Algorithms and Error Bounds for Multivariate Piecewise Constant Approximation

Abstract

Summary. We review the surprisingly rich theory of approximation of functions of many variables by piecewise constants. This covers for example the Sobolev-Poincaré inequalities, parts of the theory of nonlinear approximation, Haar wavelets and tree approximation, as well as recent results about approximation orders achievable on anisotropic partitions.

Optimal approximation order of piecewise constants on convex partitions

“…If this condition is not satisfied, then we say that the partitions in the sequence are "anisotropic". It was shown in [4,5] (and earlier in [9] for d = 2) that on a wider set of all convex partitions D N significantly better order of approximation can be achieved. More precisely, for 1 p ∞,…”

“…under the same restrictions (2). No improvement of the order N − 1 d is possible for any smooth non-constant function f on "isotropic" partitions (see [4]). Recall that a sequence of convex partitions…”

“…In this paper we continue investigations in [4,5] and study approximation properties of piecewise constants on arbitrary convex partitions that are freely adjusted to functions in Sobolev spaces. This adjustment, and notably the use of anisotropic convex polyhedral partitions allows to nearly double the approximation order in comparison to "isotropic" partitions employed in [3].…”

“…Since the partition △ is allowed to depend on f , the optimal or near-optimal approximations s do not belong to a linear space of functions when N is fixed, and the methods of constructing them are referred to as nonlinear approximation [7]. There are very few results in the literature in this "free partition" setting, see [7,Section 6.5] and [4]. In what follows we will consider estimates for the order of approximation of functions f ∈ W r q (Ω) with q < ∞ only.…”

“…No improvement of the order N − 1 d is possible for any smooth non-constant function f on "isotropic" partitions (see [4]). Recall that a sequence of convex partitions {△ N } ∞ N =1 is said to be isotropic if there exists γ > 0 such that diam (ω) γρ(ω) for all ω ∈ N ∈N △ N , where diam (ω) is the diameter of the cell ω and ρ(ω) is the diameter of the largest d-dimensional ball contained in ω.…”