Construction Of Highly Accurate Quasi-Interpolation Operators With Radial Basis Functions And Its Application

Quasi-interpolation method is one of the popular methods in approxima-tion theory. Compared to interpolation, on the one hand, one major advantageof quasi-interpolation is that it can yield an approximant without the needto solve any large-scale linear system of equations; on the other hand, somequasi-interpolation can possess shape-preserving properties (such as the MQquasi-interpolation, the B-spline quasi-interpolation).These advantages makequasi-interpolation has some significant characteristics such as stability of com-putation, a small amount of computation, etc.Radial basis function method is a common method to handle large scaledscattered data problems. Based on its simple form, Quasi-interpolation methodwith radial basis function both in theory and in application is a powerful tool,which has been widely concerned. In particular, MQ functions As a special caseof radial basis functions are an infinite time diferentiable and as a function ofthe kernel function can approximate any smooth function. A review by Frankeshowed that the MQ outperformed some29methods in terms of accuracy andefciency.Therefore, MQ quasi-interpolation is favored by the majority of scholars.For practical purposes, MQ quasi-interpolation operators on a finite range have been proposed and widely applied in the practical problems such as partial dif-ferential equation. In recent years, a batch of MQ formats have been proposed. However, construction of the existing highly accurate MQ quasi-interpolation operator often need derivative information. In the actual problem, derivative information is difficult to obtain directly. How to construct the highly accurate MQ quasi-interpolation operator without any derivative information is a focus in the study of this dissertation.The main results of this paper are as follows:(I) Discuss the even order Bernoulli-type MQ quasi-interpolation operator on the univariate finite interval.To solve the practical univariate problems(For example, people only mea-sure function of a finite number of values), We hope that the constructed ap-proximation format plays a role on the finite interval, need only a partial node information without any derivative value and has high order approximation accuracy.First, We give a kind of very practical quasi-interpolation operator Lvr with univariate MQ on the finite interval.Definition1Given a set of nodes X={xj}nj=0on the finite interval satisfies Based on above nodes, a family of even order Bernoulli-type MQ quasi-interpolation operator without any derivative value is defined as follows where Bk(x) is Bernoulli polynomial of order k. and D2i-1Axjf(xj),D2i-1Axj+1f(xj+1)(j=0,...,n,i=1,...,m)are shown in Theorem3.1.5,chapter3,§3.1.3.Second,we give error analysis about the univariate operator Lv,Theorem1let where D is a positive constant and l is a positive integer.(i)If f(x)∈C2m…[a,b],then where and C"’ is a positive constant independent of f,x, and X.(ii)if f(x)∈C2m+1[a,b],then where C is a positive constant independent of f,x,X.Finally,in§3.3,chapter3,numerical experiments show that compared with Shepard-Bernoulli interpolation operator SBm and MQ quasi-interpolation LH2m-1our operator Lvm can reach up to high convergence order:In§3.4,chap-ter3,operator Lvm.is applied to data fitting. (II)Discuss the Lidstone-type MQ quasi-interpolation operator on the univariate finite interval.First,based on the cnstruction technology of even order Bernouli-type MQ quasi-interpolation,we also give a kind of Lidstone-type quasi-interpolation operator L∧m with univariate MQ on the finite intervalDefinition2Based on above nodes in definition1,Lidstone-type MQ quasi-interpolation operator L∧m without any derivative value is defined as follows where xn+1=xn-1and D2kAxj f(xj),D2kAxj+1f(xj+1)(j=0,…,n,k=1,…,m-1)are shown in Theorem4.2.2,§4.2,chapter4.Second,we give error analysis about the univariate operator LAmTheorem2let c,l,M,X as Theorem1.(i)If f(x)∈C2m-1[a,b],then where E"’is a positive constant independent of f,x,X;(ii)If f(x)∈C2m[a,b],then where E is a positive constant independent of f, x, X.Finally, In§4.4, chapter4, numerical experiments show the obtained ap-proximations are comparable between Lidstone-type MQ quasi-interpolation operator L∧m and even order Bernoulli-type MQ quasi-interpolation operator Lvr(Ⅲ) Discuss the Waldron-type MQ quasi-interpolation operator on the multivariate finite fieldTo solve the practical multivariate problems(For example, people only measure function of a finite number of values on gridded nodes), We hope that the constructed approximation format plays a role on the multivariate finite field, need only a partial node information without any derivative value and has high order approximation accuracy.First, We give a kind of Waldron-type MQ quasi-interpolation operator Φr+1without any derivative value on the multivariate finite field.Definition3Given nodes x0<x1<...<xn, y0<y1<...<ym and data collection way we give the definition of the modified scheme where DAare shown in Theorem5.2.2,§5.2, chapter5.Second, we give error analysis about the multivariate operatorΦr+1. Theorem3Let any function f(x)∈Cr+2(Ω) and its corresponding order of derivatives be bounded on Ω. Then for r∈N, r≥2, we have whereNow let h=max{h1,h2}, we simplify Theorem3, then get the following Theorem.Theorem4If f(x,y) satisfies the conditions of Theorem3, then there exists constant p≥1such thatFinally, in§5.4, chapter5, numerical experiments show our multivariate operator not only reproduce r+1multivariate polynomials but also reach up to a rate of r+2.