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

Over the last 25 years or so radial functions have emerged as an increasingly popular method for multivariate scattered data approximation and interpolation. Some of the most commonly used radial functions are multiquadrics, thin plate splines, and Gaussians. For all three of these methods the solution of the scattered data approximation problem often leads to a large dense system of linear equations (on the order of several thousand equations). Frequently this system is also ill-conditioned.

These difficulties have led us to look at moving least-squares approximation based on radial weights. In doing this, one now needs to solve many small linear systems (on the order of three or four equations).

The main result presented in this talk will be an application of the theory of approximate approximations (due to Maz'ya and Schmidt) to moving least squares approximation. This results in a new quasi-interpolation method which does not require any linear systems to be solved. Since one can achieve this with the same numerical accuracy as the usual moving least squares method, a fast and accurate approximation method for multivariate scattered data is available.