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2022-2023秋季学期 计算和应用数学研讨会(CAM seminar)

本CAM seminar 拟每周邀请一位计算和应用数学领域专家,介绍计算和应用数学领域的最新研究成果,感兴趣的师生可持续关注!

1. 2022年12月7日, 9.30am -11.30 am,汪波(湖南师范大学),Convergence analysis of the Multi-scale Deep Neural Network (MscaleDNN)(腾讯会议:会议号:  密码: )

摘要:In this talk, we will present a numerical analysis for the convergence of the machine learning algorithm with Multi-scale neural network. We prove that the training process for some one layer neural networks with gradient descent optimization algorithm tends to diffusion process in the Fourier spectral domain as the learning rate goes to zero. Consequently, the multi-scale neural network is shown to have diffusion coefficients covering a wider range of frequency compared to fully connected neural network.



2. 2022年12月22日, 9.30-10.30,赵燕翔(The George Washington University),Nonlocal Effects on a Generalized Ohta-Kawasaki Model (腾讯会议:会议号:。。 密码:。。)

摘要: We propose a nonlocal Ohta-Kawasaki model to study the nonlocal effect on the pattern formation of some binary systems with general long-range interactions. While the nonlocal Ohta-Kawasaki model displays similar bubble patterns as the standard Ohta-Kawasaki model, by performing Fourier analysis, we find that the optimal number of bubbles for the nonlocal model may have an upper bound no matter how large the repulsive strength is. The existence of such an upper bound is characterized by the eigenvalues of the nonlocal kernels. Additionally we explore the conditions under which the nonlocal horizon parameter may promote or demote the bubble splitting, and apply the analysis framework to several case studies for various nonlocal operators.




1. 2022年8月19日,马满满,Electrokinetic flow and deformation of a drop in strong electrolytes


A hybrid or multiscale method is introduced to describe the deformation of an immiscible fluid drop in the two-phase flow of ionic fluid electrolytes in the presence of an applied DC electric field. The starting point is the Poisson-Nernst-Planck equations in the Stokes regime, followed by their asymptotic reduction in the limit when the thickness of the Debye layers that form adjacent to the interface is much less than the initial undeformed size of the drop. This leads to the formulation of boundary integral equations for the electrostatic potential and the fluid and interface velocities that contain the coupling between the electrostatic and fluid fields within the thin Debye layers. The model summary and results of the numerical simulations will be presented.

2. 2022年9月22日, 4pm, 权超禹(南方科技大学),On the Strang splitting methods for Allen-Cahn equations



3. 2022年9月28日,4pm-5pm,  仲杏慧 (浙江大学) Energy-Conserving Discontinuous Galerkin Methods for Vlasov Systems



4. 2022年10月6日,4pm-5pm, 胡光辉(澳门大学),含时科恩-沈方程数值方法探讨



5. 2022年10月11日,陈华杰(北京师范大学),  Convergence of the Planewave Approximations for Quantum Incommensurate Systems



6. 2022年10月19日,4pm-5pm, 黄纪祖(中国科学院)Energy stable schemes for gradient flows based on the DVD method



7. 2022年10月27日,上午,廖洪林(南京航空航天大学)变步长BDF 格式的稳定性与收敛性



8. 2022年11月3日,9am-10am, 王柱(南卡罗来纳大学),Level Set Learning for Dimensionality Reduction



9. (吴文俊数学中心综合报告2022年11月7日,9am-10am,柳春(Illinois Institute of Technology, Chicago),Topics in transport dynamics: boundaries and temperature



10. 2022年11月9日,9.30am-10.30am, 徐劼(中国科学院数学与系统科学研究院),Quasi-entropy and applications in liquid crystals



11. 2022年11月14日, 3.30pm -5.30pm,陈黄鑫(厦门大学), Physically consistent numerical methods for flows in porous media(腾讯会议:会议号:184631176 密码:832586 )

摘要:Simulation of multi-phase flow in porous media has wide applications such as petroleum fields and water resources. In this talk we will introduce physically consistent numerical methods for the simulation of incompressible and immiscible two-phase flow in heterogeneous porous media with capillary pressure. The new algorithm is unbiased and locally mass conservative for both of phases. We will also introduce an efficient energy stable numerical method for the Maxwell-Stefan-Darcy two-phase flow model in porous media, which can preserve multiple important physical properties of the model. Moreover, a fully discrete scheme for the stochastic Stokes-Darcy equations with multiplicative noise and its numerical analysis will also be discussed.



12. 2022年11月17日, 2pm -3pm,商晓成(University of Birmingham), Structure-preserving integrators for dissipative systems based on reversible–irreversible splitting (腾讯会议:会议号:774375728 密码:771376 )

摘要: We study the optimal design of numerical integrators for dissipative systems, for which there exists an underlying thermodynamic structure known as GENERIC (general equation for the nonequilibrium reversible–irreversible coupling). We present a framework to construct structure-preserving integrators by splitting the system into reversible and irreversible dynamics. The reversible part, which is often degenerate and reduces to a Hamiltonian form on its symplectic leaves, is solved by using a symplectic method (e.g. Verlet) with degenerate variables being left unchanged, for which an associated modified Hamiltonian (and subsequently a modified energy) in the form of a series expansion can be obtained by using backward error analysis. The modified energy is then used to construct a modified friction matrix associated with the irreversible part in such a way that a modified degeneracy condition is satisfied. The modified irreversible dynamics can be further solved by an explicit midpoint method if not exactly solvable. Our findings are verified by various numerical experiments, demonstrating the superiority of structure-preserving integrators over alternative schemes in terms of not only the accuracy control of both energy conservation and entropy production but also the preservation of the conformal symplectic structure in the case of linearly damped systems.



13. 2022年11月24日, 4.00pm -5.00pm,蒋维(武汉大学),A new sharp-interface model for simulating solid-state dewetting problems(腾讯会议:会议号:159320797 密码:625401

摘要:Solid-state dewetting is a ubiquitous phenomenon in materials science, and it describes the agglomeration of solid thin films into arrays of isolated particles on a substrate. In recent years, solid-state dewetting has found wide applications in modern technology, and much efforts have been devoted to understanding this important phenomenon. One of the effective tools for modeling and simulations of solid-state dewetting was the sharp-interface model proposed by my group [see Phys. Rev. B, 91:045303, 2015; SIAM J. Appl. Math., 80:1654, 2020]. However, these sharp-interface models belong to free boundary problems, which explicitly include surface diffusion and contact line migration. It is very difficult to design efficient numerical algorithms for solving the sharp-interface models. On the other hand, lots of pinch-off events will occur when a long island film evolves by solid-state dewetting, but the previous sharp-interface models can not automatically handle with topological events. In order to tackle these difficulties, we propose a new sharp-interface model with the thickness-dependent surface energy for simulating solid-state dewetting by removing the singularity of the triple point among the film, vapor and substrate phases. The new model is only needed to solve in a fixed domain, and it can automatically capture “topological events”. Numerical results demonstrate the high performance of the new model.



14. 2022年11月30日,10am-11am,王成(UMass Dartmouth), Linearized numerical schemes for nonlocal Cahn-Hilliard equation and its convergence analysis

摘要:A stabilized linear semi-implicit numerical scheme is considered for the nonlocal Cahn-Hilliard equation, and a detailed convergence analysis is presented. This theoretical analysis follows from consistency and stability estimates for the numerical error function. Due to the complicated form of the nonlinear term, we adopt the discrete H^{-1} norm for the error function to establish the convergence result. In addition, an assumption on the uniform maximum bound of the numerical solution is required for the theoretical justification of the energy stability, and such a bound is derived by conducting the higher order consistency analysis. Taking the view that the numerical solution is indeed the exact solution with a perturbation, the error function is uniformly bounded under a loose constraint of the time step size, which then leads to the uniform maximum-norm bound of the numerical solution. The second order accurate numerical schemes, either in the BDF2 or Crank-Nicolson approximation, are also analyzed as well.



15. 2022年12月1日, 4pm -5pm,熊涛(厦门大学), Asymptotic preserving and uniformly conditionally   stable finite difference schemes for kinetic transport equations(腾讯会议:会议号:695798063 密码:892385

摘要:In this work, uniformly unconditionally stable first and second order finite difference schemes are developed for kinetic transport equations in the diffusive scaling. We derived an approximate evolution equation for the macroscopic density, from the formal solution of the distribution function, which is then discretized by following characteristics for the transport part with a backward finite difference semi-Lagrangian approach, while the diffusive part is discretized implicitly. The resulting schemes can be shown to be asymptotic preserving (AP) in the diffusive limit. Uniformly unconditional stabilities are verified from a Fourier analysis. Numerical experiments, including high dimensional problems, have demonstrated the corresponding orders of accuracy both in space and in time, uniform stability, AP property, and good performances of our proposed approach.