一种新的Multiquadric拟插值

A New Multiquadric Quasi-Interpolation

Multiquadric(MQ)是由Hardy提出来的一种径向基函数,至今它已在大地测量学、微分方程数值解等许多方面得到了应用。目前已知的multiquadric拟插值有4种,即,Beatson和Powell的LA、LB和LC以及Wu和Schaback的LD,其中,LB是常数再生的,LC和LD是线性再生的。该文首先给出形如LD的拟插值线性再生时其基函数应具有的性质,然后构造了一种具有线性再生性和保单调性的multiquadric拟插值,并对其逼近误差进行了理论分析。最后,通过两个实例进行数值实验,从算例的结果来看,该拟插值具有良好的逼近精度。

Multiquadric（MQ） is a kind of radial basis function developed by Hardy.So far,it has been used to many realms,such as geodesy and numerical solution of differential equations.Up till now,there are known four kinds of multiquadric quasi-interpolation,namely,LA、LB and LC raised by Beatson and Powell and LD by Wu and Schaback,in which,LB is constant reproducing,LC and LD are linear reproducing.Here,firstly,the properties possessed by the basis functions of a quasi-interpolation in the form of LD are given when this quasi-interpolation is linear reproducing.Then,a multiquadric quasi-interpolation is constructed which has the properties of linear reproducing and preserving monotonicity.Moreover,the theoretical analysis to its approximation error is shown.Lastly,the effect is illustrated by two examples.The numerical results indicate that this quasi-interpolation possesses high approximation precision.