报告题目一：Constructing least-squares multivariate polynomial approximation
报告人简介: 中国科学院数学与系统科学研究院研究员，主要研究方向为随机微分方程数值解、谱方法等，在Math. Comput.、SIAM J. Sci. Comput、SIAM J. Numer. Anal. 等计算数学顶级刊物上发表多篇论文。担任 Commun. Comput. Phys、 Int. J. Uncertainty Quantification、 NMTMA等SCI期刊编委。主持过国家优青项目、国家自然科学基金重大研究计划重点项目等。获得过中科院“陈景润未来之星”等称号。
报告摘要： Polynomial approximations constructed using a least squares approach is a ubiquitous technique in numerical computations. One of the simplest ways to generate data for the least squares problems is with random sampling of a function. We discuss theory and algorithms for stability of the least-squares problem using random samples. The main lesson from our discussion is that the intuitively straightforward (``standard") density for sampling frequently yields suboptimal approximations, whereas sampling from a non-standard density either by the so-called induced distribution or the asymptotic equilibrium measure, yields near-optimal approximations. We present recent theory that demonstrates why sampling from such measures is optimal, and provide several computational experiments that support the theory. New applications of the equilibrium measure sampling will also be discussed.
报告题目二：Numerical methods for Klein-Gordon equation in the non-relativistic limit
报告人简介：北京计算科学研究中心特聘研究员，研究方向为偏微分方程的数值方法及其应用。在SIAM Journal on Applied Math.、SIAM Journal of Numerical Analysis、Mathematics of Computation等计算数学顶级刊物上发表多篇论文，主持多项国家级项目等，为国家“青年千人计划”入选者。
报告摘要： Klein-Gordon (KG) equation describes the motion of spinless particle. In the non-relativistic limit $\varepsilon\to 0^+ $ ($\varepsilon$ inversely proportional to the speed of light), the solution to the KG equation propagates waves with amplitude at O(1) and wavelength at $O(\varepsilon^2)$ in time and O(1) in space, which causes significantly numerical burdens due to the high oscillation in time. By the analysis of the non-relativistic limit of the KG equation, the KG equation can be asymptotically reduced to the nonlinear Schroedinger equations (NLS) with wave operator (NLSW) perturbed by the wave operator with strength described by a dimensionless parameter $\varepsilon\in(0,1]$. Starting with the error analysis of finite difference methods for NLSW and the uniform bounds w.r.t. $\varepsilon$, we will show the error analysis of an exponential wave integrator sine pseudospectral method for NLSW, with improved uniform error bounds. Finally, a uniformly accurate multi scale time integrator method will be constructed for solving the KG equation in the non-relativistic limit based on the NLSW expansion, and rigorous error bounds are established.
报告题目三：Discontinuous Galerkin Methods for Nonlinear Delay Differential Equations
报告人简介：北京工业大学数理学院教授，研究方向为有限元方法、积分微分方程的高精度算法等。在SIAM Numer. Anal.， SIAM Sci. Compt.等计算数学顶级刊物上发表多篇论文，主持多项国家级项目，入选“北京市科技新星”计划、“北京市教委青年拔尖人才”培育计划，获得贵州省科技进步二等奖。
报告摘要： In this report, we investigate discontinuous Galerkin (DG) methods for nonlinear vanishing delay and state dependent delay differential equations. The optimal global convergence and local superconvergence results are established. By suitable designing partitions, the optimal nodal superconvergence of the discontinuous Galerkin solutions is obtained. Numerical examples are provided to illustrate the theoretical results.
报告题目四：Better Approximations of High Dimensional Smooth Functions by Deep Neural Networks with Rectified Power Units
报告人简介：中国科学院数学与系统科学研究院研究员，主要研究方向为复杂流体建模与计算、谱方法等，在J. Fluid Mech.、SIAM J. Sci. Comput、SIAM J. Numer. Anal. 等计算数学顶级刊物上发表多篇论文，主持多项国家级项目，获得过中科院“陈景润未来之星”等称号。
报告摘要： Deep neural networks with rectified linear units (ReLU) are recently getting very popular due to its universal representation power and easier to train. Some theoretical progresses on deep ReLU network approximation power for functions in Sobolev space and Korobov space have recently been made by several groups. In this talk, we show that deep networks with rectified power units (RePU) can give better approximations for smooth functions than deep ReLU networks. Our analyses base on classical polynomial approximation theory and some efficient algorithms we proposed to convert polynomials into deep RePU networks of optimal size without any approximation error. Our constructive proofs reveal clearly the relation between the depth of the RePU network and the “order” of polynomial approximation. Taking into account some other good properties of RePU networks, such as being high-order differentiable, we advocate the use of deep RePU networks for problems where the underlying high dimensional functions are smooth or derivatives are involved in the loss function.
报告题目五：$C^1$- and $curl^2$-conforming quadrilateral spectral element methods
报告人简介：中国科学院软件研究所研究员。主要研究领域为高性能科学计算与数学软件、数值PDE的谱方法、特征值问题的高性能计算方法等。在SIAM Numer. Anal.， SIAM Sci. Compt.等计算数学顶级刊物上发表多篇论文，主持多项国家级项目等。
报告摘要： This talk is oriented for conforming spectral element methods for solving fourth order elliptic equations and quad-curl equations on quadrilated meshes. We start with the structure exploration of the $C^1$-conforming piecewise polynomial space on quadrilateral meshes. Interior, edge and vertex modes of the $C^1$-conforming basis functions are technically constructed through a bilinear mapping with the help of generalized Jacobi polynomials. In the sequel, we resort to the contravariant transformation, the de Rham complex and the generalized Jacobi polynomials to construct of the basis functions of $curl^2$-conforming quadrilateral spectral elements. Finally, numerical experiments are demonstrated to show the effectiveness and accuracy of our conforming quadrilateral spectral element methods.