Local Linear Independence Research Articles

Algorithms and Data Structures for Cs-smooth RMB-splines of Degree 2s + 1

The simple mesh refinement algorithm of Groiss et al. (2023) generates T-meshes admitting Reachable Minimally supported (RM) B-splines that possess the property of local linear independence and form a non-negative partition of unity. The construction was first presented for the bilinear case and has later been extended to Cs-smooth splines of degree p=2s+1. The present paper is devoted to algorithms and data structures for RMB-splines. We prove that the memory consumption of the data structures for representing a T-mesh and the associated RMB-splines is linear with respect to the mesh size, and we describe the details of the underlying refinement algorithm. Moreover, we introduce a novel evaluation algorithm for RMB-spline surfaces, which is based solely on repeated convex combinations of the control points, thereby generalizing de Boor's algorithm for tensor-product splines. Numerical experiments are included to demonstrate the advantageous behaviour of the proposed data structures and algorithms with respect to their efficiency. We observe that the total computational time (which includes also error estimation and spline coefficient computation) scales roughly linearly with the number of degrees of freedom for the meshes considered.

Adaptive isogeometric methods with C1 (truncated) hierarchical splines on planar multi-patch domains

Isogeometric analysis is a powerful paradigm which exploits the high smoothness of splines for the numerical solution of high order partial differential equations. However, the tensor-product structure of standard multivariate B-spline models is not well suited for the representation of complex geometries, and to maintain high continuity on general domains special constructions on multi-patch geometries must be used. In this paper, we focus on adaptive isogeometric methods with hierarchical splines, and extend the construction of [Formula: see text] isogeometric spline spaces on multi-patch planar domains to the hierarchical setting. We replace the hypothesis of local linear independence for the basis of each level by a weaker assumption, which still ensures the linear independence of hierarchical splines. We also develop a refinement algorithm that guarantees that the assumption is fulfilled by [Formula: see text] splines on certain suitably graded hierarchical multi-patch mesh configurations, and prove that it has linear complexity. The performance of the adaptive method is tested by solving the Poisson and the biharmonic problems.

Local linear independence of bilinear (and higher degree) B-splines on hierarchical T-meshes

We generate hierarchical T-meshes by repeatedly inserting new line segments, in order to adapt both the size and the shape of the cells to the specific requirements of the underlying application. The associated spaces of bilinear spline functions are spanned by the locally refined (LR) B-splines of Dokken et al. (2013), which are products of univariate B-splines defined over local knot vectors. Our method guarantees that these functions are linearly independent for the obtained class of hierarchical T-meshes. Furthermore, they even possess the property of local linear independence, since exactly 4 functions take non-zero values on each cell of the mesh, and they form a partition of unity without the need to perform additional scaling or truncation.The correctness of our refinement algorithm is verified by enumerating the (newly introduced) local configurations, which represent the possible topologies of the mesh in the vicinity of a cell. It is then shown that the method works well in all the resulting situations. The fairly large number (385) of cases sheds some light on the challenges associated with linear independence in the context of LR B-spline refinement.Additionally we apply our method to meshes that possess the additional property local semi-regularity (first studied by Weller and Hagen, 1995), where only 49 topologically different situations may arise. However, we observe that the resulting class of hierarchical T-meshes is less flexible, which leads to a substantial amount of excess refinement in applications.Finally we note that the same construction can be applied to generate basis functions for Cs–smooth spline spaces of degree p=2s+1, again with the property of local linear independence.

Effective grading refinement for locally linearly independent LR B-splines

We present a new refinement strategy for locally refined B-splines which ensures the local linear independence of the basis functions. The strategy also guarantees the spanning of the full spline space on the underlying locally refined mesh. The resulting mesh has nice grading properties which grant the preservation of shape regularity and local quasi uniformity of the elements in the refining process.

Adaptive refinement for unstructured T-splines with linear complexity

We present an adaptive refinement algorithm for T-splines on unstructured 2D meshes. While for structured 2D meshes, one can refine elements alternatingly in horizontal and vertical direction, such an approach cannot be generalized directly to unstructured meshes, where no two unique global mesh directions can be assigned. To resolve this issue, we introduce the concept of direction indices, i.e., integers associated to each edge, which are inspired by theory on higher-dimensional structured T-splines. Together with refinement levels of edges, these indices essentially drive the refinement scheme. We combine these ideas with an edge subdivision routine that allows for I-nodes, yielding a very flexible refinement scheme that nicely distributes the T-nodes, preserving global linear independence, analysis-suitability (local linear independence) except in the vicinity of extraordinary nodes, sparsity of the system matrix, and shape regularity of the mesh elements. Further, we show that the refinement procedure has linear complexity in the sense of guaranteed upper bounds on a) the distance between marked and additionally refined elements, and on b) the ratio of the numbers of generated and marked mesh elements.

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.

On the linear independence of truncated hierarchical generating systems

Motivated by the necessity to perform adaptive refinement in geometric design and numerical simulation, the construction of hierarchical splines from generating systems spanning nested spaces has been recently studied in several publications. Linear independence can be guaranteed with the help of the local linear independence of the spline basis at each level. The present paper extends this framework in several ways. Firstly, we consider spline functions that are defined on domain manifolds, while the existing constructions are limited to domains that are open subsets of Rd. Secondly, we generalize the approach to generating systems containing functions which are not necessarily non-negative. Thirdly, we present a more general approach to guarantee linear independence and present a refinement algorithm that maintains this property. The three extensions of the framework are then used in several relevant applications: doubly hierarchical B-splines, hierarchical Zwart-Powell elements, and three different types of hierarchical subdivision splines.

Adaptively refined multi-patch B-splines with enhanced smoothness

A spline space suitable for Isogeometric Analysis (IgA) on multi-patch domains is presented. Our construction is motivated by emerging requirements in isogeometric simulations. In particular, IgA spaces should allow for adaptive mesh refinement and they should guarantee the optimal smoothness of the discretized solution, even across interfaces of adjacent patches.Given a domain manifold M consisting of individual patches (isomorphic to the unit square or cube) that are glued together along interfaces, we present a construction of multi-patch B-splines defined on them. Their smoothness is enhanced by locally modifying or merging basis functions around the boundary of each patch. The resulting multi-patch B-splines with enhanced smoothness (MPBES) possess the property of local linear independence and form a non-negative partition of unity. Moreover, their span can be characterized as the linear space of all piecewise polynomial functions on the domain manifold that possess certain smoothness properties.Subsequently, adaptively refined MPBES are obtained by generalizing the construction of truncated hierarchical (TH) B-splines. More precisely, a nested sequence of spaces spanned by MPBES is considered, corresponding to steps of local enrichment. In addition, an inversely nested sequence of subdomains (which are submanifolds of M) is used to specify the local refinement level of functions in these spaces. Finally, truncated hierarchical MPBES are obtained by means of the selection and truncation mechanism of THB-splines. The desired properties of linear independence and convex partition of unity are maintained.The paper presents several numerical examples which demonstrate potential applications of the new basis in isogeometric analysis.

Linear independence of nearest-neighbor valence bond states in several two-dimensional lattices

We show for several two-dimensional lattices that the nearest-neighbor valence bond states are linearly independent. To do so, we utilize and generalize a method that was recently introduced and applied to the kagome lattice by one of the authors. This method relies on the choice of an appropriate cell for the respective lattice, for which a certain local linear independence property can be demonstrated. Whenever this is achieved, linear independence follows for arbitrarily large lattices that can be covered by such cells, for both open and periodic boundary conditions. We report that this method is applicable to the kagome, honeycomb, square, squagome, two types of pentagonal, square-octagonal, star, two types of Archimedean, three types of ``martini'', and fullerene-type lattices, e.g., the well known `buckyball.'' Applications of the linear independence property, such as the derivation of effective quantum dimer models or the construction of new solvable spin-$1/2$ models, are discussed.

On the Local Linear Independence of Generalized Subdivision Functions

Characterizing the linear and local linear independence of the functions that span a linear space is a key task if the space is to be used computationally. Given a control net, the spanning functions of one spatial coordinate of a generalized subdivision surface are called nodal functions. They are the limit, under subdivision, of associating the value one with one control net node and zero with all others. No characterization of independence of nodal functions has been published to date, even for the two most popular generalized subdivision algorithms, Catmull–Clark subdivision and Loop’s subdivision. This paper provides a road map for the verification of linear and local linear independence of generalized subdivision functions. It proves the conjectured global independence of the nodal functions of both algorithms, disproves local linear independence (for higher valences), and establishes linear independence on every surface region corresponding to a facet of the control net. Subtle exceptions, even to global independence, underscore the need for a detailed analysis to provide a sound basis for a number of recently developed computational approaches.

Supports of Locally Linearly Independent M -Refinable Functions, Attractors of Iterated Function Systems and Tilings

This paper is devoted to a study of supports of locally linearly independent M-refinable functions by means of attractors of iterated function systems, where M is an integer greater than (or equal to) 2. For this purpose, the local linear independence of shifts of M-refinable functions is required. So we give a complete characterization for this local linear independence property by finite matrix products, strictly in terms of the mask. We do this in a more general setting, the vector refinement equations. A connection between self-affine tilings and L 2 solutions of refinement equations without satisfying the basic sum rule is pointed out, which leads to many further problems. Several examples are provided to illustrate the general theory.

Local polynomial property and linear independence of refinable distributions

In this paper, local polynomial property, global linear independence, and local linear dependence of the convolution of a B-spline and a refinable distribution supported on a Cantor-like set are studied.

Linear independence of the integer translates of compactly supported distributions and refinable vectors

In this paper, the global and local linear independence of any compactly supported distributions by using time domain spaces, and of refinable vectors by invariant linear spaces are investigated.

Local linear independence of refinable vectors of functions

This paper is devoted to a study of local linear independence of refinable vectors of functions. A vector of functions is said to be refinable if it satisfies the vector refinement equation where a is a finitely supported sequence of r × r matrices called the refinement mask. A complete characterization for the local linear independence of the shifts of ϕ1,…,ϕr is given strictly in terms of the mask. Several examples are provided to illustrate the general theory. This investigation is important for construction of wavelets on bounded domains and nonlinear approximation by wavelets.

Affine similarity of refinable functions

In this paper, we consider certain affine similarity of refinable functions and establish, certain concection between some local and global properties of refinable functions, such as local and global linear independence, local smoothness and B-spline, local and global Holder continuity.

Remarks on the Wronskian and on sums decompositions

We make a number of remarks on some theorems and considerations contained in the recent and valuable monograph [9]. The connection of the Wronskian with the linear dependence, the local linear dependence and the local linear independence is considered. The necessary and/or sufficient conditions in order that a function $ h(x,y) $ has the form $ \sum^n_{i=1} f_i(x)g_i(y) $ , in terms of the Wronski matrix, are discussed. The d'Alembert equation $ hh_{xy} - h_xh_y = 0 $ is investigated, too. We formulate also some open problems.

Exponential Box-Like Splines on Nonuniform Grids

We generalize the exponential box spline by allowing it to have arbitrarily spaced knots in any of its directions and derive the corresponding recurrence and differentiation rules. The corresponding spline space is spanned by the shifts of finitely many such splines and contains the usual family of exponential polynomials. The (local) linear independence of the spanning set is equivalent to a geometric condition closely related to unimodularity.

Connections between the Support and Linear Independence of Refinable Distributions

The purpose of this paper is to study the relationships between the support of a refinable distributionφand the global and local linear independence of the integer translates ofφ. It has been shown elsewhere that a compactly supported distributionφhas globally independent integer translates if and only ifφhas minimal convex support. However, such a distribution may have “holes” in its support. By insisting thatφ∈L2(R) and generates a multiresolution analysis, Lemarié and Malgouyres have ensured that no such holes can occur. In this article we generalize this result to refinable distributions. We also give a result on the local linear independence of the integer translates ofφ. We work with integer dilation factorN⩾2 throughout this paper.

Least supported bases and local linear independence

We introduce the concept of least supported basis, which is very useful for numerical purposes. We prove that this concept is equivalent to the local linear independence of the basis. For any given locally linearly independent basis we characterize all the bases of the space sharing the same property. Several examples for spline spaces are given.

Subdivision Schemes Determined by Coefficients of a Hurwitz Polynomial

The functional equation \[E(t) = \sum_{i = 0}^k {a_i E(2t - i),\qquad \sum_{i = - \infty }^\infty {E(t - i) = 1} } \] is known to have a unique compactly supported continuous solution, given that $\{ a_i \} _{i = 0}^k $ is a positive sequence satisfying $\Sigma _i a_{2i} = \Sigma _i a_{2i - 1} = 1$. This solution, $E(t)$, is also obtained by a subdivision algorithm. This paper discusses the particular case where $\Sigma _{i = 0}^k a_i z^i $ is a Hurwitz polynomial, i.e., all zeros are in the left half plane.The problem of Lagrange interpolation from the space $\{ E( \cdot - i)\} ,i \in Z$, was recently analyzed by Goodman and Micchelli [SIAM J. Math. Anal., 23 (1992), pp. 766–784]. In particular, they showed that the problem is solvable under a condition similar to the B-spline case.Here an alternative analysis is introduced, and results on the problem of Hermite interpolation are stated. The local linear independence of $\{ E( \cdot - i)\} ,i \in Z$, is discussed also for the case where $\Sigma _{i = 0} ...