In this paper, we focus on the stochastic inverse eigenvalue problem of reconstructing a stochastic matrix from the prescribed spectrum. We directly reformulate the stochastic inverse eigenvalue problem as a constrained optimization problem over several matrix manifolds to minimize the distance between isospectral matrices and stochastic matrices. Then we propose a geometric Polak–Ribi`ere–Polyak-based nonlinear conjugate gradient method for solving the constrained optimization problem. The global convergence of the proposed method is established. Our method can also be extended to the stochastic inverse eigenvalue problem with prescribed entries. An extra advantage is that our models yield new isospectral flow methods. Finally, we report some numerical tests to illustrate the efficiency of the proposed method for solving the stochastic inverse eigenvalue problem and the case of prescribed entries.
Li-Ping ZhangDepartment of Mathematical Sciences, Tsinghua UniversityLiqun QiDepartment of Applied Mathematics, Hong Kong Polytechnic UniversityGuanglu ZhouDepartment of Mathematics and Statistics, Curtin University, Perth, Australia
Numerical Linear Algebramathscidoc:1804.26001
SIAM Journal on Matrix Analysis and Applications, 35, (2), 437-452, 2014.4
We introduceM-tensors. This concept extends the concept ofM-matrices. We denote
Z-tensors as the tensors with nonpositive off-diagonal entries. We show that M-tensors must be Ztensors
and the maximal diagonal entry must be nonnegative. The diagonal elements of a symmetric
M-tensor must be nonnegative. A symmetric M-tensor is copositive. Based on the spectral theory
of nonnegative tensors, we show that the minimal value of the real parts of all eigenvalues of an Mtensor
is its smallest H+-eigenvalue and also is its smallest H-eigenvalue. We show that a Z-tensor is
an M-tensor if and only if all its H+-eigenvalues are nonnegative. Some further spectral properties of
M-tensors are given. We also introduce strong M-tensors, and some corresponding conclusions are
given. In particular, we show that all H-eigenvalues of strong M-tensors are positive. We apply this
property to study the positive definiteness of a class of multivariate forms associated with Z-tensors.
We also propose an algorithm for testing the positive definiteness of such a multivariate form.
So-Hsiang ChouBowling Green State UniversityTsung-Ming HuangNational Taiwan Normal UniversityTiexiang LiSoutheast UniversityJia-Wei LinNational Chiao Tung UniversityWen-Wei LinNational Chiao Tung University
Numerical Linear Algebramathscidoc:1802.26001
Journal of Computational Physics, 386, 611-631, 2019.6
The standard Yee's scheme for the Maxwell eigenvalue problems places the discrete electric field variable at the midpoints of the edges of the grid cells. It performs well when the permittivity is a scalar field. However, when the permittivity is a Hermitian full tensor filed it would generate un-physical complex eigenvalues or frequencies. In this paper, we propose a finite element method which can be interpreted as a modified Yee's scheme to overcome this difficulty. This interpretation enables us to create a fast FFT eigensolver that can compute very effectively the band structure of the anisotropic photonic crystal with SC and FCC lattices. Furthermore, we overcome the usual
large null space associated with the Maxwell eigenvalue problem by deriving a null-space free discrete eigenvalue problem which involves a crucial Hermitian positive definite linear system to be solved in each of the iteration steps. It is demonstrated that the CG method without preconditioning converges in 37 iterations even when the dimension of the matrix is as large as $5,184,000$.