Approximation by sums of piecewise linear polynomials

Abstract

We present two partitioning algorithms that allow a sum of piecewise linear polynomials over a number of overlaying convex partitions of the unit cube Ω in Rd to approximate a function f∈Wp3(Ω) with the order N−6/(2d+1) in the Lp-norm, where N is the total number of cells of all partitions, which makes a marked improvement over the N−2/d order achievable on a single convex partition. The gradient of f is approximated with the order N−3/(2d+1). The first algorithm creates d convex partitions and relies on the knowledge of the eigenvectors of the average Hessians of f over the cells of an auxiliary uniform partition, whereas the second algorithm with d+12 convex partitions is independent of f. In addition, we also give an f-independent partitioning algorithm for a sum of d piecewise constants that achieves the approximation order N−2/(d+1).