Hierarchical Scattered Data Filtering for Multilevel Interpolation Schemes Ky ? Xk Mathematical Methods for Curves and Surfaces: Oslo 2000 211

Multilevel scattered data interpolation requires decomposing the given data into a hierarchy of nested subsets. This paper concerns the eecient construction of such hierarchies. To this end, a recursive lter scheme for scattered data is proposed which generates hierarchies of locally optimal nested subsets. The scheme is a composition of greedy thinning, a recursive point removal strategy, and exchange, a local optimization procedure. The utility of the lter scheme for multilevel interpolation using radial basis functions is shown by numerical examples. x1. Introduction Scattered data approximation requires recovering a function f : IR d ! IR, d 1, from a given data vector D Z (f) = (f(z 1); : : :; f(z N)) T 2 IR N of function values sampled from f at a nite set Z = fz 1 ; : : : ; z N g IR d of locations. Especially when N is extremely large, and the points in Z are unevenly distributed , multilevel interpolation schemes are appropriate techniques. One such scheme was introduced in 4], where compactly supported radial basis functions were used. For a recent survey on radial basis functions, we recommend 1]. The starting point in 4] is a decomposition of the given data into a hierarchy (1) of L+1 nested subsets. As connrmed in Section 4, the performance of the mul-tilevel interpolation scheme 4] heavily depends on the choice of the hierarchy (1). Due to available error estimates for radial basis function interpolation, we wish to keep for each level index 1 j L the covering radius r(X j ; X j?1) = max y2X j?1 d X j (y) of X j on X j?1 small. Throughout this paper, we use the notation d X (y) = min x2X All rights of reproduction in any form reserved.