arXiv subjects

Hendrik Speleers

Publications and source records attributed to Hendrik Speleers.

At least 19 recordsLinked to original sources

arXiv2026

Isogeometric discretizations for the spectrum of the Laplace operator: outlier-free spline bases

Optimal spline subspaces are an elegant and efficient tool to remove spurious outliers in isogeometric Galerkin discretizations for the approximation of the spectrum of the Laplace operator. For practical purposes, it is valuable to have a basis construction for such spaces with good computational and spectral properties. We provide a characterization of the bases that enjoy a B-spline-like support structure and whose mass and stiffness matrices are simultaneously diagonalizable. It turns out that these mass and stiffness matrices admit explicitly known closed-form expressions for their eigenvalues, implying that the considered bases are outlier-free. A numerical procedure for constructing such bases is also presented.

math.NA↗

arXiv2026

On the completeness of generalized hierarchical spline spaces

We introduce a general theoretical approach to hierarchical spline spaces that replaces the classical constructive definition - based on basis selection - with a descriptive formulation in terms of regularity constraints. Specifically, we define generalized hierarchical spline spaces on multi-level domains as collections of piecewise functions satisfying hierarchical contact conditions across interfaces between refinement levels. The proposed framework applies to a broad class of local function spaces and relies on a minimal abstract requirement, the extension assumption, rather than on specific polynomial properties. Within this framework, we identify rules under which the hierarchical selection mechanism yields a complete basis, in the sense that it spans exactly the space characterized by the contact conditions. As an application, we consider Tchebycheffian spline spaces. We show that spaces generated by extended complete Tchebycheff (ECT) systems fit in this framework, thereby establishing the completeness of hierarchical Tchebycheffian splines. This demonstrates that the proposed theory naturally extends beyond the polynomial setting and provides a unified foundation for hierarchical constructions in more general spline spaces.

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arXiv2026

A Construction of $C^{r}$ Conforming Finite Elements on the Alfeld Split in Any Dimension

Constructing $C^r$ conforming finite element spaces in any dimension is a long-standing problem. For general triangulations, this problem was recently addressed by Hu-Lin-Wu (2024), under certain conditions on supersmoothness and polynomial degree. In this paper, a first unified construction on the Alfeld split in any dimension is given, where the supersmoothness conditions and the polynomial degree requirement are relaxed.

math.NA↗

arXiv2026

Isogeometric analysis with $C^1$ cubic Powell-Sabin splines

In this paper, we consider $C^1$ cubic Powell-Sabin splines for the numerical solution of boundary value problems on planar and spatial surface domains. We first review the construction and basic properties of polynomial and rational $C^1$ cubic Powell-Sabin spline representations on unstructured triangulations. Then, we discuss how these flexible representations can be exploited to create geometry mappings suited for a precise description of (classes of) surface domains. This is illustrated with several examples. Finally, the obtained domain descriptions are utilized in the isogeometric analysis framework for solving various Poisson and biharmonic problems. It is demonstrated that $C^1$ cubic Powell-Sabin splines form a powerful alternative to $C^0$ cubic Lagrange elements and bicubic NURBS.

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arXiv2025

On the normalization of trigonometric and hyperbolic B-splines

Trigonometric and hyperbolic B-splines can be computed via recurrence relations analogous to the classical polynomial B-splines. However, in their original formulation, these two types of B-splines do not form a partition of unity and consequently do not admit the notion of control polygons with the convex hull property for design purposes. In this paper, we look into explicit expressions for their normalization and provide a recursive algorithm to compute the corresponding normalization weights. As example application, we consider the exact representation of a circle in terms of $C^{2n-1}$ trigonometric B-splines of order $m=2n+1\geq3$, with a variable number of control points. We also illustrate the approximation power of trigonometric and hyperbolic splines.

math.NA↗

arXiv2025

Outlier-free isogeometric discretizations for Laplace eigenvalue problems: closed-form eigenvalue and eigenvector expressions

We derive explicit closed-form expressions for the eigenvalues and eigenvectors of the matrices resulting from isogeometric Galerkin discretizations based on outlier-free spline subspaces for the Laplace operator, under different types of homogeneous boundary conditions on bounded intervals. For optimal spline subspaces and specific reduced spline spaces, represented in terms of B-spline-like bases, we show that the corresponding mass and stiffness matrices exhibit a Toeplitz-minus-Hankel or Toeplitz-plus-Hankel structure. Such matrix structure holds for any degree p and implies that the eigenvalues are an explicitly known sampling of the spectral symbol of the Toeplitz part. Moreover, by employing tensor-product arguments, we extend the closed-form property of the eigenvalues and eigenvectors to a d-dimensional box. As a side result, we have an algebraic confirmation that the considered optimal and reduced spline spaces are indeed outlier-free.

math.NA↗

arXiv2024

A parsimonious approach to $C^2$ cubic splines on arbitrary triangulations: Reduced macro-elements on the cubic Wang-Shi split

We present a general method to obtain interesting subspaces of the $C^2$ cubic spline space defined on the cubic Wang-Shi refinement of a given arbitrary triangulation $\mathcal{T}$. These subspaces are characterized by specific Hermite degrees of freedom associated with only the vertices and edges of $\mathcal{T}$, or even only the vertices of $\mathcal{T}$. Each subspace still contains cubic polynomials while saving a consistent number of degrees of freedom compared with the full space. The dimension of the considered subspaces can be as small as six times the number of vertices of $\mathcal{T}$. The method fits in the setting of macro-elements: any function of such a subspace can be constructed on each triangle of $\mathcal{T}$ separately by specifying the necessary Hermite degrees of freedom. The explicit local representation in terms of a local simplex spline basis is also provided. This simplex spline basis intrinsically takes care of the complex geometry of the Wang-Shi split, making it transparent to the user.

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arXiv2024

Quadrature rules for splines of high smoothness on uniformly refined triangles

In this paper, we identify families of quadrature rules that are exact for sufficiently smooth spline spaces on uniformly refined triangles in $\mathbb{R}^2$. Given any symmetric quadrature rule on a triangle $T$ that is exact for polynomials of a specific degree $d$, we investigate if it remains exact for sufficiently smooth splines of the same degree $d$ defined on the Clough-Tocher 3-split or the (uniform) Powell-Sabin 6-split of $T$. We show that this is always true for $C^{2r-1}$ splines having degree $d=3r$ on the former split or $d=2r$ on the latter split, for any positive integer $r$. Our analysis is based on the representation of the considered spline spaces in terms of suitable simplex splines.

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arXiv2024

Using Geometric Symmetries to Achieve Super-Smoothness for Cubic Powell-Sabin Splines

In this paper, we investigate $C^2$ super-smoothness of the full $C^1$ cubic spline space on a Powell-Sabin refined triangulation, for which a B-spline basis can be constructed. Blossoming is used to identify the $C^2$ smoothness conditions between the functionals of the dual basis. Some of these conditions can be enforced without difficulty on general triangulations. Others are more involved but greatly simplify if the triangulation and its corresponding Powell-Sabin refinement possess certain symmetries. Furthermore, it is shown how the $C^2$ smoothness constraints can be integrated into the spline representation by reducing the set of basis functions. As an application of the super-smooth basis functions, a reduced spline space is introduced that maintains the cubic precision of the full $C^1$ spline space.

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arXiv2023

Construction of $C^2$ cubic splines on arbitrary triangulations

In this paper, we address the problem of constructing $C^2$ cubic spline functions on a given arbitrary triangulation $\mathcal{T}$. To this end, we endow every triangle of $\mathcal{T}$ with a Wang-Shi macro-structure. The $C^2$ cubic space on such a refined triangulation has a stable dimension and optimal approximation power. Moreover, any spline function in such space can be locally built on each of the macro-triangles independently via Hermite interpolation. We provide a simplex spline basis for the space of $C^2$ cubics defined on a single macro-triangle which behaves like a Bernstein/B-spline basis over the triangle. The basis functions inherit recurrence relations and differentiation formulas from the simplex spline construction, they form a nonnegative partition of unity, they admit simple conditions for $C^2$ joins across the edges of neighboring triangles, and they enjoy a Marsden-like identity. Also, there is a single control net to facilitate control and early visualization of a spline function over the macro-triangle. Thanks to these properties, the complex geometry of the Wang-Shi macro-structure is transparent to the user. Stable global bases for the full space of $C^2$ cubics on the Wang-Shi refined triangulation $\mathcal{T}$ are deduced from the local simplex spline basis by extending the concept of minimal determining sets.

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arXiv2023

Extraction and application of super-smooth cubic B-splines over triangulations

The space of $C^1$ cubic Clough-Tocher splines is a classical finite element approximation space over triangulations for solving partial differential equations. However, for such a space there is no B-spline basis available, which is a preferred choice in computer aided geometric design and isogeometric analysis. A B-spline basis is a locally supported basis that forms a convex partition of unity. In this paper, we explore several alternative $C^1$ cubic spline spaces over triangulations equipped with a B-spline basis. They are defined over a Powell-Sabin refined triangulation and present different types of $C^2$ super-smoothness. The super-smooth B-splines are obtained through an extraction process, i.e., they are expressed in terms of less smooth basis functions. These alternative spline spaces maintain the same optimal approximation power as Clough-Tocher splines. This is illustrated with a selection of numerical examples in the context of least squares approximation and finite element approximation for second and fourth order boundary value problems.

math.NA↗

arXiv2022

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.

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arXiv2022

ExSpliNet: An interpretable and expressive spline-based neural network

In this paper we present ExSpliNet, an interpretable and expressive neural network model. The model combines ideas of Kolmogorov neural networks, ensembles of probabilistic trees, and multivariate B-spline representations. We give a probabilistic interpretation of the model and show its universal approximation properties. We also discuss how it can be efficiently encoded by exploiting B-spline properties. Finally, we test the effectiveness of the proposed model on synthetic approximation problems and classical machine learning benchmark datasets.

cs.LG↗

arXiv2022

Ritz-type projectors with boundary interpolation properties and explicit spline error estimates

In this paper we construct Ritz-type projectors with boundary interpolation properties in finite dimensional subspaces of the usual Sobolev space and we provide a priori error estimates for them. The abstract analysis is exemplified by considering spline spaces and we equip the corresponding error estimates with explicit constants. This complements our results recently obtained for explicit spline error estimates based on the classical Ritz projectors in [Numer. Math. 144(4):889--929, 2020].

math.NA↗

arXiv2021

Application of optimal spline subspaces for the removal of spurious outliers in isogeometric discretizations

We show that isogeometric Galerkin discretizations of eigenvalue problems related to the Laplace operator subject to any standard type of homogeneous boundary conditions have no outliers in certain optimal spline subspaces. Roughly speaking, these optimal subspaces are obtained from the full spline space defined on certain uniform knot sequences by imposing specific additional boundary conditions. The spline subspaces of interest have been introduced in the literature some years ago when proving their optimality with respect to Kolmogorov $n$-widths in $L^2$-norm for some function classes. The eigenfunctions of the Laplacian -- with any standard type of homogeneous boundary conditions -- belong to such classes. Here we complete the analysis of the approximation properties of these optimal spline subspaces. In particular, we provide explicit $L^2$ and $H^1$ error estimates with full approximation order for Ritz projectors in the univariate and in the multivariate tensor-product setting. Besides their intrinsic interest, these estimates imply that, for a fixed number of degrees of freedom, all the eigenfunctions and the corresponding eigenvalues are well approximated, without loss of accuracy in the whole spectrum when compared to the full spline space. Moreover, there are no spurious values in the approximated spectrum. In other words, the considered subspaces provide accurate outlier-free discretizations in the univariate and in the multivariate tensor-product case. This main contribution is complemented by an explicit construction of B-spline-like bases for the considered spline subspaces. The role of such spaces as accurate discretization spaces for addressing general problems with non-homogeneous boundary behavior is discussed as well.

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arXiv2021

On the matrices in B-spline collocation methods for Riesz fractional equations and their spectral properties

In this work, we focus on a fractional differential equation in Riesz form discretized by a polynomial B-spline collocation method. For an arbitrary polynomial degree $p$, we show that the resulting coefficient matrices possess a Toeplitz-like structure. We investigate their spectral properties via their symbol and we prove that, like for second order differential problems, also in this case the given matrices are ill-conditioned both in the low and high frequencies for large $p$. More precisely, in the fractional scenario the symbol has a single zero at $0$ of order $α$, with $α$ the fractional derivative order that ranges from $1$ to $2$, and it presents an exponential decay to zero at $π$ for increasing $p$ that becomes faster as $α$ approaches $1$. This translates in a mitigated conditioning in the low frequencies and in a deterioration in the high frequencies when compared to second order problems. Furthermore, the derivation of the symbol reveals another similarity of our problem with a classical diffusion problem. Since the entries of the coefficient matrices are defined as evaluations of fractional derivatives of the B-spline basis at the collocation points, we are able to express the central entries of the coefficient matrix as inner products of two fractional derivatives of cardinal B-splines. Finally, we perform a numerical study of the approximation behavior of polynomial B-spline collocation. This study suggests that, in line with non-fractional diffusion problems, the approximation order for smooth solutions in the fractional case is $p+2-α$ for even $p$, and $p+1-α$ for odd $p$.

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arXiv2021

Best low-rank approximations and Kolmogorov n-widths

We relate the problem of best low-rank approximation in the spectral norm for a matrix $A$ to Kolmogorov $n$-widths and corresponding optimal spaces. We characterize all the optimal spaces for the image of the Euclidean unit ball under $A$ and we show that any orthonormal basis in an $n$-dimensional optimal space generates a best rank-$n$ approximation to $A$. We also present a simple and explicit construction to obtain a sequence of optimal $n$-dimensional spaces once an initial optimal space is known. This results in a variety of solutions to the best low-rank approximation problem and provides alternatives to the truncated singular value decomposition. This variety can be exploited to obtain best low-rank approximations with problem-oriented properties.

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arXiv2021

Computation of multi-degree Tchebycheffian B-splines

Multi-degree Tchebycheffian splines are splines with pieces drawn from extended (complete) Tchebycheff spaces, which may differ from interval to interval, and possibly of different dimensions. These are a natural extension of multi-degree polynomial splines. Under quite mild assumptions, they can be represented in terms of a so-called MDTB-spline basis; such basis possesses all the characterizing properties of the classical polynomial B-spline basis. We present a practical framework to compute MDTB-splines, and provide an object-oriented implementation in Matlab. The implementation supports the construction, differentiation, and visualization of MDTB-splines whose pieces belong to Tchebycheff spaces that are null-spaces of constant-coefficient linear differential operators. The construction relies on an extraction operator that maps local Tchebycheffian Bernstein functions to the MDTB-spline basis of interest.

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