Highly accurate operator factorization methods for the integral fractional Laplacian and its generalization

In this paper, we propose a new class of operator factorization methods to discretize the integral fractional Laplacian $(-\Delta)^\frac{\alpha}{2}$ for $\alpha \in (0, 2)$. The main advantage of our method is to easily increase numerical accuracy by using high-degree Lagrange basis functions, but remain the scheme structure and computer implementation unchanged. Moreover, our discretization of the fractional Laplacian results in a symmetric (multilevel) Toeplitz differentiation matrix, which not only saves memory cost in simulations but enables efficient computations via the fast Fourier transforms. The performance of our method in both approximating the fractional Laplacian and solving the fractional Poisson problems was detailedly examined. It shows that our method has an optimal accuracy of ${\mathcal O}(h^2)$ for constant or linear basis functions, while ${\mathcal O}(h^4)$ if quadratic basis functions are used, with $h$ a small mesh size. Note that this accuracy holds for any $\alpha \in (0, 2)$ and can be further increased if higher-degree basis functions are used. If the solution of fractional Poisson problem satisfies $u \in C^{m, l}(\bar{\Omega})$ for $m \in {\mathbb N}$ and $0 < l < 1$, then our method has an accuracy of ${\mathcal O}\big(h^{\min\{m+l,\, 2\}}\big)$ for constant and linear basis functions, while ${\mathcal O}\big(h^{\min\{m+l,\, 4\}}\big)$ for quadratic basis functions. Additionally, our method can be readily applied to study generalized fractional Laplacians with a symmetric kernel function, and numerical study on the tempered fractional Poisson problem demonstrates its efficiency.Comment: 21 pages, 7 figure

Understanding Patient Journeys with Telehealth: A Poisson-Factor-Marked Hawkes Process

The emerging telehealth platforms connect patients with physicians using telecommunication technologies and are transforming the traditional healthcare delivery process. Meanwhile, patient care journeys spreading across online and offline health service channels call for new research methodologies. Using a dataset from a telehealth platform, we develop a novel Poisson-factor-marked Hawkes process to model such a journey and quantify the mutual-modulating effects of various patient activities. Our estimation results demonstrate the disparate impacts of the patient’s health conditions and physician characteristics on choosing care channels. Taking advantage of the self-generation property of our model, we simulate policy and strategic interventions, which highlights the practical value of the proposed model and offers implications for better patient routing and service design for telehealth platforms

Eat4Thought: A Design of Food Journaling

Food journaling is an effective method to help people identify their eating patterns and encourage healthy eating habits as it requires self-reflection on eating behaviors. Current tools have predominately focused on tracking food intake, such as carbohydrates, proteins, fats, and calories. Other factors, such as contextual information and momentary thoughts and feelings that are internal to an individual, are also essential to help people reflect upon and change attitudes about eating behaviors. However, current dietary tracking tools rarely support capturing these elements as a way to foster deep reflection. In this work, we present Eat4Thought -- a food journaling application that allows users to track their emotional, sensory, and spatio-temporal elements of meals as a means of supporting self-reflection. The application enables vivid documentation of experiences and self-reflection on the past through video recording. We describe our design process and an initial evaluation of the application. We also provide design recommendations for future work on food journaling.Comment: 8 page

A Comprehensive Study and Comparison of the Robustness of 3D Object Detectors Against Adversarial Attacks

Recent years have witnessed significant advancements in deep learning-based 3D object detection, leading to its widespread adoption in numerous applications. As 3D object detectors become increasingly crucial for security-critical tasks, it is imperative to understand their robustness against adversarial attacks. This paper presents the first comprehensive evaluation and analysis of the robustness of LiDAR-based 3D detectors under adversarial attacks. Specifically, we extend three distinct adversarial attacks to the 3D object detection task, benchmarking the robustness of state-of-the-art LiDAR-based 3D object detectors against attacks on the KITTI and Waymo datasets. We further analyze the relationship between robustness and detector properties. Additionally, we explore the transferability of cross-model, cross-task, and cross-data attacks. Thorough experiments on defensive strategies for 3D detectors are conducted, demonstrating that simple transformations like flipping provide little help in improving robustness when the applied transformation strategy is exposed to attackers. Finally, we propose balanced adversarial focal training, based on conventional adversarial training, to strike a balance between accuracy and robustness. Our findings will facilitate investigations into understanding and defending against adversarial attacks on LiDAR-based 3D object detectors, thus advancing the field. The source code is publicly available at \url{https://github.com/Eaphan/Robust3DOD}.Comment: 30 pages, 14 figure