Local Polynomial Reproduction and Moving Least Squares Approximation

 

Abstract

We discuss multivariate approximation methods based on local polynomial reproduction. The goal of these methods is to reconstruct a function that is only known at certain, in general scattered, centers. Methods based on local polynomial reproductions have the following advantages: They are local, this means that for the reconstruction of the function at a certain point x only the neighbouring centers are necessary. The methods are capable of dealing with scattered data and are therefore so-called meshless methods. They lead to the expected convergence orders based on the polynomial reproduction. Finally, in most cases the computational effort is linear in both the number of centers and the number of evaluation points, which makes these methods very atractive for higher dimensional problems. As an example we take a closer look at the Moving Least Squares Approximation.