Sharp error estimates for spline approximation: explicit constants, nn-widths, and eigenfunction convergence

In this paper we provide a priori error estimates in standard Sobolev (semi-)norms for approximation in spline spaces of maximal smoothness on arbitrary grids. The error estimates are expressed in terms of a power of the maximal grid spacing, an appropriate derivative of the function to be approximated, and an explicit constant which is, in many cases, sharp. Some of these error estimates also hold in proper spline subspaces, which additionally enjoy inverse inequalities. Furthermore, we address spline approximation of eigenfunctions of a large class of differential operators, with a particular focus on the special case of periodic splines. The results of this paper can be used to theoretically explain the benefits of spline approximation under $k$-refinement by isogeometric discretization methods. They also form a theoretical foundation for the outperformance of smooth spline discretizations of eigenvalue problems that has been numerically observed in the literature, and for optimality of geometric multigrid solvers in the isogeometric analysis context.Comment: 31 pages, 2 figures. Fixed a typo. Article published in M3A

Computation of multi-degree B-splines

Multi-degree splines are smooth piecewise-polynomial functions where the pieces can have different degrees. We describe a simple algorithmic construction of a set of basis functions for the space of multi-degree splines, with similar properties to standard B-splines. These basis functions are called multi-degree B-splines (or MDB-splines). The construction relies on an extraction operator that represents all MDB-splines as linear combinations of local B-splines of different degrees. This enables the use of existing efficient algorithms for B-spline evaluations and refinements in the context of multi-degree splines. A Matlab implementation is provided to illustrate the computation and use of MDB-splines

Whole breast radiotherapy in prone and supine position: is there a place for multi-beam IMRT?

Background: Early stage breast cancer patients are long-term survivors and finding techniques that may lower acute and late radiotherapy-induced toxicity is crucial. We compared dosimetry of wedged tangential fields (W-TF), tangential field intensity-modulated radiotherapy (TF-IMRT) and multi-beam IMRT (MB-IMRT) in prone and supine positions for whole-breast irradiation (WBI). Methods: MB-IMRT, TF-IMRT and W-TF treatment plans in prone and supine positions were generated for 18 unselected breast cancer patients. The median prescription dose to the optimized planning target volume (PTVoptim) was 50 Gy in 25 fractions. Dose-volume parameters and indices of conformity were calculated for the PTVoptim and organs-at-risk. Results: Prone MB-IMRT achieved (p= 600 cc heart dose was consistently lower in prone position; while for patients with smaller breasts heart dose metrics were comparable or worse compared to supine MB-IMRT. Doses to the contralateral breast were similar regardless of position or technique. Dosimetry of prone MB-IMRT and prone TF-IMRT differed slightly. Conclusions: MB-IMRT is the treatment of choice in supine position. Prone IMRT is superior to any supine treatment for right-sided breast cancer patients and left-sided breast cancer patients with larger breasts by obtaining better conformity indices, target dose distribution and sparing of the organs-at-risk. The influence of treatment techniques in prone position is less pronounced; moreover dosimetric differences between TF-IMRT and MB-IMRT are rather small

Tchebycheffian B-splines in isogeometric Galerkin methods

Tchebycheffian splines are smooth piecewise functions whose pieces are drawn from (possibly different) Tchebycheff spaces, a natural generalization of algebraic polynomial spaces. They enjoy most of the properties known in the polynomial spline case. In particular, under suitable assumptions, Tchebycheffian splines admit a representation in terms of basis functions, called Tchebycheffian B-splines (TB-splines), completely analogous to polynomial B-splines. A particularly interesting subclass consists of Tchebycheffian splines with pieces belonging to null-spaces of constant-coefficient linear differential operators. They grant the freedom of combining polynomials with exponential and trigonometric functions with any number of individual shape parameters. Moreover, they have been recently equipped with efficient evaluation and manipulation procedures. In this paper, we consider the use of TB-splines with pieces belonging to null-spaces of constant-coefficient linear differential operators as an attractive substitute for standard polynomial B-splines and rational NURBS in isogeometric Galerkin methods. We discuss how to exploit the large flexibility of the geometrical and analytical features of the underlying Tchebycheff spaces according to problem-driven selection strategies. TB-splines offer a wide and robust environment for the isogeometric paradigm beyond the limits of the rational NURBS model.Comment: 35 pages, 18 figure

Explicit error estimates for spline approximation of arbitrary smoothness in isogeometric analysis

In this paper we provide a priori error estimates with explicit constants for both the $L^2$-projection and the Ritz projection onto spline spaces of arbitrary smoothness defined on arbitrary grids. This extends the results recently obtained for spline spaces of maximal smoothness. The presented error estimates are in agreement with the numerical evidence found in the literature that smoother spline spaces exhibit a better approximation behavior per degree of freedom, even for low smoothness of the functions to be approximated. First we introduce results for univariate spline spaces, and then we address multivariate tensor-product spline spaces and isogeometric spline spaces generated by means of a mapped geometry, both in the single-patch and in the multi-patch case.Comment: 39 pages, 4 figures. Improved the presentation. Article published in Numerische Mathemati

On multivariate polynomials in Bernstein–Bézier form and tensor algebra

AbstractThe Bernstein–Bézier representation of polynomials is a very useful tool in computer aided geometric design. In this paper we make use of (multilinear) tensors to describe and manipulate multivariate polynomials in their Bernstein–Bézier form. As an application we consider Hermite interpolation with polynomials and splines

Adaptive isogeometric analysis with hierarchical box splines

Isogeometric analysis is a recently developed framework based on finite element analysis, where the simple building blocks in geometry and solution space are replaced by more complex and geometrically-oriented compounds. Box splines are an established tool to model complex geometry, and form an intermediate approach between classical tensor-product B-splines and splines over triangulations. Local refinement can be achieved by considering hierarchically nested sequences of box spline spaces. Since box splines do not offer special elements to impose boundary conditions for the numerical solution of partial differential equations (PDEs), we discuss a weak treatment of such boundary conditions. Along the domain boundary, an appropriate domain strip is introduced to enforce the boundary conditions in a weak sense. The thickness of the strip is adaptively defined in order to avoid unnecessary computations. Numerical examples show the optimal convergence rate of box splines and their hierarchical variants for the solution of PDEs

Adaptive refinement with locally linearly independent LR B-splines: Theory and applications

In this paper we describe an adaptive refinement strategy for LR B-splines. The presented strategy ensures, at each iteration, local linear independence of the obtained set of LR B-splines. This property is then exploited in two applications: the construction of efficient quasi-interpolation schemes and the numerical solution of elliptic problems using the isogeometric Galerkin method.Comment: 23 pages, 14 figure