Applied Math
PhD Seminar

2026 Spring

The seminar of this semester is organized by Jiamao Wu and Zhuangao He, and co-organized by the graduate student union in the School of Mathematical Sciences at Fudan. This section is partially sponsored by Shanghai Key Laboratory for Contemporary Applied Mathematics.

2026-03-12 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand In this talk, we consider the numerical approximation of the Abels-Garcke-Grün model for a binary mixture of two viscous incompressible fluids with unmatched densities and viscosities. The system consists of a Navier-Stokes equation for the volume-averaged fluid velocity and a convective Cahn-Hilliard equation with Flory-Huggins potential for the phase-field variable. Based on the pressure stabilization method and an additional first-order stabilization term to the advective velocity in the Cahn-Hilliard equation, the scheme is fully decoupled and preserves the physical properties, i.e. the positivity-preserving property, which means that the discrete solution of the phase-field always stays in the physical interval (-1, 1) at a point-wise level, and the unconditional energy stability. Additionally, we perform a detailed optimal rate convergence analysis and derive error estimates. Numerical results are presented to validate the convergence rate and energy stability.

2026-03-19 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand We study optimal learning rate schedules (LRSs) under the functional scaling law (FSL) framework, where the loss is written explicitly as a functional of the schedule. This formulation reveals a sharp phase transition governed by two exponents: a source exponent $s>0$ controlling signal learning and a capacity exponent $\beta>1$ controlling noise forgetting. In the easy-task regime, the optimal schedule follows a power decay to zero, with exponent determined by $\beta$. In the hard-task regime, the optimal schedule takes a warmup-stable-decay (WSD) form: it maintains the largest admissible learning rate for most of training and decays only near the end, with a vanishing decay fraction. We further analyze shape-fixed schedules, showing how the tail exponent governs both their optimality and their limitations through capacity saturation. This yields a principled evaluation of commonly used schedules such as cosine and linear decay. Finally, we apply the power-decay LRS to one-pass SGD for kernel regression and show that the last iterate attains the exact minimax-optimal rate.

Past Presentations

2026-03-05 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand Currently, several finite difference methods are used to solve the Dirac equation in the semiclassical regime. However, finite difference methods have certain limitations in terms of accuracy and stability. The CNFD method, often employed to enhance stability, usually requires solving large linear systems, leading to significant computational costs. In this work, we conduct a rigorous analysis of the numerical error for the time-splitting methods applied to the Dirac equation in the semiclassical region. The dimensionless parameter $\epsilon \in (0,1]$, representing the reduced Planck constant, causes the solution field to exhibit rapid oscillations with characteristic wavelengths of order $O(\epsilon)$. We demonstrate that both $S_1$ and $S_2$ schemes preserve total mass conservation in a discrete sense. Furthermore, we establish error estimates for the time-splitting methods, clearly describing how the approximation error varies with time resolution $\tau$, spatial discretization step $h$, and the parameter $\epsilon$. We also derive error bounds for physical observables, and through numerical experiments, we validate the reliability of the error estimates.