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High performance solution of skew-symmetric eigenvalue problems with applications in solving the Bethe-Salpeter eigenvalue problem

机译:求解贝特 - 排雷特征值问题的应用高性能解

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We present a high-performance solver for dense skew-symmetric matrix eigenvalue problems. Our work is motivated by applications in computational quantum physics, where one solution approach to solve the Bethe-Salpeter equation involves the solution of a large, dense, skew-symmetric eigenvalue problem. The computed eigenpairs can be used to compute the optical absorption spectrum of molecules and crystalline systems. One state-of-the art high-performance solver package for symmetric matrices is the ELPA (Eigenvalue SoLvers for Petascale Applications) library. We exploit a link between tridiagonal skew-symmetric and symmetric matrices in order to extend the methods available in ELPA to skew-symmetric matrices. This way, the presented solution method can benefit from the optimizations available in ELPA that make it a well-established, efficient and scalable library. The solution strategy is to reduce a matrix to tridiagonal form, solve the tridiagonal eigenvalue problem and perform a back-transformation for eigenvectors of interest. ELPA employs a one-step or a two-step approach for the tridiagonalization of symmetric matrices. We adapt these to suit the skew-symmetric case. The two-step approach is generally faster as memory locality is exploited better. If all eigenvectors are required, the performance improvement is counteracted by the additional back transformation step. We exploit the symmetry in the spectrum of skew-symmetric matrices, such that only half of the eigenpairs need to be computed, making the two-step approach the favorable method. We compare performance and scalability of our method to the only available high-performance approach for skew-symmetric matrices, an indirect route involving complex arithmetic. In total, we achieve a performance that is up to 3.67 times higher than the reference method using Intel's ScaLAPACK implementation. Our method is freely available in the current release of the ELPA library. (C) 2020 Elsevier B.V. All rights reserved.
机译:我们为密集偏斜矩阵特征值问题提供了一种高性能求解器。我们的作品受到计算量子物理学中的应用,其中解决贝特 - Salpeter方程的一种解决方案方法涉及大,密集,偏差的特征值问题的解决方案。计算的特征方可用于计算分子和结晶系统的光学吸收光谱。用于对称矩阵的一个最先进的高性能求解器包是ELPA(用于PETASCALE应用的特征值溶剂)文库。我们利用了三角形歪曲对称和对称矩阵之间的链路,以扩展ELPA中可用的方法到歪曲对称矩阵。这样,所呈现的解决方案方法可以从ELPA中提供的优化中受益,使其成为一个完善,高效和可扩展的库。解决方案策略是将矩阵减少到三角形形式,解决三角形特征值问题并对感兴趣的特征向量进行后转换。 ELPA采用一步或两步方法,用于对称矩阵的三角形。我们适应这些以适应歪斜对称案例。两步方法通常更快,因为更好地利用内存局部性。如果需要所有特征向量,则额外的后转换步骤抵消了性能改进。我们利用歪曲对称矩阵频谱的对称性,使得只需要计算特征环的一半,使得两步方法是有利的方法。我们将我们方法的性能和可扩展性与涉及复杂算法的唯一可用的高性能方法进行比较唯一可用的高性能方法。总之,我们实现了使用英特尔的缩放方法的参考方法高达3.67倍的性能。我们的方法在ELPA库的当前版本中自由提供。 (c)2020 Elsevier B.v.保留所有权利。

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