報告題目一：Constructing least-squares multivariate polynomial approximation
報告時間：2019年9月28日9:00-10:00
報告地點：必一体育平台一樓報告廳
報告人📿：周濤研究員
報告人簡介: 中國科必一數學與系統科學研究院研究員，主要研究方向為隨機微分方程數值解、譜方法等🤹🏽♂️，在Math. Comput.、SIAM J. Sci. Comput、SIAM J. Numer. Anal. 等計算數學頂級刊物上發表多篇論文。擔任 Commun. Comput. Phys🪂💂🏼、 Int. J. Uncertainty Quantification、 NMTMA等SCI期刊編委𓀂🔞。主持過國家優青項目、國家自然科學基金重大研究計劃重點項目等。獲得過中科院“陳景潤未來之星”等稱號。
報告摘要： 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.
報告題目二🏌🏿♂️：Numerical methods for Klein-Gordon equation in the non-relativistic limit
報告時間：2019年9月28日10:00-11:00
報告地點👨🏿🍼：必一体育平台一樓報告廳
報告人🏖：蔡勇勇研究員
報告人簡介：北京計算科學研究中心特聘研究員👇🏻，研究方向為偏微分方程的數值方法及其應用。在SIAM Journal on Applied Math.、SIAM Journal of Numerical Analysis、Mathematics of Computation等計算數學頂級刊物上發表多篇論文🫔，主持多項國家級項目等😲🩰。
報告摘要🫂🕌： 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.
報告題目三🤞🏽：Discontinuous Galerkin Methods for Nonlinear Delay Differential Equations
報告時間：2019年9月28日13:00-14:00
報告地點：必一体育平台一樓報告廳
報告人：黃秋梅教授
報告人簡介：北京工業大學數必一体育平台教授🔪，研究方向為有限元方法、積分微分方程的高精度算法等。在SIAM Numer. Anal.， SIAM Sci. Compt.等計算數學頂級刊物上發表多篇論文，主持多項國家級項目，入選“北京市科技新星”計劃、“北京市教委青年拔尖人才”培育計劃🤍，獲得貴州省科技進步二等獎。
報告摘要： 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.
報告題目四：Better Approximations of High Dimensional Smooth Functions by Deep Neural Networks with Rectified Power Units
報告時間：2019年9月28日13:00-14:00
報告地點🌻：必一体育平台一樓報告廳
報告人🧍🦦：於海軍研究員
報告人簡介👨🏼🎨：中國科必一數學與系統科學研究院研究員，主要研究方向為復雜流體建模與計算👮🏻♀️、譜方法等🚵🏻，在J. Fluid Mech.、SIAM J. Sci. Comput、SIAM J. Numer. Anal. 等計算數學頂級刊物上發表多篇論文🥅，主持多項國家級項目🤜🏽，獲得過中科院“陳景潤未來之星”等稱號👿。
報告摘要🫅🏽： 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.
報告題目五：$C^1$- and $curl^2$-conforming quadrilateral spectral element methods
報告時間：2019年9月28日10:00-11:00
報告地點：必一体育平台一樓報告廳
報告人：李會元研究員
報告人簡介🧑🏻🎓：中國科必一軟件研究所研究員。主要研究領域為高性能科學計算與數學軟件、數值PDE的譜方法、特征值問題的高性能計算方法等。在SIAM Numer. Anal.🛅， SIAM Sci. Compt.等計算數學頂級刊物上發表多篇論文，主持多項國家級項目等。
報告摘要： 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.