Weide 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.

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.

Jin Yuan, Xuelan Qiu, Jinran Wu et al.

The interest in predicting online learning performance using ML algorithms has been steadily increasing. We first conducted a scientometric analysis to provide a systematic review of research in this area. The findings show that most existing studies apply the ML methods without considering learning behavior patterns, which may compromise the prediction accuracy and precision of the ML methods. This study proposes an integration framework that blends learning behavior analysis with ML algorithms to enhance the prediction accuracy of students' online learning performance. Specifically, the framework identifies distinct learning patterns among students by employing clustering analysis and implements various ML algorithms to predict performance within each pattern. For demonstration, the integration framework is applied to a real dataset from edX and distinguishes two learning patterns, as in, low autonomy students and motivated students. The results show that the framework yields nearly perfect prediction performance for autonomous students and satisfactory performance for motivated students. Additionally, this study compares the prediction performance of the integration framework to that of directly applying ML methods without learning behavior analysis using comprehensive evaluation metrics. The results consistently demonstrate the superiority of the integration framework over the direct approach, particularly when integrated with the best-performing XGBoosting method. Moreover, the framework significantly improves prediction accuracy for the motivated students and for the worst-performing random forest method. This study also evaluates the importance of various learning behaviors within each pattern using LightGBM with SHAP values. The implications of the integration framework and the results for online education practice and future research are discussed.