General Algorithms for Multiscale Approximation

We describe efficient algorithms for adaptive multiscale approximation of functions that are sampled with uneven density and/or have important small-scale detail limited to small portions of their domain. Our algorithm constructs the approximation from the top down, using a special least squares fitting of the residual at each level, followed by a basis reduction procedure to discard elements that contribute very little to the approximation. An important feature of these algorithms is their generality, since they are independent of domain dimension and shape, mesh, and approximation basis. Another important feature is that they do not need to generate the full basis at each level.