报告题目：Error analysis of hyperbolic cross approximation
It is well-known that sparse grid algorithm has been widely accepted as an efficient tool to overcome the "curse of dimensionality" in some degree. Hyperbolic cross approximation is one of the sparse grid, whose error analysis will be discussed in this talk. The first part of this talk is devoted to the error estimate of hyperbolic cross (HC) approximations with generalized Hermite functions. The exponential convergence in both regular and optimized HC approximations has been shown. Furthermore, the error estimate of Hermite spectral method to high-dimensional linear parabolic PDEs with HC approximations has been investigated in the properly weighted Korobov spaces. In the second part of this talk, we shall give the error estimate of HC approximations with all sorts of Askey polynomials. These polynomials are useful in generalized polynomial chaos (gPC) in the field of uncertainty quantification. Under some smooth condition of the random input, the HC approximation yields exponential convergence as before. Moreover, we apply gPC to numerically solve the ordinary differential equations with slightly higher dimensional random inputs. Finally, we shall investigate the connection between the standard ANOVA approximation and Galerkin approximation, so that HC approximation can be naturally combined with ANOVA approximation. Part of this talk is the joint work with Stephen S.-T. Yau.
报告人简介：罗雪，现任北京航空航天大学数学与系统科学学院副教授。2013年毕业于美国伊利诺伊大学芝加哥分校，获理学博士学位。主要从事非线性滤波理论和计算、谱方法算法在非线性滤波中的应用以及偏微分方程分析等领域的研究。2015年底获升IEEE资深会员（senior member）。2016年获丘成桐新世界数学奖博士论文银奖。近年来，在国际著名期刊 Automatica、IEEE Trans. Automat. Control、Comm. Partial Differential Equations、SIAM J. Numer. Anal. 等上发表论文20余篇。主持北京市自然科学基金一项、国家自然科学基金一项。