FEAST As A Subspace Iteration Eigensolver Accelerated By Approximate Spectral Projection
FEAST As A Subspace Iteration Eigensolver Accelerated By Approximate Spectral Projection
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DOI：
10.1137/13090866x
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发表时间：
2013-02
期刊：
影响因子：
--
通讯作者：
P. Tang;E. Polizzi
中科院分区：
文献类型：
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作者：
P. Tang;E. Polizzi
厄米特矩阵或矩阵束的一段特征值及其对应的特征向量的计算有很多应用。最近提出了一种新的基于密度矩阵的算法，并开发了FEAST软件。密度矩阵方法允许FEAST的实现利用现代计算机体系结构的一个关键优势，即多级并行。因此，该软件包得到了很好的接受，特别是在电子结构界。然而，对盛宴的理论分析却相对滞后。例如，FEAST算法尚未被证明是收敛的。本文对FEAST进行了详细的数值分析。特别是，我们证明了FEAST算法可以理解为与Rayleigh-Ritz过程相结合的加速子空间迭代算法。FEAST的新奇之处在于它的加速器，它是一个有理矩阵函数，将光谱投影仪近似到所讨论的特征空间。对这种近似光谱投影仪的数值性质和FEAST算法中生成的子空间的分析证实了该算法的收敛。本文展示了FEAST对舍入误差的恢复能力，并建立了可用于增强算法稳健性的性质。最后，我们提出了FEAST的扩展，以处理非厄米问题，并提出了一些未来的研究方向。
The calculation of a segment of eigenvalues and their corresponding eigenvectors of a Hermitian matrix or matrix pencil has many applications. A new density-matrix-based algorithm has been proposed recently and a software package FEAST has been developed. The density-matrix approach allows FEAST's implementation to exploit a key strength of modern computer architectures, namely, multiple levels of parallelism. Consequently, the software package has been well received, especially in the electronic structure community. Nevertheless, theoretical analysis of FEAST has lagged. For instance, the FEAST algorithm has not been proven to converge. This paper offers a detailed numerical analysis of FEAST. In particular, we show that the FEAST algorithm can be understood as an accelerated subspace iteration algorithm in conjunction with the Rayleigh-Ritz procedure. The novelty of FEAST lies in its accelerator which is a rational matrix function that approximates the spectral projector onto the eigenspace in question. Analysis of the numerical nature of this approximate spectral projector and the resulting subspaces generated in the FEAST algorithm establishes the algorithm's convergence. This paper shows that FEAST is resilient against rounding errors and establishes properties that can be leveraged to enhance the algorithm's robustness. Finally, we propose an extension of FEAST to handle non-Hermitian problems and suggest some future research directions.