地支持半精度浮点运算。这些低精度为显着加速线性代数计算提供了新的机会。这项“小额资助”提案中的研究是下一代高效且数值稳定的特征求解器的概念证明，该特征求解器利用现代硬件的不同算术精度，同时保持数值稳定性。我们的研究集中在对称特征值问题上，该问题的特征值是实数，具有全套正交特征向量，但任何进步都将对非对称特征问题、广义特征问题和奇异值分解的未来算法产生直接影响。这些算法将在 MATLAB 中开发为原型，使用模拟半 精确。我们将分析它们的数值稳定性以及算术成本和通信成本方面的效率，以确定哪些应该在最先进的数值线性代数库（例如 GPU 上免费提供的矩阵代数和多核架构 (MAGMA) 库）中完全实现。

项目成果

Performance impact of precision reduction in sparse linear systems solvers.

• DOI: 10.7717/peerj-cs.778
• 发表时间: 2022
• 期刊: PeerJ. Computer science
• 作者: Zounon M;Higham NJ;Lucas C;Tisseur F
• 通讯作者: Tisseur F

Robust Rational Approximations of Nonlinear Eigenvalue Problems

非线性特征值问题的鲁棒有理逼近

• DOI: 10.1137/20m1380533
• 发表时间: 2022
• 期刊: SIAM Journal on Scientific Computing
• 影响因子: 3.1
• 作者: Güttel S

A parametrization of structure-preserving transformations for matrix polynomials

矩阵多项式的结构保持变换的参数化

• DOI: 10.1016/j.laa.2023.05.024
• 发表时间: 2023
• 期刊: Linear Algebra and its Applications
• 影响因子: 1.1
• 作者: Garvey S