Computing the full spectrum of large sparse palindromic quadratic eigenvalue problems arising from surface Green's function calculations

Tsung-Ming Huang Department of Mathematics, National Taiwan Normal University Wen-Wei Lin Department of Applied Mathematics, National Chiao Tung University Tian Heng Department of Applied Mathematics, National Chiao Tung University Guan-Hua Chen Department of Chemistry, The University of Hong Kong

Numerical Analysis and Scientific Computing Numerical Linear Algebra mathscidoc:1810.25001

Journal of Computational Physics, 356, 340-355
Full spectrum of a large sparse $\top$-palindromic quadratic eigenvalue problem ($\top$-PQEP) is considered arguably for the first time in this article. Such a problem is posed by calculation of surface Green's functions (SGFs) of mesoscopic transistors with a tremendous non-periodic cross-section. For this problem, general purpose eigensolvers are not efficient, nor is advisable to resort to the decimation method \etc to obtain the Wiener-Hopf factorization. After reviewing some rigorous understanding of SGF calculation from the perspective of $\top$-PQEP and nonlinear matrix equation, we present our new approach to this problem. In a nutshell, the unit disk where the spectrum of interest lies is broken down adaptively into pieces small enough that they each can be locally tackled by the generalized $\top$-skew-Hamiltonian implicitly restarted shift-and-invert Arnoldi (G$\top$SHIRA) algorithm with suitable shifts and other parameters, and the eigenvalues missed by this divide-and-conquer strategy can be recovered thanks to the accurate estimation provided by our newly developed scheme. Notably the novel non-equivalence deflation is proposed to avoid as much as possible duplication of nearby known eigenvalues when a new shift of G$\top$SHIRA is determined. We demonstrate our new approach by calculating the SGF of a realistic nanowire whose unit cell is described by a matrix of size $4000\times 4000$ at the density functional tight binding level, corresponding to a $8\times 8\; \mbox{nm}^{2}$ cross-section. We believe that quantum transport simulation of realistic nano-devices in the mesoscopic regime will greatly benefit from this work.
Palindromic quadratic eigenvalue problem, GTSHIRA, Non-equivalence deflation, Surface Green's function, Quantum transport
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@inproceedings{tsung-mingcomputing,
  title={Computing the full spectrum of large sparse palindromic quadratic eigenvalue problems arising from surface Green's function calculations},
  author={Tsung-Ming Huang, Wen-Wei Lin, Tian Heng, and Guan-Hua Chen},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20181002221230782568160},
  booktitle={Journal of Computational Physics},
  volume={356},
  pages={340-355},
}
Tsung-Ming Huang, Wen-Wei Lin, Tian Heng, and Guan-Hua Chen. Computing the full spectrum of large sparse palindromic quadratic eigenvalue problems arising from surface Green's function calculations. Vol. 356. In Journal of Computational Physics. pp.340-355. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20181002221230782568160.
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