Approximation with diversified B-splines

Abstract

When approximating functions defined on some domain $\Omega \subset {\mathbb{R}}^d$, standard tensor product splines reveal sub-optimal behavior, in particular, if Ω is non-convex. As an alternative, we suggest a natural diversification strategy for the B-spline basis ${\{B_i\}}_i$. It is grounded on employing a separate copy $B_{i,\gamma }$ of $B_i$ for every connected component γ of its support $\mathrm{supp}{B}_i\cap \Omega$. In the bivariate case, which is important for applications, this process enhances the spline space to a crucial extent. Concretely, we prove that the error in uniform tensor product spline approximation of a function $f:{\mathbb{R}}^2\supset \Omega \to \mathbb{R}$ can be bounded in terms of the pure partial derivatives of f, where the constant depends neither on the shape of Ω nor on the knot grid. An example shows that a similar result cannot hold true for higher dimensions, even if the domain is convex and has a smooth boundary.

Introduction

Spline approximation is a fundamental issue in theory and applications like reverse engineering (Várady et al., 1997) or simulation (Cottrell et al., 2009, Höllig, 2003, Höllig et al., 2001). However, our current knowledge on the subject is leaving some important questions unanswered when it comes to approximation of multivariate functions defined on subsets $\Omega \subset {\mathbb{R}}^d$. Open issues include the appropriate choice of the spline space itself and the dependence of constants in error estimates on the shape of Ω or the chosen knot sequence.

Let ${\mathcal{S}}_n(T,\Omega )$ denote the span of tensor product B-splines of coordinate order $n=(n_1,\dots ,n_d)$ with knots $T=(T_1,\dots ,T_d)$ restricted to Ω. In the fundamental work (Dahmen et al., 1980), it is shown that

$$\underset{s\in {\mathcal{S}}_n(T,\Omega )}{\inf }{\Vert f-s\Vert }_{\Omega ,L^p}\le C\sum _{i=1}^d{h}_i^{n_i}{\Vert {\partial }_i^{n_i}f\Vert }_{\Omega ,L^p}$$

for some $C>0$, where $h=(h_1,\dots ,h_d)$ is the maximal spacing of knots. In Mößner and Reif (2009) and Reif (2012), this result is elaborated for the special cases of interpolation and approximation with polynomials, respectively. The estimate suggests that the pure partial derivatives of f alone should be sufficient to bound the error, and that a fine knot sequence in a distinct coordinate direction should be sufficient to compensate for large derivatives in that direction. Unfortunately, such a conclusion may not be drawn imprudently from the results in Dahmen et al. (1980), in particular for the following reasons: First, the domain Ω is assumed to be coordinate-wise convex, which is a severe restriction of the range of applicability. Second, Ω has to live up to a series of technical assumptions which may be hard to verify in a concrete setting. Third, and perhaps most impedingly, a detailed analysis of statements and arguments reveals a hidden dependence of the number C on the aspect ratio

$$\varrho :=\underset{i,j}{\max }\frac{h_i}{h_j}$$

of the knot grid, even in the case of uniform splines. Thus, the previously mentioned compensation of a large value of ${\Vert {\partial }_i^{n_i}f\Vert }_{\Omega ,L^p}$ by an exclusive refinement of the knot sequence $T_i$ is questioned, as decreasing $h_i$ alone is increasing ϱ. A similar dependence of C on the aspect ratio can also be observed in other approaches to the topic, like Höllig et al. (2001) or Mößner and Reif (2008). So it is plausible, but by no means evident, that this phenomenon is not an artifact of insufficient proof techniques, but a matter of fact.

In the second section of this paper, we present examples in two and three variables which actually prove that C cannot be independent of ϱ. While the 2d example exploits the non-convexity of the domain, the 3d case gets along with a domain which is strictly convex and has a perfectly smooth boundary. The special structure of the 3d counterexample might be useful to identify a subclass of domains where (1) is valid with uniform C. However, this topic is not addressed here.

Instead, in the third section, we propose a remedy to the problems observed in the 2d case. It is based on the observation that the spline space ${\mathcal{S}}_n(T,\Omega )$ is not rich enough to deal adequately with non-convexity. Let $S_i$ denote the support of the B-spline $B_i$. Then its relevant part $S_i\cap \Omega$ might consist of several connected components. In the standard setting, the single B-spline $B_i$ is overcharged by possibly conflicting demands coming from simultaneous error minimization on all these components. So it is a natural approach to use a separate copy $B_{i,\gamma }$ of $B_i$ for each connected component γ of $S_i\cap \Omega$. This process, called diversification, provides a significant amount of extra flexibility. Our main theorem on approximation with diversified B-splines states that (1) holds true for a broad class of domains $\Omega \subset {\mathbb{R}}^2$ with a constant C which is independent of the aspect ratio and the shape of Ω.

For the proof of our main theorem, another new concept, called condensation, is introduced to address the notorious problem of lacking stability of the basis when working on domains with boundary. Condensation is replacing a given knot sequence by a finer one without changing the span of B-splines on the considered domain. Thus, the size of the support and the knot spacing can be made comparable, facilitating the construction of stable quasi-interpolants.

To focus on the essence of ideas, we confine ourselves to the case of uniform knot sequences and to error measurements with respect to the sup-norm, i.e., $p=\infty$. Arbitrary knot sequences and exponents p can be dealt with in a similar fashion, but require a significantly increased complexity of notations and arguments.