理想插值算子的离散逼近研究及其应用

Polynomial interpolation is a basic problem in approximation theory, it plays a crucial role both in scientific computation and engineering application. In this project, we will focus on an important class of multivariate polynomial interpolation problems, namely the so-called ideal interpolation. Ideal interpolation is a generalization of the univariate Hermite interpolation. Many practical problems in reality can be transformed into multivariate approximation model. However, the interpolation basis of ideal interpolation cannot be computed as easily as Lagrange interpolation, and the corresponding error analysis is difficult to establish. In this project, we will use Lagrange interpolation to approximate ideal interpolation, which is equal to discrete a given ideal interpolation problem to Lagrange interpolation. We call this problem the discrete approximation problem. If we approximate differential operator with difference operator, then the interpolation basis of ideal interpolation can be obtained from Lagrange interpolation basis directly under certain conditions, and the error formulas of Lagrange interpolation can be used to analyze the error formulas of the original interpolation problem. Thus the discrete approximation problem boils down to the discretization of the D-invariant subspace. Moreover, with the aid of algebraic geometry and the analyzation of the structure of D-invariant subspaces, we will discuss the relationship between ideal projectors and Lagrange projectors.

多项式插值作为逼近理论的基本问题，在科学计算及工程实践中都发挥着重要的作用。本项目将研究一类重要的多元多项式插值问题，即所谓的理想插值。理想插值是Hermite插值的推广，实际问题中涉及的插值多数以理想插值的形式呈现。但是一般的理想插值未能像Lagrange插值那样可以很容易地计算出插值基，相应的误差分析等也很困难。本项目拟考虑利用Lagrange插值逼近理想插值（等价于将理想插值离散为Lagrange插值的极限，称之为离散逼近问题），亦即利用差分算子逼近微分算子，这样在一定条件下可以由Lagrange插值基直接得到理想插值的插值基，并可以利用Lagrange插值的误差来分析理想插值的误差。因此，问题归结为将离散逼近问题转化为插值条件中微分闭子空间的离散，进一步利用代数几何工具并结合微分闭子空间的结构分析，深入探讨理想投影算子与Lagrange投影算子之间的关系。