插值法在逼近散乱数据时需要通过求解大规模的线性方程组来确定插值系数，此时不仅计算规模大且不稳定。拟插值方法直接给出逼近函数式，避免了求解大规模线性代数方程组，正成为大规模数据逼近的重要工具。基于散乱数据的拟插值方法比基于网格点集的拟插值方法有着更广泛的应用价值，但基于散乱数据的多元拟插值的构造工作目前处于起步之中。本项目拟在基于高维散乱数据的高精度拟插值格式的构造及其在双曲方程求解中的应用开展创新性研究工作，重点研究基于散乱数据的高精度多元全局和局部两种形式的拟插值格式的构造及其逼近分析，并将其用于双曲方程的数值求解，以尝试发展一种能适应于混合网格且具有统一形式的双曲方程求解格式。为此拟解决以下关键问题：具有低次多项式恢复和保形特性的基本型多元拟插值格式的构造；具有某些特性的权函数的构造；有限散乱点集上的高阶精度的数值微分公式；人工粘性的添加方法。目的在于完善拟插值方法与理论，推广其应用。

The interpolation method determines the coefficients of the interpolation function by solving a large linear systems whose interpolation matric might be ill-conditioned. Consequently, not only the computation expense is very large but also the process is unstable. The quasi-interpolation method gives the expression of approximation function directly, does not require solution of any linear system and hence is gradually becoming an important approximation approach to a large set of scattered data. The quasi-interpolation scheme based on scattered data has more extensive application value than one based on uniform grids, but then the study on the multivariate quasi-interpolation method based on scattered data is now being placed on start in. The project plans to develop the work of innovation in the construction of high-accuracy quasi-interpolation schemes based on high-dimension scattered data and its application to the solution of hyperbolic equations. The project particularly study the constructions and the approximation error analysis of high-accuracy multivariate gobal quasi-interpolation schemes based on scattered data and high-accuracy multivariate local quasi-interpolation schemes based on scattered data, and use these quasi-interpolation schemes to solve hyperbolic equations numerically, so as to attempt to develop a solution scheme of hyperbolic equations with uniform form for mixed mesh. For the purpose, the project plans to solve the following problems: the construction of basic multivariate quasi-interpolation function with the reproduction of few degree polynomial and shade-preserving,the construction of weighted functions with specific properties, the construction of high-order numerical differential formula on finite set of scattered data and the addition of artificial viscosity. The aim of the project is to supplement the technique and theory of quasi-interpolation approximation method and generalize its application.