Title 1: Constructing least-squares multivariate polynomial approximation
Date: 2019.9.28 9:00-10:00
Place: School of Science Meeting Room 1st
Reporter: ZHOU Tao
Abstract: 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.
Title 2: Numerical methods for Klein-Gordon equation in the non-relativistic limit
Date: 2019.9.28 10:00-11:00
Place: School of Science Meeting Room 1st
Reporter: CAI Yongyong
Abstract: 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.
Title 3: Discontinuous Galerkin Methods for Nonlinear Delay Differential Equations
Date: 2019.9.28 13:00-14:00
Place: School of Science Meeting Room 1st
Reporter: HUANG Qiuhai
Abstract: 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.
Title 4: Better Approximations of High Dimensional Smooth Functions by Deep Neural Networks with Rectified Power Units
Date: 2019.9.28 13:00-14:00
Place: School of Science Meeting Room 1st
Reporter: YU Haijun
Abstract: 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.
Title 5: $C^1$- and $curl^2$-conforming quadrilateral spectral element methods
Date: 2019.9.28 10:00-11:00
Place: School of Science Meeting Room 1st
Reporter: LI Huiyuan
Abstract: 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.
