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[CT096]

Session Date & Time : 3E (Aug.23, 17:40-19:20)
Type : Contributed Talk
Abstract : The Eberlein method is a Jacobi-type process for solving the eigenvalue problem of an arbitrary matrix. In each iteration two transformations are applied on the underlying matrix, a plane rotation and a non-unitary elementary transformation. In this talk we present the method under the broad class of generalized serial pivot strategies. We provide the proof of the global convergence and give several numerical examples.
Classification : 65F15
Author(s) :
- Erna Begovic (University of Zagreb)
- Ana Perkovic (University of Zagreb)

Session Date & Time : 3E (Aug.23, 17:40-19:20)
Type : Contributed Talk
Abstract : In this article, we propose the Fast Algorithms for Maxwell's Equations (FAME) package for solving Maxwell's equations for modeling three-dimensional photonic crystals. FAME combines the null-space free method with fast Fourier transform (FFT)-based matrix-vector multiplications to solve the generalized eigenvalue problems (GEPs) arising from Yee's discretization. Numerical results demonstrate the potential of our proposed package to enable large-scale numerical simulations for novel physical discoveries and engineering applications of photonic crystals.
Classification : 65F15, 74E10, 74E15
Author(s) :
- Xing-Long Lyu (Southeast University)
- Tiexiang Li (Southeast University)
- Tsung-Ming Huang (National Taiwan Normal University)
- Jia-Wei Lin (National Yang Ming Chiao Tung University)
- Heng Tian (Sichuan University of Science and Engineering)
- Wen-Wei Lin (National Yang Ming Chiao Tung University)

Session Date & Time : 3E (Aug.23, 17:40-19:20)
Type : Contributed Talk
Abstract : In this talk, a Fast Linear Elastic Eigenvalue Problem Solver (FLEEPS) is developed to calculate band structures of 3D isotropic phononic crystals. FLEEPS is an iterative eigensolver of quasi-linear complexity to compute the smallest few eigenvalues of the linear elastic eigenvalue problem. The weighted SVD-preconditioned CG method in FLEEPS convergences faster than the AMG-preconditioned CG method by more than 60 times. Band structure calculations of several 3D isotropic phononic crystals demonstrate the strengths of FLEEPS.
Classification : 65F15, 74E10, 74E15
Author(s) :
- Tiexiang Li (Southeast University)
- Heng Tian (Sichuan University of Science and Engineering)
- Xing-Long Lyu (Southeast University)
- Wen-Wei Lin (National Yang Ming Chiao Tung University)

Session Date & Time : 3E (Aug.23, 17:40-19:20)
Type : Contributed Talk
Abstract : In this paper, we study the Helmholtz transmission eigenvalue problem for inhomogeneous anisotropic media in two and three dimension. Starting with a nonlinear fourth order formulation established by Cakoni, Colton and Haddar in 2009, by introducing some auxiliary variables, we present an equivalent mixed formulation for this problem, followed up with the finite element discretization. Using the proposed scheme, we rigorously show that the optimal convergence rate for the transmission eigenvalues both on convex and nonconvex domains can be expected. Moreover, by this scheme, we will obtain a sparse generalized eigenvalue problem whose size is so demanding even with a coarse mesh that its smallest few real eigenvalues fail to be solved by the shift and invert method. We partially overcome this critical issue by deflating the almost all of the ∞ eigenvalue of huge multiplicity, resulting in a drastic reduction of the matrix size without deteriorating the sparsity. Extensive numerical examples are reported to demonstrate the effectiveness and efficiency of the proposed scheme.
Classification : 65F15, 65M60, 65N25, 78M10
Author(s) :
- Qing Liu (School of Mathematics, Southeast University)
- Tiexiang Li (School of Mathematics, Southeast University)
- Shuo Zhang (Academy of Mathematics and System Sciences, Chinese Academy of Sciences)

Session Date & Time : 3E (Aug.23, 17:40-19:20)
Type : Contributed Talk
Abstract : This work discusses solutions of large linear systems of algebraic equations relevant to establish a solution to the discrete ordinates approximation of the two-dimensional linear Boltzmann equation. Direct and iterative methods are investigated, along with domain decomposition techniques and parallel implementation. The type of the quadrature scheme describing the directions and the class of problems to be solved, neutron or radiation problems, directly affect the final choice of the numerical algorithm.
Classification : 65F22, 65F05, 65F10, 65MXX
Author(s) :
- Rudnei Dias da Cunha (Universidade Federal do Rio Grande do Sul)
- Liliane Basso Barichello (Universidade Federal do Rio Grande do Sul)