An Improved Radial Basis Function Quasi-interpolation Method

With the development of Radial Basis Function, its high efficiency and simplestorage and computing in the computer has been growing attention. Therefore, radial basisfunction is widely used in computation geometry, neural networks, and Numerical solutionof partial differential equation. In recent years, scholars represented by Schaback,Wendland, Powell, Beaston and Wu Zongmin have conducted large numbers of researcheson theory and application of radial basis function and achieved important results, whichgives the radial basis function a wider range of application. In many applications of radialbasis function, there are many examples are completed by doing radial basis functioninterpolation approximation. But for multivariate interpolation of scattered data, when thedata is too large, even though its coefficient matrix is not ill-conditioned, it will be verydifficult to calculate.As a result, some scholars think that if they don’t solve any system of linearequations directly approximation function is given. This leads to the quasi-interpolationtheory. With further research, some quasi-interpolation can even possess shape-preservingpropertie（ssuch as the MQ quasi-interpolation, the spline quasi-interpolation, etc.）Besides,compared with interpolation, quasi-interpolation has some other advantages such asstability of computation, a small amount of computation, etc. Therefore, the radial basisfunction quasi-interpolation theory developed rapidly, especially in multi-quadric functionquasi-interpolation. Based on the four quasi-interpolation operator、、、, wehave obtained many operators with good properties. But cubic multi-quadric functionquasi-interpolation theory developed slowly. This paper attempts to apply the idea ofquasi-interpolation operator successful improved algorithm to the cubic multi-quadric function quasi-interpolation operator. Theoretical proof and numerical resultsshow that the character of the quasi-interpolation operator was very good. In д metric, when selecting identical maximum step h and the shape parameter c, itsmagntiudes of the error function are smaller than them of quasi-interpolation operator.Further, we can get its convergence order higher than quasi-interpolation operator. In this paper, the basic structure is as follows. First, we introduce the basicknowledge of radial basis function, RBF interpolation, quasi-interpolation, etc. especiallythe case with multi-quadric function and the cubic multi-quadric. Next, we introduce thecubic multi-quadric function quasi-interpolation operator and the improved algorithmof quasi-interpolation operator to do the theoretical basis for the derivation of the newquasi-interpolation operator in this paper. Then we construct the new quasi-interpolationoperator, and prove the related properties, especially the improvement of theconvergence order of the quasi-interpolation operator. Finally, we constructed toverify the character of the new quasi-interpolation operator by numerical experiments, andlook forward to the next phase of work.