arXiv subjects

Carla Manni

Publications and source records attributed to Carla Manni.

17 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.

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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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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.

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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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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

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].

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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

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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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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arXiv2020

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.

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arXiv2020

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.

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arXiv2020

A Tchebycheffian extension of multi-degree B-splines: Algorithmic computation and properties

In this paper we present an efficient and robust approach to compute a normalized B-spline-like basis for spline spaces with pieces drawn from extended Tchebycheff spaces. The extended Tchebycheff spaces and their dimensions are allowed to change from interval to interval. The approach works by constructing a matrix that maps a generalized Bernstein-like basis to the B-spline-like basis of interest. The B-spline-like basis shares many characterizing properties with classical univariate B-splines and may easily be incorporated in existing spline codes. This may contribute to the full exploitation of Tchebycheffian splines in applications, freeing them from the restricted role of an elegant theoretical extension of polynomial splines. Numerical examples are provided that illustrate the procedure described.

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arXiv2019

Sharp error estimates for spline approximation: explicit constants, $n$-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.

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arXiv2018

Isogeometric analysis for 2D and 3D curl-div problems: Spectral symbols and fast iterative solvers

Alfv\'en-like operators are of interest in magnetohydrodynamics, which is used in plasma physics to study the macroscopic behavior of plasma. Motivated by this important and complex application, we focus on a parameter-dependent curl-div problem that can be seen as a prototype of an Alfv\'en-like operator, and we discretize it using isogeometric analysis based on tensor-product B-splines. The involved coefficient matrices can be very ill-conditioned, so that standard numerical solution methods perform quite poorly here. In order to overcome the difficulties caused by such ill-conditioning, a two-step strategy is proposed. First, we conduct a detailed spectral study of the coefficient matrices, highlighting the critical dependence on the different physical and approximation parameters. Second, we exploit such spectral information to design fast iterative solvers for the corresponding linear systems. For the first goal we apply the theory of (multilevel block) Toeplitz and generalized locally Toeplitz sequences, while for the second we use a combination of multigrid techniques and preconditioned Krylov solvers. Several numerical tests are provided both for the study of the spectral problem and for the solution of the corresponding linear systems.

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arXiv2014

Convergence of univariate non-stationary subdivision schemes via asymptotical similarity

A new equivalence notion between non-stationary subdivision schemes, termed asymptotical similarity, which is weaker than asymptotical equivalence, is introduced and studied. It is known that asymptotical equivalence between a non-stationary subdivision scheme and a convergent stationary scheme guarantees the convergence of the non-stationary scheme. We show that for non-stationary schemes reproducing constants, the condition of asymptotical equivalence can be relaxed to asymptotical similarity. This result applies to a wide class of non-stationary schemes of importance in theory and applications.

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