Stability of multiquadric quasi-interpolation to approximate high order derivatives  

作　　者：MA LiMin 1,2 & WU ZongMin 1,1 Shanghai Key Laboratory for Contemporary Applied Mathematics,School of Mathematical Sciences,Fudan University,Shanghai 200433,China 2 Department of Information and Computer Science,Zhejiang Gongshang University,Hangzhou 310018,China 

出　　处：《Science China Mathematics》2010年第4期985-992,共8页中国科学：数学（英文版）

基　　金：supported by the Major State Basic Research Development Program of China (973 Program) (Grant No.2006CB303102);the Science and Technology Commission of Shanghai Municipality (Grant No.09DZ2272900)

摘　　要：Numerical simulation of the high order derivatives based on the sampling data is an important and basic problem in numerical approximation,especially for solving the differential equations numerically.The classical method is the divided difference method.However,it has been shown strongly unstable in practice.Actually,it can only be used to simulate the lower order derivatives in applications.To simulate the high order derivatives,this paper suggests a new method using multiquadric quasi-interpolation.The stability of the multiquadric quasi-interpolation method is compared with the classical divided difference method.Moreover,some numerical examples are presented to confirm the theoretical results.Both theoretical results and numerical examples show that the multiquadric quasi-interpolation method is much stabler than the divided difference method.This property shows that multiquadric quasi-interpolation method is an efficient tool to construct an approximation of high order derivatives based on scattered sampling data even with noise.Numerical simulation of the high order derivatives based on the sampling data is an important and basic problem in numerical approximation,especially for solving the differential equations numerically.The classical method is the divided difference method.However,it has been shown strongly unstable in practice.Actually,it can only be used to simulate the lower order derivatives in applications.To simulate the high order derivatives,this paper suggests a new method using multiquadric quasi-interpolation.The stability of the multiquadric quasi-interpolation method is compared with the classical divided difference method.Moreover,some numerical examples are presented to confirm the theoretical results.Both theoretical results and numerical examples show that the multiquadric quasi-interpolation method is much stabler than the divided difference method.This property shows that multiquadric quasi-interpolation method is an efficient tool to construct an approximation of high order derivatives based on scattered sampling data even with noise.

关 键 词：numerical differential radial basis functions (RBFs) Hardy’s MULTIQUADRIC (MQ) quasiinterpolation divided difference method white noise EXPECTATION variance 

分 类 号：TP391.41[自动化与计算机技术—计算机应用技术]