Near-best $$C^2$$ C 2 quartic spline quasi-interpolants on type-6 tetrahedral partitions of bounded domains

Abstract

In this paper, we present new quasi-interpolating spline schemes defined on 3D bounded domains, based on trivariate C 2 quartic box splines on type-6 tetrahedral partitions and with approximation order four. Such methods can be used for the reconstruction of gridded volume data. More precisely, we propose near-best quasi-interpolants, i.e. with coefficient functionals obtained by imposing the exactness of the quasi-interpolants on the space of polynomials of total degree three and minimizing an upper bound for their infinity norm. In case of bounded domains the main problem consists in the construction of the coefficient functionals associated with boundary generators (i.e. generators with supports not completely inside the domain), so that the functionals involve data points inside or on the boundary of the domain.We give norm and error estimates and we present some numerical tests, illustrating the approximation properties of the proposed quasiinterpolants, and comparisons with other known spline methods. Some applications with real world volume data are also provided.

On spline quasi-interpolation through dimensions

“…Concerning the solution of integral equations, we mention the papers [2,13,38,46]. In particular, spline QIPs of the form (1), with s = 1 and a bounded interval, are used for the numerical solution of linear [46] and non linear [13] integral equations of the second kind…”

“…the approximants here considered are piecewise polynomials [18,19,21,76,77,82]. In particular, the aim of this paper is to collect the main results obtained by the authors, also in collaboration with other researchers, in such a topic, highlighting the approximation properties and the reconstruction of functions and data [3, 5, 6, 8-10, 26, 32-34, 36, 43, 45, 47, 57, 58, 67, 69-71], the applications in numerical integration and differentiation [22, 25, 27-31, 35, 37, 39-42, 44, 56, 68] and the numerical solution of integral and differential problems [2,13,24,38,46]. The above results can also be considered through spline dimension: 1D [2, 13, 25, 28, 29, 35-37, 39-42, 46, 57, 68], 2D [6, 8, 9, 22, 24, 26, 27, 30-34, 38, 43, 45, 56-58, 67, 70] and 3D [3,5,10,44,47,69,71].…”

On the construction of trivariate near-best quasi-interpolants based on C2 quartic splines on type-6 tetrahedral partitions

“…Other methods based on trivariate C 1 splines of total degree have been proposed, in [16,29] and [25] on type-6 tetrahedral partitions, in [23] on truncated octahedral partitions, in [27,28,30] on Powell-Sabin (Worsey-Piper) split, and in [24] by using quadratic trivariate super splines on uniform tetrahedral partitions. Furthermore, higher smoothness C 2 has been considered in [10][11][12]18,20], where the reconstruction of volume data is provided in the space of C 2 quartic splines.…”

Trivariate near-best blending spline quasi-interpolation operators

“…In [1], the construction of a type-1 NB QIO into the linear space S 2 4 (τ ) spanned by the integer translates of three C 2 -quartic B-splines on the four-direction mesh τ of the plane is presented. The extension to the three-dimensional case is done in [14,21,22,23]. In [9], the construction of trivariate near-best quasi-interpolants based on C 2 quartic splines on type-6 tetrahedral partitions is addressed.…”