Approximate Moving Least-Squares Approximation: A Fast and Accurate Multivariate Approximation Method

We propose a fast and accurate approximation method for large sets of multivariate data using radial functions. In the traditional radial basis function approach this task is usually accomplished by solving a large system of linear equations stemming from an interpolation formulation. In the traditional moving least-squares method one needs to solve a small linear system for each evaluation of the approximant. We present an approximation scheme – based on the work on approximate approximation by Maz’ya and Schmidt – that has approximation properties similar to the moving least-squares method, but completely avoids the solution of linear systems. Moreover, the sums required for the evaluation of the approximant can be processed quickly. We establish a connection to traditional radial basis function approximation by using appropriate radial generating functions. Examples of locally supported as well as globally supported functions with arbitrary approximation orders are given. §