Tiexiang Li

Tsung-Ming Huang, Tiexiang Li, Wei-De Li et al.

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 with all 14 Bravais lattices can be efficiently computed in a unified framework. We uncover the delicate machinery behind several key results of our work and on the basis of this new understanding we drastically simplify the derivations, proofs and arguments in our framework. In this work particular effort is made on reformulating the Bloch boundary condition for all 14 Bravais lattices in the redefined orthogonal coordinate system, and establishing eigen-decomposition of discrete partial derivative operators by systematic use of commutativity among them, which has been overlooked previously, and reducing eigen-decomposition of double-curl operator to the canonical form of a 3x3 complex skew-symmetric matrix under unitary congruence. With the validity of the novel nullspace free method in the broad context, we perform some calculations on one benchmark system to demonstrate the accuracy and efficiency of our algorithm.

Jingkai Zhang, Xing-Long Lyu, Tiexiang Li et al.

Band-structure calculations for three-dimensional anisotropic phononic crystals require the repeated solution of large elastic generalized eigenvalue problems along Bloch paths. In standard staggered-grid discretizations, anisotropic coupling may involve derivative components located at incompatible grid positions, so additional interpolation or averaging closures are often introduced. This paper proposes a parity-blocked rotated staggered discretization based on four Bloch-periodic body-diagonal differences. The directional derivatives are reconstructed from these diagonal differences, leading to a Hermitian $B_hC_hB_h^H$ generalized eigenvalue formulation that incorporates anisotropic derivative coupling without separate interpolation closures. On even grids, when the stiffness and mass matrices are nodewise local multiplication matrices, the body-diagonal shifts preserve two independent parity invariants. The discrete velocity space is then decomposed exactly into four mutually independent block subspaces, and the full discrete spectrum can be recovered by solving the four smaller eigenvalue problems and merging their spectra. The full and block formulations are further organized in a unified Fourier SVD framework, which supports $Γ$-point zero-mode treatment, shift-invert Krylov iteration, inner PCG solves, and GPU matrix-vector products. Numerical experiments for a three-dimensional two-phase anisotropic phononic crystal show that the block implementation preserves the full-space spectrum while substantially reducing the wall-clock time. The results demonstrate that the proposed method provides a structured and efficient solver for large-scale band-structure computations of three-dimensional anisotropic phononic crystals.

Teng-Chao Sun, Tiexiang Li, Wen-Wei Lin et al.

In this paper, we propose a spectral framework that embeds 1D and 2D quasiperiodic Helmholtz eigenvalue problems into higher-dimensional (2D and 4D) periodic spaces via the projection method \cite{jiang2014numerical, jiang2024numerical}. To effectively map the elevated high-dimensional states back to the original physical space, we establish a novel validation framework based on the weighted expectation of pointwise Rayleigh quotients. Supported by comprehensive error and spectral analysis, we demonstrate that the eigenvalues derived from this expectation align more authentically with the original quasiperiodic model, ultimately yielding a more appropriate and reliable eigenpair solution. Numerical experiments on continuous media demonstrate that our approach offers an accurate, robust, and scalable tool for solving quasiperiodic Helmholtz eigenvalue problems.

Huang Tsung-Ming, Li Tiexiang, Li Wei-De et al.

There are many numerical methods for simulate three-dimensional photonic crystals, after comparison, we choose Yee's scheme to be our discrete method. So far, this method can only be applied to simple cubic lattice and face-centered cubic lattice, but there are 14 Bravais lattices in three-dimensional space. In this paper we extended Yee's scheme to all of the 14 Bravais lattices. After discretization, we get a general eigenvalue problem, and we will analyze this general eigenvalue problem. The second important task of this paper is to find the eigen-decomposition of the discrete curl operator, after a series of complicated calculations, we find that all the lattices can be summed up into two kinds of decomposition. We are interested in finding the several few smallest real eigenvalues, but the large dimension of null space is 1/3 of all, this seriously affected the convergence of calculation, we use a technique which is called nullspace-free method to avoid this trouble. But this technique transforms the sparse matrix in our problem to a dense matrix, fortunately, the eigenvectors we found before are related to discrete Fourier transformation. The efficiency of calculation has been significantly improved by using the fast Fourier transformation. Finally, we calculate the band structures of photonic crystals on various lattices, and implement high performance calculations on the GPU.

Zhen-Chen Guo, Tiexiang Li, Ying-Ying Zhou

To solve the Bethe-Salpeter eigenvalue problem with distinct sizes, two efficient methods, called ΓQR algorithm and Γ-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 and the associated eigenvectors still hold the properties similar to those of $H$. Theorems are given to demonstrate the validity of the proposed two algorithms in theory. Numerical results are presented to illustrate the superiorities of our methods.

Wen-Jie Wu, Tiexiang Li, Wen-Wei Lin

We develop a unified DeepONet framework for logarithmically stable infinite-dimensional inverse problems, with inverse acoustic scattering as a model application. The framework is formulated at the operator level by separating the learned inverse map into measurement encoding, finite-dimensional neural approximation, and functional reconstruction components. For inverse maps satisfying a logarithmic stability estimate, we establish quantitative a priori error bounds giving separate estimates for the encoder error, the neural approximation error, and the reconstruction error, thereby characterizing the dependence on the encoder dimension, the network size, and the reconstruction dimension. For comparison, we also record the corresponding Lipschitz-stable estimate arising from the same error decomposition. The abstract theory is then specialized to the recovery of a medium contrast from fixed-frequency far-field measurements. Numerical experiments in two and three dimensions illustrate stable reconstructions under measurement noise.

Jingkai Zhang, Tiexiang Li, Shuo Zhang

This paper develops a natural decomposition method (NDM)for imposing essential boundary conditions in noninterpolatory meshfree Galerkin spaces without boundary parameter tuning or auxiliary constraint construction. In such spaces, algebraic coefficients generally do not coincide with boundary values; hence coefficient assignment or nodal boundary prescription is not equivalent to imposing the continuous trace required by the variational problem. NDM introduces boundary data before discretization through a natural transfer mechanism: a source subproblem accounts for the forcing term, a weighted curl correction transfers the remaining trace mismatch, and a scalar recovery step reconstructs the solution from the corrected weighted gradient. For topologically trivial single domains with connected boundary, the reconstructed solution is equivalent, at the continuous level, to the solution satisfying the prescribed essential boundary data. The discrete analysis separates the approximation defect of the recovery space from the upstream transfer error visible to that space. Numerical experiments on benchmark problems evaluate the proposed transfer mechanism and report the associated conditioning, computational cost, and boundary perturbation behavior.