报告题目: Non-splitting Eulerian-Lagrangian WENO schemes for two-dimensional nonlinear convection-diffusion equations

报告人：蔡晓峰 (北京师范大学)

时间：2025年8月17日 9:00—9:40

地点：澳门新葡京娱乐城 425报告厅

报告摘要：

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.

报告人简介： 蔡晓峰，北京师范大学数学研究中心执行主任，副教授，北师香港浸会大学副教授（双聘），主要从事针对双曲方程和动理学模型的高效稳健可靠的数值方法研究。研究成果发表在Journal of Computational Physics、SIAM Journal on Scientific Computing、Mathematics of Computation等权威期刊

报告题目: A posteriori error estimates in H(curl) and H(div) via auxiliary space preconditioning

报告人：李雨文 (浙江大学研究员)

时间：2025年8月17日 9:40—10:20

地点：澳门新葡京娱乐城 425报告厅

报告摘要：This talk presents a posteriori error estimates of finite elements in H(curl) and H(div) problems based on auxiliary H(grad) space preconditioning, which is a continuous analogue of Hipmair-Xu preconditioner on the continuous level. In particular, we obtain novel parameter-robust and polynomial-degree-robust a posteriori error estimates in H(curl) and H(div) spaces. The material is based on joint work with Ludmil Zikatanov.

报告人简介：李雨文在南京大学取得理学学士和硕士学位，在加利福尼亚大学圣迭戈分校获得数学博士学位，2019-2022年在宾州州立大学数学系担任乔拉研究助理教授，之后加入浙江大学数学科学学院担任百人计划研究员。他的主要研究方向是微分方程数值解，包括有限元方法、保结构方法、模型降阶、逼近理论，研究成果发表于SINUM、MathComp、FoCM的计算数学高水平期刊，主持一项国家自然科学基金面上项目。

报告题目: Adaptive Space-Time Methods for Multiscale Flow Problems Using a Partially Explicit Splitting Scheme

报告人：梁永达 (香港城市大学教授)

时间：2025年8月17日 10:40—11:20

地点：澳门新葡京娱乐城 425报告厅

报告摘要： In this talk, I will present a space-time adaptive framework for efficiently simulating flow problems in multiscale media with high-contrast coefficients. These problems pose significant challenges due to the need to capture non-local effects across scales and the restrictive time step sizes required by explicit schemes in the presence of large coefficient variations. To address these issues, we propose a partially explicit temporal splitting scheme combined with an adaptive multiscale method. Our approach constructs two multiscale subspaces to separately handle fast and slow flow components, using implicit discretization for one and explicit for the other. A multirate time-stepping strategy is employed to accommodate the disparate flow rates across regions. To ensure both accuracy and efficiency, we derive a posteriori error estimators in space and time that guide local enrichment of  the multiscale spaces and adaptive refinement of time steps. Starting with a minimal number of basis functions and coarse temporal resolution, the algorithm adaptively improves the solution by introducing energy-minimizing basis functions where needed. I will discuss the stability and convergence analysis of the method, and present numerical results that illustrate its effectiveness in capturing multiscale features while significantly reducing computational cost.

报告人简介： 梁永达教授于2010年获得香港中文大学数学学士学位。2012年获得香港中文大学数学硕士学位。2017年在美国德克萨斯A&M大学获得数学博士学位。在2022年加入香港城市大学之前，他曾在加州大学尔湾分校担任访问助理教授。他的研究兴趣主要集中在多尺度问题的数值方法。

报告题目:  pETNNs: Partial Evolutionary Tensor Neural Networks for Solving Time-dependent Partial Differential Equations

报告人：赵进 (首都师范大学交叉科学研究院特聘副研究员)

时间：2025年8月17日 11:20—12:00

地点：澳门新葡京娱乐城 425报告厅

报告摘要：In this talk, we will introduce our recent work for solving time-dependent partial differential equations with both of high accuracy and remarkable extrapolation, called partial evolutionary tensor neural networks (pETNNs). Our proposed architecture leverages the inherent accuracy of tensor neural networks, while incorporating evolutionary parameters that enable remarkable extrapolation capabilities. By adopting innovative parameter update strategies, the pETNNs achieve a significant reduction in computational cost while maintaining precision and robustness. Notably, the pETNNs enhance the accuracy of conventional evolutional deep neural networks and empowers computational abilities to address highdimensional problems. Numerical experiments demonstrate the superior performance of the pETNNs in solving time-dependent complex equations, including the Navier-Stokes equations, high-dimensional heat equation, highdimensional transport equation and Korteweg-de Vries type equation.

报告人简介：赵进，首都师范大学交叉科学研究院特聘副研究员。中国工程物理研究院博士，曾在北京大学数学科学学院从事博士后研究工作。他的研究方向包含数值方法，数学建模，机器学习等，包括将机器学习应用到偏微分方程数值解和数学建模中。相关工作发表在SINUM，SISC, JCP, PRE, INT J HEAT MASS TRAN等权威学术期刊。

报告题目:  高维密度函数估计方法及求解偏微分方程

报告人：廖奇峰

时间：2025年8月17日 15:00—15:40

地点：澳门新葡京娱乐城 425报告厅

报告摘要：概率密度函数估计仍是计算科学与工程中的一个难题。通过耦合 Knothe-Rosenblatt (KR) 重排和基于流的生成模型，我们开发了一种可逆传输映射，称为 KRnet，用于高维密度估计。本报告对Krnet就行概述，并介绍其用于求解高维偏微分方程的自适应版本。

报告人简介：廖奇峰目前为上海科技大学信息科学与技术学院常任副教授、研究员、博士生导师，视觉与数据智能研究中心联合主任、中国数学会计算数学分会理事。于2010在英国曼彻斯特大学澳门新葡京娱乐城 获得数值计算博士学位，于2006年获得四川大学澳门新葡京娱乐城 学士学位。2011年1月至2012年6月，在美国马里兰大学计算机系从事博士后研究工作；2012年7月至2015年2月，在美国麻省理工学院航空航天系从事博士后研究工作；2015年3月，作为研究员加入上海科技大学信息科学与技术学院。

报告题目:  A ROM-enhanced Preoconditioner for a Kinetic Transport Equation

报告人：彭志超(香港科技大学数学系助理教授)

时间：a 15:40—16:20

地点：澳门新葡京娱乐城 425报告厅

报告摘要： We consider a linear kinetic transport equation that models particles propagating through and interacting with a background medium. In multi-query applications, such as uncertainty quantification, sensitivity analysis, design optimization and medical imaging, this equation may be solved multiple times for various parameters. Reduced order model (ROM) is a technique to reduce degrees of freedom for these parametric problems by finding and leveraging low-rank structures across parameters. In this talk, we will briefly discuss our recent work on applications of ROMs for this equation.

We will design a ROM-enhanced preconditioner for parametric problems. Classical diffusion synthetic acceleration (DSA) preconditioner builds on the diffusion limit of the kinetic transport equation. However, without sufficiently strong scattering effect, the kinetic equation may be far from this limit. Additionally, low-rank structures across parameters are not utilized by DSA. To address these issues, we enhance the DSA preconditioner with a data-driven ROM that starts from the original kinetic description and leverages low-rank structures across parameters.

报告人简介： 彭志超，现任香港科技大学数学系助理教授一职。2015年在北京大学取得学士学位，2020年在美国伦斯勒理工大学取得博士学位，曾任美国密歇根州立大学访问助理教授。主要研究兴趣是偏微分方程数值解和数据驱动的模型约化。

报告题目: A Randomized GMsFEM with Data-Driven Predictors for Parametric Flow Problems in Multiscale Heterogeneous Media

报告人：刘松玮(香港城市大学)

时间：2025年8月17日 16:20—17:00

地点：澳门新葡京娱乐城 425报告厅

报告摘要：In this report, we propose a randomized generalized multiscale finite element method (Randomized GMsFEM) for flow problems with parameterized inputs and high-contrast heterogeneous media. The method employs a data-driven predictor to construct multiscale basis functions in two stages: offline and online. In the offline stage, a snapshot space is generated via spectral decompositions, and a reduced matrix is obtained using SVD to predict eigenfunctions. In the online stage, these eigenfunctions are evaluated for new parameter realizations to construct the multiscale space. Furthermore, our approach addresses the complexity of multiple permeability fields with random inputs and multiple multiscale information,  providing accurate and efficient approximations. Moreover, we conduct a rigorous convergence analysis for our Randomized GMsFEM. Finally, we present extensive numerical examples, demonstrating its superior performance compared to the traditional GMsFEM.

报告人简介： 刘松玮，2024年本科毕业于湖南大学，现就读于香港城市大学数学系，导师梁永达。研究方向为多尺度问题，数值分析，模型降阶。