PKUAMLSemniar

PKU Applied Math Lunch Seminar

应用数学青年讨论班（午餐会）日程表

本学期（2024春）应用数学拔尖计划将开展午餐会，本学期由金则宇和李天佑同学负责。

本学期日程如下：

Mar 14

庞彤瑶（清华大学丘成桐数学中心）

Image Restoration: from Sparse Approximation to Deep Learning

具体信息

摘要: Image restoration refers to recovering high-quality images from degraded or limited measurements, which has applications in many fields, such as science and medicine. Recently, deep learning has emerged as a prominent tool for many problems including image restoration. Most of the deep learning methods are supervised which requires large amount of paired training data including truth images. In this talk, I will introduce several self-supervised methods which only use the on-hand measurements for training while still showing comparable performance to supervised learning. These proposed self-supervised methods have great potential for real-world image restoration tasks, where it can be difficult to collect clean images and build high-quality training datasets.

报告人信息: 报告人简介：报告人现为清华大学丘成桐数学中心助理教授，2014年本科毕业于北京大学元培学院，2019年博士毕业于新加坡国立大学数学系，并继续在该系从事博士后研究至2023年底，导师为沈佐伟和纪辉教授。其主要研究方向包括无监督深度学习算法，图像处理，贝叶斯估计等。
Mar 28

廖世晨（中国科学院大学数学科学学院）

Subspace Newton Method for Sparse Group L0 Optimization Problem

具体信息

摘要: In this talk, we focus on sparse optimization problem involving sparse group L0 norm regularization, which is an important class of nonconvex and discontinuous optimization problem that has a wild range of application. In terms of theory, we analyze the optimality conditions of the sparse group optimization problem, leveraging the notion of a $\kappa$-stationary point, whose linkage to local and global minimizer is established. In terms of algorithms, several classical algorithms for directly solving L0 regularized problem will be reviewed. We then propose a subspace Newton algorithm for sparse group L0 optimization problem and prove its global convergence property as well as local second-order convergence rate. Numerical experiments on signal recovery and image reconstruction demonstrate the efficacy of our methods.

报告人信息: 报告人简介：报告人为中国科学院大学数学科学学院运筹学与控制论专业博士生，导师为郭田德老师，研究方向为稀疏优化、机器学习中的优化方法与理论。研究生期间曾获华罗庚奖学金等。
Apr 10

李逸飞（新加坡国立大学数学系）

A Structure-Preserving Parametric Finite Element Method of Anisotropic Geometric Flows

具体信息

摘要: Designing a numerical scheme that can preserve the geometric structure for anisotropic geometric flows with an arbitrary anisotropic surface energy is a long-standing problem. In this talk, for anisotropic mean curvature flow and anisotroic surface diffusion, we propose and analyze a structure-preserving parametric finite element methods (SP-PFEM) for the evolution of a closed curve in 2D, which preserve two geometric structures – area conservation and energy dissipation – at the full-discretized level, The SP-PFEM innovates with a novel surface energy matrix and the Cahn-Hoffman ξ-vector, leading to a new geometric identity for dealing with the weighted mean curvature. This new geometric identity allows our SP-PFEM to be easily extended to various geometric flows with anisotropic effects. Extensive numerical results demonstrate its efficiency, stability, and success in other geometric flows.

报告人信息: 李逸飞新加坡国立大学数学系博士后。本科毕业于北京大学，博士毕业于新加坡国立大学。主要研究几何流的保结构参数化有限元算法。
Apr 25

李昊轩（北京大学前沿交叉学科研究院）

用于解决选择偏差的因果推断方法

具体信息

摘要: 选择偏差是指在研究中，由于用于模型训练的样本的选择不够随机或者不够代表性，导致最终预测结果产生偏差。例如，在推荐系统的交互矩阵中，用户可以自由选择对哪些物品进行评分，导致收集到的评分并不是交互矩阵中所有评分的代表性样本。为了解决选择偏差，许多统计方法已经被提出，例如基于插补的方法，基于逆概率加权的方法，基于双稳健的方法等。其中基于双稳健的方法可以在插补模型或倾向模型正确指定时实现无偏估计，目前被广泛用于业界的真实场景中。在本次报告中，我们将介绍在存在未观测混杂时去除选择偏差的方法（WWW 23，NeurIPS 23），多重稳健方法（AAAI 23），放宽双稳健无偏性假设的方法（ICML 24），自适应均衡函数选择方法（ICML 23，ICLR 24），以及在非独立同分布场景下的双稳健方法（ICLR 24）等。提出的方法与现有方法相比能够实现更好的统计泛化理论保证，并且能够在大规模真实世界数据集上提升去偏效果。

报告人信息: 报告人为北京大学大数据科学研究中心，数据科学直博三年级博士生，CCF会员、IEEE会员、ACM会员，获首批国家自然科学基金青年学生基础研究项目（博士研究生）资助。研究兴趣为因果推断与大语言模型，缺失数据，可信人工智能，分布外泛化，数据融合等。已在ICML、NeurIPS、ICLR、KDD等多个CCF-A顶尖会议以第一作者发表13篇论文，其中3篇论文被评选为Spotlight或Oral，现为ICML、NeurIPS、ICLR、KDD、WWW等多个顶会PC Member和Area Chair，以及TKDE、TOIS、TKDD、The Innovation、《中国科学：信息科学》等多个顶级期刊审稿人。
May 15

薛逸丹（卡迪夫大学）

Computation of two-dimensional Stokes flows via lightning and AAA rational approximation

具体信息

摘要: Most micro-scale fluid flows, where viscous effects dominate over inertial effects, can be described by the Stokes equations. In this talk, I will present an algorithm for computing two-dimensional Stokes problems combining a complex variables method and rational approximation techniques [Y. Xue, S. L. Waters, and L. N. Trefethen, SIAM J. Sci. Comput., 46 (2024), pp. A1214-A1234]. The computations usually take less than a second and give solutions with at least 6-digit accuracy. Examples and demos of applications in various scenarios will be presented to showcase the speed and accuracy of the algorithm.

报告人信息: 薛逸丹，英国卡迪夫大学数学系博士后副研究员。2019年本科毕业于爱丁堡大学机械工程专业，2023年在牛津大学获得工程科学（生物医学工程）博士学位。2022年10月至2023年12月在牛津大学数学系担任EPSRC博士后副研究员（独立课题负责人）。所涉及的研究领域包括脑血流与新陈代谢模拟、基于计算机模拟的临床实验、生物力学、计算生物学、流体力学和科学计算。本科毕业论文发表于Cardiovasc. Eng. Technol. (Springer), 博士和博士后工作发表于多个领域的权威期刊包括J. Biomech., PLOS Comput. Biol.和SIAM J. Sci. Comput. 曾获得爱丁堡大学机械工程专业奖章和英国机械工程师学会最佳本科毕业生奖。
May 30

马志婷（BIMSA）

Uniform accuracy of implicit-explicit backward differentiation formulas (IMEX-BDF) for linear hyperbolic relaxation systems

具体信息

摘要: This work is concerned with the uniform accuracy of implicit-explicit backward differentiation formulas for general linear hyperbolic relaxation systems satisfying the structural stability condition proposed previously by the third author. We prove the uniform stability and accuracy of a class of IMEX-BDF schemes discretized spatially by a Fourier spectral method. The result reveals that the accuracy of the fully discretized schemes is independent of the relaxation time in all regimes. It is verified by numerical experiments on several applications to traffic flows, rarefied gas dynamics and kinetic theory.

报告人信息: Zhiting Ma obtained the B.S. degree from Lanzhou University in 2015 and Ph.D. degree from Department of Mathematical Sciences at Tsinghua University in 2021. Then, she worked as a postdoc at School of Mathematical Sciences, Peking University. Currently, she is an Assistant Professor in Beijing Institute of Mathematical Sciences and Applications (BIMSA).
Jun 12

李博文（中国科学院数学与系统科学研究院）

Convergence Analysis of Forward-Backward Accelerated Algorithms

具体信息

摘要: A significant milestone in modern gradient-based optimization is the development of Nesterov’s accelerated gradient descent (NAG) method. This forward-backward technique has been further enhanced by its proximal generalization, known as the fast iterative shrinkage-thresholding algorithm (FISTA), which finds extensive applications in image science and engineering. In this talk, I will present a tighter inequality for the proximal gradient step of iteration points. By incorporating this tighter inequality into a well-constructed Lyapunov function, we achieve proximal subgradient norm minimization for convex objective functions using the phase-space representation. This approach provides a unified framework to prove the convergence of forward-backward algorithms. A key question in the literature is whether both NAG and FISTA exhibit linear convergence for strongly convex functions without needing prior knowledge of the strongly convex modulus. We address this question using the high-resolution ordinary differential equation (ODE) framework. Our analysis introduces a new Lyapunov function with a dynamically adapting coefficient of kinetic energy that evolves throughout the iterations. This advancement offers deeper insights into the convergence behavior of these algorithms.

报告人信息: 李博文，中国科学院数学与系统科学研究院博士生，主要研究方向为最优化理论和算法。目前的工作重点是通过微分方程框架研究一阶优化算法。对Nesterov加速梯度法和快速迭代收缩阈值算法（FISTA）提出了统一的分析框架，并且在此框架下得到了新的线性收敛结果。对于交替方向乘子法（ADMM）和原始对偶混合梯度法（PDHG）提出了对应的高精度微分方程，并研究了相应的Lyapunov函数和收敛性结果。

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