Tiexiang Li

Tiexiang Li
Southeast University (China) | SEU · School of Mathematics

PhD

About

59
Publications
4,821
Reads
How we measure 'reads'
A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. Learn more
313
Citations
Additional affiliations
April 2008 - April 2020
Southeast University (China)
Position
  • Professor
January 2008 - July 2019
Southeast University (China)
Position
  • Professor
September 2003 - January 2008
Peking University
Position
  • PhD Student

Publications

Publications (59)
Article
Full-text available
Utilizing the optimal mass transportation (OMT) technique to convert an irregular 3D brain image into a cube, a required input format for a U-net algorithm, is a brand new idea for medical imaging research. We develop a cubic volume-measure-preserving OMT (V-OMT) model for the implementation of this conversion. The contrast-enhanced histogram equal...
Article
In this paper, we present a unified finite difference framework to efficiently compute band structures of three dimensional linear non-dispersive isotropic photonic crystals with any of 14 Bravais lattice structures to a reasonable accuracy. Specifically, we redefine a suitable orthogonal coordinate system, and meticulously reformulate the Bloch co...
Preprint
Full-text available
Utilizing the optimal mass transportation (OMT) technique to convert an irregular 3D brain image into a cube, a required input format for the U-net algorithm, is a brand new idea for medical imaging research. We develop a cubic volume-measure-preserving OMT (V-OMT) model for the implementation of this conversion. The contrast-enhanced histogram equ...
Article
This paper focuses on studying the bifurcation analysis of the eigenstructure of the $\gamma $-parameterized generalized eigenvalue problem ($\gamma $-GEP) arising in three-dimensional source-free Maxwell’s equations with Pasteur media, where $\gamma $ is the magnetoelectric chirality parameter. The weakly coupled case, namely, $\gamma < \gamma _{\...
Article
Full-text available
Optimal mass transportation has been widely applied in various fields, such as data compression, generative adversarial networks, and image processing. In this paper, we adopt the projected gradient method, combined with the homotopy technique, to find a minimal volume-measure-preserving solution for a 3-manifold optimal mass transportation problem...
Article
Full-text available
In this work, we study the interior transmission eigenvalues for elastic scattering in an inhomogeneous medium containing an obstacle. This problem is related to the reconstruction of the support of the inhomogeneity without the knowledge of the embedded obstacle by the far-field data or the invisibility cloaking of an obstacle. Our goal is to prov...
Article
Full-text available
Optimal mass transport (OMT) theory, the goal of which is to move any irregular 3D object (i.e., the brain) without causing significant distortion, is used to preprocess brain tumor datasets for the first time in this paper. The first stage of a two-stage OMT (TSOMT) procedure transforms the brain into a unit solid ball. The second stage transforms...
Article
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...
Article
This paper focuses on studying the eigenstructure of generalized eigenvalue problems (GEPs) arising in the three-dimensional source-free Maxwell equations for bi-anisotropic complex media with a 3-by-3 permittivity tensor ε>0, a permeability tensor μ>0, and scalar magnetoelectric coupling constants ξ=ζ̄=ıγ. The bi-Lebedev scheme is appealing becaus...
Preprint
Full-text available
This paper focuses on studying the bifurcation analysis of the eigenstructure of the $\gamma$-parameterized generalized eigenvalue problem ($\gamma$-GEP) arising in three-dimensional (3D) source-free Maxwell's equations with Pasteur media, where $\gamma$ is the magnetoelectric chirality parameter. For the weakly coupled case, namely, $\gamma < \gam...
Article
In the paper, we study a class of biharmonic equations with singular weight functions as follows: Δ2u−βΔpu+Vλ(x)u=fxuq−2uinRN,u∈H2(RN),where N≥3,Δ2u=Δ(Δu), Δpu=div(|∇u|p−2∇u),β≥0 is a parameter, 2<p,q<2NN−2 and Vλ(x)=λa(x)−b(x) with λ>0. Under some suitable assumptions on a,b and f, we obtain the existence and multiplicity of nontrivial solutions f...
Article
Full-text available
This work is devoted to the numerical computation of complex band structure \(\mathbf {k}=\mathbf {k}(\omega )\in {\mathbb {C}}^3\), with \(\omega \) being positive frequencies, of three dimensional isotropic dispersive or non-dispersive photonic crystals from the perspective of structured quadratic eigenvalue problems (QEPs). Our basic strategy is...
Article
We study the non-existence, existence and multiplicity of positive solutions to the following nonlinear Kirchhoff equation:{−M(∫RN|∇u|2dx)Δu+μV(x)u=Q(x)|u|p−2u+λf(x)u in RN,u∈H1(RN), where N≥3,2<p<2⁎:=2NN−2, M(t)=at+b (a,b>0), the potential V is a nonnegative function in RN and the weight function Q∈L∞(RN) with changes sign in Ω‾:={V=0}. We mainly...
Preprint
Full-text available
We study the non-existence, existence and multiplicity of positive solutions to the following nonlinear Kirchhoff equation:% \begin{equation*} \left\{ \begin{array}{l} -M\left( \int_{\mathbb{R}^{3}}\left\vert \nabla u\right\vert ^{2}dx\right) \Delta u+\mu V\left( x\right) u=Q(x)\left\vert u\right\vert ^{p-2}u+\lambda f\left( x\right) u\text{ in }\m...
Article
The standard Yee's scheme for the Maxwell eigenvalue problem 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 field it would generate un-physical complex eigenvalues or frequencies. In this...
Article
Full-text available
This article focuses on numerically studying the eigenstructure behavior of generalized eigenvalue problems (GEPs) arising in three dimensional (3D) source-free Maxwell's equations with magnetoelectric coupling effects which model 3D reciprocal chiral media. It is challenging to solve such a large-scale GEP efficiently. We combine the null-space fr...
Article
In this paper, we consider the two-dimensional Maxwell’s equations with the TM mode in pseudo-chiral media. The system can be reduced to the acoustic equation with a negative index of refraction. We first study the transmission eigenvalue problem (TEP) for this equation. By the continuous finite element method, we discretize the reduced equation an...
Preprint
Calculation of band structure of three dimensional photonic crystals amounts to solving large-scale Maxwell eigenvalue problems, which are notoriously challenging due to high multiplicity of zero eigenvalue. In this paper, we try to address this problem in such a broad context that band structure of three dimensional isotropic photonic crystals wit...
Article
To solve the Bethe-Salpeter eigenvalue problem with distinct sizes, two efficient methods, called {\Gamma}QR algorithm and {\Gamma}-Lanczos algorithm, are proposed in this paper. Both algorithms preserve the special structure of the initial matrix $H=\begin{bmatrix}A & B-\overline{B} & -\overline{A}\end{bmatrix}$, resulting the computed eigenvalues...
Article
In this paper, we study intermittent behaviors of coupled piecewise-expanding map lattices with two nodes and a weak coupling. We show that the successive phase transition between ordered and disordered phases occurs for almost every orbit. That is, we prove $\liminf_{n\rightarrow \infty}| x_1(n)-x_2(n)|=0$ and $\limsup_{n\rightarrow \infty}| x_1(n...
Article
We propose an efficient eigensolver for computing densely distributed spectra of the two-dimensional transmission eigenvalue problem (TEP), which is derived from Maxwell's equations with Tellegen media and the transverse magnetic mode. The governing equations, when discretized by the standard piecewise linear finite element method, give rise to a l...
Article
In this paper, we present an efficient ΓQR algorithm for solving the linear response eigenvalue problem , where is -symmetric with respect to . Based on newly introduced Γ-orthogonal transformations, the ΓQR algorithm preserves the -symmetric structure of throughout the whole process, and thus guarantees the computed eigenvalues to appear pairwise...
Article
Full-text available
The transmission eigenvalue problem, besides its critical role in inverse scattering problems, deserves special interest of its own due to the fact that the corresponding differential operator is neither elliptic nor self-adjoint. In this paper, we provide a spectral analysis and propose a novel iterative algorithm for the computation of a few posi...
Article
Full-text available
In this paper, we study the existence of homoclinic solutions for a class of fourth order differential equations. By using variational methods, the existence and the non-existence of nontrivial homoclinic solutions are obtained, depending on a parameter.
Article
Full-text available
In this paper, a structure-preserving algorithm is developed for the computation of a semi-stabilizing solution of a Generalized Algebraic Riccati Equation (GARE). The semi-stabilizing solution of GAREs has been used to characterize the solvability of the (J, J ')-spectral factorization problem in control theory for general rational matrices which...
Article
Full-text available
We consider the solution of large-scale Lyapunov and Stein equations. For Stein equations, the well-known Smith method will be adapted, with $A_k = A^{2^k}$ not explicitly computed but in the recursive form $A_k = A_{k-1}^{2}$ , and the fast growing but diminishing components in the approximate solutions truncated. Lyapunov equations will be fi...
Article
We study the generalized eigenvalue problems (GEPs) derived from modeling the surface acoustic wave in piezoelectric materials with periodic inhomogeneity. The eigenvalues appear in the reciprocal pairs due to periodic boundary conditions in the modeling. By transforming the GEP into a T-palindromic quadratic eigenvalue problem (TPQEP), the recipro...
Article
In this paper, we are concerned with the computation of a few eigenpairs with smallest eigenvalues in absolute value of quadratic eigenvalue problems. We first develop a semiorthogonal generalized Arnoldi method where the name comes from the application of a pseudo inner product in the construction of a generalized Arnoldi reduction for a generaliz...
Article
Full-text available
We consider the pole assignment problems for time‐invariant linear and quadratic control systems, with time‐delay in the control. Closed‐loop eigenvectors in X = [x 1, x 2, … ] are chosen from their corresponding invariant subspaces, possibly optimizing some robustness measure, and explicit expressions for the feedback matrices are given in terms o...
Article
Full-text available
We consider the solution of large-scale algebraic Riccati equations with numerically low-ranked solutions. For the discrete-time case, the structure-preserving doubling algorithm has been adapted, with the iterates for AA not explicitly computed but in the recursive form Ak=Ak−12−Dk(1)Sk−1[Dk(2)]⊤, with Dk(1) and Dk(2) being low-ranked and Sk−1 bei...
Article
In this paper, we study the multiplicity of positive solutions for the nonhomogeneous elliptic problem: −Δu+λu=f(x)up−1+μh(x)−Δu+λu=f(x)up−1+μh(x) in RNRN. We will show how the shape of the graph of f(x)f(x) affects the number of positive solutions.
Article
Full-text available
Consider the nonlinear matrix equation X + A*X-q A = Q where 0 < q <= 1. A new sufficient condition for this equation to have positive definite solution is provided and two iterative methods for the maximal positive definite solution are proposed. Applying the theory of condition number developed by Rice, an explicit expression of the condition num...
Article
The recovery of Bacillus thuringiensis (Bt) based biopesticides from fermented sludge broth via cross-flow microfiltration (CFMF) was evaluated. Three types of microfiltration membranes, WX-model cellulose, cellulose acetate, and polyethersulfone, were tested for their permeate flux and restoration capacity. The cellulose acetate membrane with a po...
Article
Full-text available
For the steady-state solution of a differential equation from a one-dimensional multistate model in transport theory, we shall derive and study a nonsymmetric algebraic Riccati equation B− – XF− – F+X + XB+X = 0, where F ± ≡ (I – F)D ± and B ± ≡ BD ± with positive diagonal matrices D ± and possibly low-ranked matrices F and B. We prove the existenc...
Article
Full-text available
x = Ax + Bu with linear state-feedback u = Fx . An algorithm using the Schur form has been proposed, producing suboptimal solutions which can be refined further using optimization. In this paper, the algorithm is improved, with a weighted sum of the feedback gain and the departure from normality being used as the robustness measure. Newton refineme...
Article
Full-text available
In this paper, we propose the palindromic doubling algorithm (PDA) for the palindromic generalized eigenvalue problem (PGEP) A x = Ax . We establish a complete convergence theory of the PDA for PGEPs without unimodular eigenvalues, or with unimodular eigenval- ues of partial multiplicities two (one or two for eigenvalue 1). Some important applicati...
Article
Full-text available
We extend the Rayleigh–Ritz method to the eigen-problem of periodic matrix pairs. Assuming that the deviations of the desired periodic eigenvectors from the corresponding periodic subspaces tend to zero, we show that there exist periodic Ritz values that converge to the desired periodic eigenvalues unconditionally, yet the periodic Ritz vectors may...
Chapter
Full-text available
In Chu (Syst Control Lett 56:303–314, 2007), the pole assignment problem was considered for the control system [(x)\dot] = Ax + Bu\dot{x} = Ax + Bu with linear state-feedback u = Fx.u = Fx. An algorithm using the Schur form has been proposed, producing good suboptimal solutions which can be refined further using optimization. In this paper, the a...
Conference Paper
Full-text available
We consider the solution of the rational matrix equations, or generalized algebraic Riccati equations with ra- tional terms, arising in stochastic optimal control in continuous- and discrete-time. The modified Newton's methods, the DARE- and CARE-type iterations for continuous- and discrete-time rational Riccati equations respectively, will be cons...
Conference Paper
Full-text available
We consider the solution of the rational matrix equations, or generalized algebraic Riccati equations with rational terms, arising in stochastic optimal control in continuous- and discrete-time. Fixed-point iteration and (modified) Newton's methods will be considered. In particular, the convergence results of a new modified Newton's method, for bot...
Conference Paper
Full-text available
We consider the solution of the rational matrix equations, or generalized algebraic Riccati equations with ra- tional terms, arising in stochastic optimal control in continuous- and discrete-time. Fixed-point iteration and (modified) Newton's methods will be considered. In particular, the convergence results of a new modified Newton's method, for b...
Article
Full-text available
For the steady-state solution of an integral–differential equation from a two-dimensional model in transport theory, we shall derive and study a nonsymmetric algebraic Riccati equation B--XF--F+X+XB+X=0, where F±≡I-sˆPD±, B-≡(bˆI+sˆP)D- and B+≡bˆI+sˆPD+ with a nonnegative matrix P, positive diagonal matrices D±, and nonnegative parameters f, bˆ≡b/(...
Article
Full-text available
In this paper, we study the decomposition of the Nehari manifold via the combination of concave and convex nonlinearities. Furthermore, we use this result and the Ljusternik–Schnirelmann category to prove that the existence of multiple positive solutions for a Dirichlet problem involving critical Sobolev exponent.
Article
We investigate the perturbation of the palindromic eigenvalue problem for the matrix quadratic P(λ)≡λ 2 A 1 T +λA 0 +A 1 , with A 0 ,A 1 ∈ℂ n×n and A 0 T =A 0 . The perturbation of eigenvalues and eigenvectors, in terms of palindromic matrix polynomials and palindromic linearizations, are discussed using Sun’s implicit function approach.
Article
Full-text available
We propose a structure-preserving doubling algorithm for a quadratic eigenvalue problem arising from the stability analysis of time-delay systems. We are particularly interested in the eigenvalues on the unit circle, which are difficult to estimate. The convergence and backward error of the algorithm are analyzed and three numerical examples are pr...
Article
Full-text available
In this paper, we study the effect of domain shape on the existence of 2–nodal solutions for a semilinear elliptic equation involving non-odd nonlinearities.
Article
Full-text available
In this paper, we propose the palindromic doubling algorithm (PDA) for the palindromic generalized eigenvalue problem (PGEP) A * x = λAx. We establish a complete convergence theory of the PDA for PGEPs without unimodular eigenvalues, or with unimodular eigenval-ues of partial multiplicities two (one or two for eigenvalue 1). Some important applicat...
Article
Full-text available
We propose an algorithm for the pole assignment problem for descriptor systems with proportional and derivative state feedback. The algorithm is the first of its kind, making use of the Schur form and minimizing the departure from normality of the closed-loop poles by Newton’s method. Three illustrative examples are given.

Network

Cited By