Tsung Ming Huang

Convergence Analysis of the Doubling Algorithm for Several Nonlinear Matrix Equations in the Critical Case

In this paper, we review two types of doubling algorithm and some techniques for analyzing them. We then use the techniques to study the doubling algorithm for three different nonlinear matrix equations in the critical case. We show that the convergence of the doubling algorithm is at least linear with rate

Structure-Preserving Algorithms for Palindromic Quadratic Eigenvalue Problems Arising from Vibration of Fast Trains

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A parallel additive Schwarz preconditioned Jacobi-Davidson algorithm for polynomial eigenvalue problems in quantum dot simulation

. As compared to earlier work on this topic, the results we present here are more general, and the analysis here is much simpler.

Preconditioning bandgap eigenvalue problems in three-dimensional photonic crystals simulations

In this paper, based on Patels algorithm (1993), we propose a structure-preserving algorithm for solving palindromic quadratic eigenvalue problems (QEPs). We also show the relationship between the structure-preserving algorithm and the URV-based structure-preserving algorithm by Schroder (2007). For large sparse palindromic QEPs, we develop a generalized

Stochastic growth dynamics and composite defects in quenched immiscible binary condensates.

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EIGENDECOMPOSITION OF THE DISCRETE DOUBLE-CURL OPERATOR WITH APPLICATION TO FAST EIGENSOLVER FOR THREE-DIMENSIONAL PHOTONIC CRYSTALS

-skew-Hamiltonian implicitly restarted shift-and-invert Arnoldi algorithm for solving the resulting

Palindromic quadratization and structure-preserving algorithm for palindromic matrix polynomials of even degree

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A Robust Numerical Algorithm for Computing Maxwell's Transmission Eigenvalue Problems

-skew-Hamiltonian pencils. Numerical experiments show that our proposed structure-preserving algorithms perform well on the palindromic QEP arising from a finite element model of high-speed trains and rails.

A Null Space Free Jacobi--Davidson Iteration for Maxwell's Operator

We develop a parallel Jacobi-Davidson approach for finding a partial set of eigenpairs of large sparse polynomial eigenvalue problems with application in quantum dot simulation. A Jacobi-Davidson eigenvalue solver is implemented based on the Portable, Extensible Toolkit for Scientific Computation (PETSc). The eigensolver thus inherits PETScs efficient and various parallel operations, linear solvers, preconditioning schemes, and easy usages. The parallel eigenvalue solver is then used to solve higher degree polynomial eigenvalue problems arising in numerical simulations of three dimensional quantum dots governed by Schrodingers equations. We find that the parallel restricted additive Schwarz preconditioner in conjunction with a parallel Krylov subspace method (e.g. GMRES) can solve the correction equations, the most costly step in the Jacobi-Davidson algorithm, very efficiently in parallel. Besides, the overall performance is quite satisfactory. We have observed near perfect superlinear speedup by using up to 320 processors. The parallel eigensolver can find all target interior eigenpairs of a quintic polynomial eigenvalue problem with more than 32 million variables within 12 minutes by using 272 Intel 3.0GHz processors.

Eigenvalue solvers for three dimensional photonic crystals with face-centered cubic lattice

To explore band structures of three-dimensional photonic crystals numerically, we need to solve the eigenvalue problems derived from the governing Maxwell equations. The solutions of these eigenvalue problems cannot be computed effectively unless a suitable combination of eigenvalue solver and preconditioner is chosen. Taking eigenvalue problems due to Yees scheme as examples, we propose using Krylov-Schur method and Jacobi-Davidson method to solve the resulting eigenvalue problems. For preconditioning, we derive several novel preconditioning schemes based on various preconditioners, including a preconditioner that can be solved by Fast Fourier Transform efficiently. We then conduct intensive numerical experiments for various combinations of eigenvalue solvers and preconditioning schemes. We find that the Krylov-Schur method associated with the Fast Fourier Transform based preconditioner is very efficient. It remarkably outperforms all other eigenvalue solvers with common preconditioners like Jacobi, Symmetric Successive Over Relaxation, and incomplete factorizations. This promising solver can benefit applications like photonic crystal structure optimization.