EIM魔数点特性研究及其最小二乘格式  

THE OPTIMUM OF EIM MAGIC POINTS AND THE LEAST-SQUARES FORM

作　　者：龚禾林 张世全[2] Yvon Maday Gong Helin;Zhang Shiquan;Yvon Maday(Paris Elite Institute of Technology,Shanghai Jiao Tong University,Shanghai 200240,China;School of Mathematics,Sichuan University,Chengdu 610064,China;Institut Universitaire de France/Sorbonne Universites,UPMC Univ Paris 06 and CNRS UMR 7598,Laboratoire Jacques-Louis Lions,Paris,F-75005,France)

机构地区：[1]上海交通大学巴黎卓越工程师学院,上海200240 [2]四川大学数学学院,成都610064 [3]法国索邦大学LJLL实验室,巴黎70005,法国

出　　处：《数值计算与计算机应用》2023年第1期25-36,共12页Journal on Numerical Methods and Computer Applications

基　　金：国家自然科学基金(11905216)资助.

摘　　要：经验插值法(empirical interpolation method,EIM)首先由Yvon Maday和他的合作者在2004年提出,旨在提升非仿射或非线性偏微分方程模型降阶(reduced basis technique)的计算效率,随后在模型降阶和数据同化领域得到了广泛应用.EIM的计算过程分为离线、在线两个过程:离线阶段,基于待插值函数空间的大量样本函数,通过EIM算法逐一计算插值基函数和插值点(魔数点);在线阶段,基于在魔数点的函数值和基函数,在线重构待插值函数.本文重点研究了EIM算法得到的插值点的空间分布特性,提出了最小二乘格式的EIM(LS-EIM)以进一步提升EIM精度和稳定性.比较了EIM算法确定的魔数点和其它各种采样方法确定的点对LS-EIM的收敛性和精度的影响.通过数值计算发现,相比随机采样和其他方法选取的采样点,EIM算法得到的魔数点用于LS-EIM可获得最快收敛速度和最优重构精度,通常仅需不到2倍于基函数维数的魔数点数,即不到2个魔数点,LS-EIM即可实现对最佳重构的逼近.The empirical interpolation method(EIM)has been introduced to extend the reduced basis technique to nonaffine and nonlinear partial differential equations(PDEs),and then widely used in the field of model order reduction and data assimilation.The efficiency of EIM is achieved through an offline-online decomposition procedure.In the offline phase,a set of the snapshots(manifold)of the underlying function are solved,and then a set of basis and interpolation points(magic point)are solved recursively based on the manifold.In the online phase,the interpolation to an unknown function is done by combining the observations of the function itself at some points and the reduced basis.In this paper,the performance of the magic points is analyzed.To further improve the accuracy and stability,we propose a least-squares framework of EIM(LS-EIM)with more points than originally required,and different kind of points sampling methods are investigated.Our numerical finding is that,with only two times of measurements from the first n EIM magic points,LS-EIM allows to reach an accuracy similar to that of best approximation in the reduce space which is spanned by the first EIM basis functions,with a high stability performance.Furthermore,these additional EIM points are better than a choice of a random sampling or other sampling methods.

关 键 词：EIM 魔数点 模型降阶 最小二乘 

分 类 号：O241.3[理学—计算数学]