Hierarchical spline spaces: quasi-interpolants and local approximation estimates
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Hierarchical spline spaces: quasi-interpolants and local approximation estimates Hendrik Speleers1
Received: 25 January 2016 / Accepted: 21 September 2016 / Published online: 24 October 2016 © Springer Science+Business Media New York 2016
Abstract A local approximation study is presented for hierarchical spline spaces. Such spaces are composed of a hierarchy of nested spaces and provide a flexible framework for local refinement in any dimensionality. We provide approximation estimates for general hierarchical quasi-interpolants expressed in terms of the truncated hierarchical basis. Under some mild assumptions, we prove that such hierarchical quasi-interpolants and their derivatives possess optimal local approximation power in the general q-norm with 1 ≤ q ≤ ∞. In addition, we detail a specific family of hierarchical quasi-interpolants defined on uniform hierarchical meshes in any dimensionality. The construction is based on cardinal B-splines of degree p and central factorial numbers of the first kind. It guarantees polynomial reproduction of degree p and it requires only function evaluations at grid points (odd p) or half-grid points (even p). This results in good approximation properties at a very low cost, and is illustrated with some numerical experiments. Keywords Local approximation · Quasi-interpolation · Hierarchical bases · Local refinement · Tensor-product B-splines Mathematics Subject Classification (2010) 41A15 · 65D07 · 65D17
Communicated by: Larry L. Schumaker Hendrik Speleers
[email protected] 1
Department of Mathematics, University of Rome ‘Tor Vergata’, Via della Ricerca Scientifica, 00133 Rome, Italy
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H. Speleers
1 Introduction Quasi-interpolation is a general term denoting the construction, with a low computational cost, of accurate approximants to a given set of data or a given function. A quasi-interpolant is usually obtained as a linear combination of a set of blending functions that form a convex partition of unity and have small local support. The computation of the corresponding coefficients only involves a local portion of the given data/function. These properties ensure both numerical stability and local control of the constructed approximant. Quasi-interpolants in polynomial spline spaces are a common and powerful approximation tool, see e.g. [1, 3, 19, 21]. In this paper we focus on quasi-interpolants in hierarchical spline spaces. Such spline spaces provide a flexible framework for local refinement coupled with a remarkable intrinsic simplicity. Hierarchical B-splines are defined in terms of a hierarchy of locally refined meshes, reflecting different levels of refinement. They were introduced in [10] as an accumulation of tensor-product B-splines with nested knot vectors. The hierarchical approach has been successfully applied in various areas, ranging from approximation theory to numerical simulation, see e.g. [6, 10, 13, 20, 26]. A similar local refinement approach can also be found in [8]. The original set of hierarchical B-splines in [10] lacks some important pr
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