ncku-talk2019-04-25

Colloquium

DATE 2019-04-25　16:10-17:00

PLACE 數學館3174教室

SPEAKER 田恆 (Tian Heng) 博士（交通大學應用數學系）

TITLE Large Sparse Palindromic Quadratic Eigenvalue Problems Arising from Computing Complete Complex Band Structure of 1D Nanowires

ABSTRACT Complete spectrum of a large sparse T-palindromic quadratic eigenvalue problem (T-PQEP) is needed in the calculation of surface Green’s functions (SGFs) of nanowires with a tremendous non-periodic cross-section. Equivalently, complete complex band structure of quasi-1D nanowires is required. 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 nonlinear matrix equation, we present our own approach to this problem. In brief, the unit disk where the spectrum of interest lies is broken down adaptively into small pieces so that they each can be locally tackled by the generalized T-skew-Hamiltonian implicitly restarted shift-and-invert Arnoldi (GTSHIRA) 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 in this process, the novel non-equivalence deflation of the T-PQEP greatly accelerates the computation. We will demonstrate the new approach by one realistic example. If possible, we will discuss our recent application of this new approach to the computation of partial complex band structure of 3D photonic crystals/metamaterials.

Colloquium

DATE	2019-04-25　16:10-17:00

PLACE	數學館3174教室

SPEAKER	田恆 (Tian Heng) 博士（交通大學應用數學系）

TITLE	Large Sparse Palindromic Quadratic Eigenvalue Problems Arising from Computing Complete Complex Band Structure of 1D Nanowires

ABSTRACT	Complete spectrum of a large sparse T-palindromic quadratic eigenvalue problem (T-PQEP) is needed in the calculation of surface Green’s functions (SGFs) of nanowires with a tremendous non-periodic cross-section. Equivalently, complete complex band structure of quasi-1D nanowires is required. 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 nonlinear matrix equation, we present our own approach to this problem. In brief, the unit disk where the spectrum of interest lies is broken down adaptively into small pieces so that they each can be locally tackled by the generalized T-skew-Hamiltonian implicitly restarted shift-and-invert Arnoldi (GTSHIRA) 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 in this process, the novel non-equivalence deflation of the T-PQEP greatly accelerates the computation. We will demonstrate the new approach by one realistic example. If possible, we will discuss our recent application of this new approach to the computation of partial complex band structure of 3D photonic crystals/metamaterials.