Nonuniform Variational Interpolatory Refinement

In this paper we investigate certain properties of the curves sk generated by a non-uniform variational refinement scheme defined over a k-window of some bi-infinite sequence of points. In particular, we show that error between sn and sk for k ≪ n over some fixed interval decays exponentially as k ↑ n, showing that sn can be well-approximated by sk for k much smaller than n. We use this result to approximate the variational refinement curves locally, and to approximate the global curve using a domain decomposition algorithm. Since the computation of the localized curves is more efficient, the new scheme results in a savings in computation for certain applications. We also generalize our non-uniform variational refinement scheme to the modeling of parametric surfaces defined over a rectangular grid.