Year | Month |
2024 | Jan Feb Mar May Jun Jul Aug |
2023 | Jan Feb Mar Apr May Jun Jul Aug Oct Nov Dec |
Title: | Non-splitting Eulerian-Lagrangian WENO schemes for two-dimensional nonlinear convection-diffusion equations |
Speaker: | Dr. Xiaofeng Cai, Research Center for Mathematics Beijing Normal University and UIC |
Time/Place: | 15:00 - 16:00 FSC1217 |
Abstract: | In this talk, we develop high-order, conservative, non-splitting Eulerian-Lagrangian (EL) Runge-Kutta (RK) finite volume (FV) weighted essentially non-oscillatory (WENO) schemes for convection-diffusion equations. The proposed EL-RK-FV-WENO scheme defines modified characteristic lines and evolves the solution along them, significantly relaxing the time-step constraint for the convection term.The main algorithm design challenge arises from the complexity of constructing accurate and robust reconstructions on dynamically varying Lagrangian meshes. This reconstruction process is needed for flux evaluations on time-dependent upstream quadrilaterals and time integrations along moving characteristics.To address this, we propose a strategy that utilizes a WENO reconstruction on a fixed Eulerian mesh for spatial reconstruction, and updates intermediate solutions on the Eulerian background mesh for implicit-explicit RK temporal integration. This strategy leverages efficient reconstruction and remapping algorithms to manage the complexities of polynomial reconstructions on time-dependent quadrilaterals, while ensuring local mass conservation. The proposed scheme ensures mass conservation due to the flux-form semi-discretization and the mass-conservative reconstruction on both background and upstream cells. Extensive numerical tests have been performed to verify the effectiveness of the proposed scheme. |
Title: | Composite Algorithms of Data-driven and Model-driven Methods in Banach Spaces |
Speaker: | Prof. Qi YE, School of Mathematical Sciences, South China Normal University |
Time/Place: | 15:00 - 16:00 FSC1217 |
Abstract: | In this presentation, we introduce a novel mathematical framework for machine learning that integrates data-driven and model-driven methods. Typically, data-driven methods are employed to implement black-box algorithms, while model-driven methods are utilized for white-box algorithms. The primary concept involves leveraging the local information from multimodal data and multiscale models to develop global approximate solutions through learning algorithms. The utilization of composite algorithms offers an alternative approach to exploring the mathematical theory of machine learning. This includes investigating interpretability through approximation theory, nonconvexity and nonsmoothness through optimization theory, and generalization and overfitting through regularization theory. For our computational medicine project focusing on pancreatic cancer, we investigate the composite algorithm involving image processing and modeling simulation. |
Title: | Mathematical Frameworks for Understanding Biological Shape Formation: From Brain Organoids to Skin Wrinkles |
Speaker: | Dr. Xiaoyi Chen, Beijing Normal University & Hong Kong Baptist University United International College |
Time/Place: | 15:00 - 16:00 FSC1217 |
Abstract: | In recent years, we studied how biological tissues generate complex morphologies through different mechanisms during growth mathematically and mechanically. Specifically, our work is concentrated in two main areas: 1. We have established a stress-free growth model to analytically study the morphological development of different biological tissues. This model provides a significant complement to the currently popular instability mechanisms. 2. We have derived asymptotically consistent morphoelastic plate/shell models that can effectively describe the large deformations of thin/slender tissues commonly found in nature. Using these models, we have explained several biological growth phenomena, including the formation of brain organic and the inversion of the Volvox embryo as well as the wrinkling of human skin. Looking ahead, we plan to develop related numerical algorithms based on the morphoelastic plate/shell theory, including the recently popular Physics-Informed Neural Networks (PINN) method. |
Title: | Bayesian Methods for Random-effects Meta-analysis of Rare Binary Events in Biomedical Research |
Speaker: | Prof. Xinlei Sherry Wang, Department of Mathematics,University of Texas at Arlington,USA |
Time/Place: | 10:00 - 11:00 FSC1217 |
Abstract: | Meta-analysis of rare binary events in biomedical research is often hampered by low statistical power in individual studies and limitations of traditional methods. This paper introduces novel Bayesian procedures that address these challenges. Our approach employs a flexible random-effects model, eliminating the need for pre-specified directions of variability, and incorporates P6lya-Gamma augmentation for efficient computation. Simulations and real-world applications (56 studies on rosiglitazone, 41 studies on stomach ulcers) demonstrate less biased and more stable estimates compared to conventional methods. Further, we propose a Bayesian goodness-of-fit test within a binomial-normal hierarchical model. This test leverages pivotal quantities and the Cauchy combination method for dependent p-values, effectively utilizing all data, including double zeros without artificial correction. Simulations and real data applications confirm well-controlled Type I error rates and increased power for detecting model misspecification. |
Title: | Original Studies on the Prevention and Treatment of Diabetic Panvascular Disease by Integrating Chinese and Western Medicine |
Speaker: | Ms. Wenting Wang, China Center for Evidence‑Based Medicine of TCM, China |
Time/Place: | 11:00 - 12:00 FSC1217 |
Abstract: | Diabetic panvascular diseases (DPDs) has become a major public health problem, and the search for preventive and curative drugs with panvascular benefits and the elucidation of their mechanisms have become a major focus and challenge in the field of Cardiometabolic disease. From a systemic and holistic perspective, Traditional Chinese medicine (TCM), with its multi-component, multi-target and systemic regulation, has unique advantages and development potential in the prevention and treatment of DPDs, and can work synergistically with Western medicine to improve patients' clinical symptoms and poor prognosis. However, the unclear biological substance of TCM symptoms and the unclear mechanism of action of TCM formulas have seriously restricted its clinical application. Focusing on the above issues, our team has conducted a series of original studies on diabetic coronary artery disease, diabetic atherosclerosis and vascular calcification, and has obtained a number of research results, which provide evidence to support the construction of Chinese and Western medicine diagnostic and treatment strategies for diabetic DPDs. |
Title: | On the Backward Error Incurred by the Compact Rational Krylov Linearization |
Speaker: | Dr. Hongjia Chen, Nanchang University, Nanchang, China |
Time/Place: | 10:00 - 11:00 FSC1217 |
Abstract: | One of the most successful methods for solving a polynomial (PEP) or rational eigenvalue problem (REP) is to recast it, by linearization, as an equivalent but larger generalized eigenvalue problem which can be solved by standard eigensolvers. In this work, we investigate the backward errors of the computed eigenpairs incurred by the application of the well-received compact rational Krylov (CORK) linearization. Our treatment is unified for the PEPs or REPs expressed in various commonly used bases, including Taylor, Newton, Lagrange, orthogonal, and rational basis functions. We construct one-sided factorizations that relate the eigenpairs of the CORK linearization and those of the PEPs or REPs. With these factorizations, we establish upper bounds for the backward error of an approximate eigenpair of the PEPs or REPs relative to the backward error of the corresponding eigenpair of the CORK linearization. These bounds suggest a scaling strategy to improve the accuracy of the computed eigenpairs. We show, by numerical experiments, that the actual backward errors can be successfully reduced by scaling and the errors, before and after scaling, are both well predicted by the bounds. |
The Department has a distinguished record in teaching and research. A number of faculty members have been recipients of relevant awards.
Learn MoreDr S. Hon recevied the Early Career Award (21/22) from the Research Grants Council.
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