Robust Discontinuity Indicators for High-Order Reconstruction of Piecewise Smooth Functions

In many applications, piecewise continuous functions are commonly interpolated over meshes. However, accurate high-order manipulations of such functions can be challenging due to potential spurious oscillations known as the Gibbs phenomena. To address this challenge, we propose a novel approach, Robust Discontinuity Indicators (RDI), which can efficiently and reliably detect both C^{0} and C^{1} discontinuities for node-based and cell-averaged values. We present a detailed analysis focusing on its derivation and the dual-thresholding strategy. A key advantage of RDI is its ability to handle potential inaccuracies associated with detecting discontinuities on non-uniform meshes, thanks to its innovative discontinuity indicators. We also extend the applicability of RDI to handle general surfaces with boundaries, features, and ridge points, thereby enhancing its versatility and usefulness in various scenarios. To demonstrate the robustness of RDI, we conduct a series of experiments on non-uniform meshes and general surfaces, and compare its performance with some alternative methods. By addressing the challenges posed by the Gibbs phenomena and providing reliable detection of discontinuities, RDI opens up possibilities for improved approximation and analysis of piecewise continuous functions, such as in data remap.Comment: 37 pages, 37 figures, submitted to Computational and Applied Mathematics (COAM

Data Mining Techniques in Analyzing Process Data: A Didactic

Due to increasing use of technology-enhanced educational assessment, data mining methods have been explored to analyse process data in log files from such assessment. However, most studies were limited to one data mining technique under one specific scenario. The current study demonstrates the usage of four frequently used supervised techniques, including Classification and Regression Trees (CART), gradient boosting, random forest, support vector machine (SVM), and two unsupervised methods, Self-organizing Map (SOM) and k-means, fitted to one assessment data. The USA sample (N = 426) from the 2012 Program for International Student Assessment (PISA) responding to problem-solving items is extracted to demonstrate the methods. After concrete feature generation and feature selection, classifier development procedures are implemented using the illustrated techniques. Results show satisfactory classification accuracy for all the techniques. Suggestions for the selection of classifiers are presented based on the research questions, the interpretability and the simplicity of the classifiers. Interpretations for the results from both supervised and unsupervised learning methods are provided