Mixed Precision Symmetric Eigensolvers: Proof of Concept

代数特征值问题的数值求解是支撑计算科学和工程的许多领域的关键技术，包括声学、航空学、控制理论、流体力学、种群建模、量子物理、机器人和结构工程。在所有这些领域，都出现了对本征值问题的快速和数值可靠解的需求。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(多核处理器和图形处理单元(GPU)等加速器)越来越支持半精度浮点算术。这些低精度为极大地加速线性代数计算提供了新的机会。这一“小拨款”方案的研究是对下一代高效和数值稳定的特征解算器的概念证明，该解算器在保持数值稳定性的同时利用现代硬件的不同算术精度。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且具有全集的正交化特征向量，但任何进展都将直接影响未来非对称特征问题、广义特征问题和奇异值分解的算法。算法将作为原型在MATLAB中开发，使用模拟的半精度。将分析它们的数值稳定性以及它们在算术成本和通信成本方面的效率，以便确定哪一个(S)应该在诸如GPU上免费提供的矩阵代数和多核架构(MAGMA)库等最先进的数值线性代数库中完全实现。

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.