In 2011, 2012, and 2013 I organized the geometry seminar at the Center of Mathematics for Applications, University of Oslo. Below you can find titles, abstracts and slides of the talks that were presented in this period.
November 6, Evely Leetma (University of Oslo)
Title: On the deviation of a parametric quadratic spline interpolant from its data polygon.
Abstract: Usually the control points of spline interpolants are obtained by solving a system of linear equations. We derive an explicit formula for the control points of periodic uniform quadratic spline interpolants. The explicit formula is used to obtain bounds for local and global deviation between the spline and its data polygon. For cubic splines these bounds are known in case of all three common parametrizations: uniform, chordal and centripetal.
October 30, Georg Muntingh (SINTEF)
Title: A Hermite subdivision scheme for C2 quintic macro-elements on the Powell-Sabin 12-split
Abstract: In order to construct a C1 quadratic spline over an arbitrary triangulation, one can split each triangle into 12 subtriangles, resulting in a finer triangulation known as the Powell-Sabin 12-split. It was shown by Dyn and Lyche that the corresponding spline surface can be plotted quickly by means of a Hermite subdivision scheme. In this talk we introduce a nodal macro-element on the 12-split for the space of quintic splines that are locally C3 and globally C2. For quickly evaluating any such spline, a Hermite subdivision scheme is derived, implemented, and tested in the computer algebra system Sage. Using the available first derivatives for Phong shading, visually appealing plots can be generated after 3-4 refinements.
October 23, Tom Lyche (University of Oslo)
Title: Smooth Simplex Splines for the Powell-Sabin 12-Split
Abstract: In a joint paper with Elaine Cohen and Richard Riesenfeld, a simplex spline basis for C1 quadratics on the well-known Powell-Sabin 12-split was constructed. This basis has all the usual properties of univariate B-splines. In particular, they form a nonnegative partition of unity, provide a recurrence relation down to hat functions, satisfy a Marden-like identity, and exhibit Lq stability for a scaled version. Furthermore, the restriction of each basis element to the boundary edges of the macro element reduces to a standard univariate B-spline. A control mesh can be formed that mimics the shape of the surface and exhibits distance O(h2) to any one of its control points from its surface, where h is the length of the longest edge. We obtain a pyramidal evaluation algorithm in terms of the control points that is strikingly reminiscent of the analogous one for triangular Bézier surfaces. In this talk we consider the possibility of using simplex splines of higher degree and smoothness as a basis for piecewise polynomials on the 12-split.
October 16, Michael Floater (University of Oslo)
Title: Lower set polynomial interpolation
Abstract: A lower set of multi-indices in d variables defines both an associated space of polynomials and, within any Cartesian grid of points, a subset of the points that admits unique interpolation from that space. This talk discusses how the interpolant can be expressed, and computed, from tensor-product interpolants. An application is the construction of nodal bases for finite element serendipity spaces in d dimensions.
June 17, Juan Gerardo Alcázar Arribas (Universidad de Alcalá, Madrid)
Title: Similarity Detection for Rational Curves
Abstract: In Pattern Recognition, there is a vast literature concerning the question how to detect whether two curves are similar. Essentially, the problem is to recognize a certain curve as the result of applying a movement to another curve in a database. Most of the strategies proposed so far deal with curves in implicit form, and ultimately resort to numerics to decide whether such curves are related by a similarity. In this talk, we present a new, fast, and deterministic algorithm to address the problem in the case when the curves are defined by a rational proper parametrization in exact arithmetic. The algorithm does not require to compute or use implicit equations of the curves, and takes advantage of the fact that the curves are similar if and only if their parametrizations are related by means of a Möbius transformation. It has been implemented and tested in the Sage computer algebra system, and shows good performance for middle inputs. This is joint work with Carlos Hermoso and Georg Muntingh.
February 25, Georg Muntingh (CMA)
Title: Exact regularity of subdivision schemes of higher arity
Abstract: In a paper of 1992, Rioul introduced a technique to compute the exact regularity of a wide class of binary subdivision schemes. In this talk we will look at the extension of this technique to subdivision schemes of higher arity. We use this technique to compute the exact regularity of various subdivision schemes encountered in the literature, including pseudo-splines of higher arity defined by Conti and Hormann. This is joint work with Michael Floater and Maria Charina.
February 11, Michael S. Floater (CMA)
Title: Exact regularity of pseudo-splines
Abstract: This talk is a follow up of our previous talk in the seminar. As discussed earlier, there is a large class of subdivision schemes whose exact regularity can be determined without needing to compute (approximately) the joint spectral radius of a pair of matrices: it is sufficient to compute the spectral radius of a single matrix. The basic idea was discovered by Rioul, but has also been observed by Bin Han. Within this class are the pseudo-spline schemes, which can be viewed as a blend between B-splines and the interpolatory Dubuc-Deslauriers family of schemes. In this talk we derive, numerically, the regularities of the primal and dual pseudo-spline schemes of low order. We then derive, mathematically, various comparisons between these regularities using the Fourier transform. We show, for example, that the regularity of the Dubuc-Deslauriers family of schemes increases with the size of the subdivision mask. This is joint work with Georg Muntingh.
January 15, Tor Dokken (SINTEF)
Title: Splines for geometry and PDEs
October 29, Andrea Bressan (CMA/University of Pavia)
Title: B-spline generators for piecewise polynomials on quadrilateral meshes
Abstract: Local refinement is needed both in geometric design and in numerical analysis. One recent approach is Locally Refinable B-splines where a set of B-splines is obtained algorithmically. The set of B-splines does not necessarily span the whole piecewise polynomial space, nor do they need to be linearly independent. In this seminar a set of equivalent properties that imply linear independence is presented.
October 22, Michael Floater (CMA)
Title: Exact regularity of symmetric univariate subdivision schemes
Abstract: In this talk we review and refine a technique of Rioul to determine the Hölder regularity of a large class of symmetric subdivision schemes from the spectral radius of a single matrix. These schemes include those of Dubuc and Deslauriers, their dual versions, and more generally the pseudo-spline and dual pseudo-spline schemes. The technique also applies to convex combinations of these, and therefore to several 'tension parameter' schemes.
October 8, Kjell Fredrik Pettersen (Sintef)
Title: Dimension formulas for the space of multivariate spline functions over a non-regular domain partition
Abstract: In 2010, Bernard Mourrain established a dimension formula for the space of bivariate spline functions on a planar rectangular domain with a rectangular subdivision. This approach can be generalized to dimension formulas for a space of d-variate spline functions over a rectangular d-dimensional box (d-box) in Rd, with an arbitrary subdivision into smaller d-boxes and arbitrary continuity constraints between the boxes. We will use homological techniques to establish spline space dimension formulas consisting of two parts. The first part is combinatorial and depends only on the topological structure of the subdivision and the order of continuity between each pair of adjacent d-boxes, while the parameterization does not matter. The second part will consist of dimensions of homology terms. We will try to investigate the role of the homology terms, and see how they vanish if the substructures of the subdivision is not too narrow. This is a joint work with Bernard Mourrain (INRIA, Nice, France)
October 1, Evely Leetma (CMA)
Title: Solution of smoothing problems with obstacles
Abstract: We consider classical smoothing problems for several variables where the values to be smoothened are the knot values of an unknown function and the solution should have the minimal seminorm on Beppo Levi space. The smoothing problem with weights needs the solution of a certain linear system but in practice the weights are not known. The smoothing problem with obstacles, where error bounds on a finite set of knots are given, is, however, completely practical. The necessary and sufficient conditions describing the solution are known but finding an algorithm to solve this problem is still an open problem. The solutions of both these problems are natural splines. A quite natural method of adding-removing knots is proposed by M. Ignatov and A. Pevnyj in their monograph but this method can generate a cycle. We present examples of cycling and counterexamples to possible use of some ideas. We also give some sufficient conditions for finiteness of the method. Another very natural idea is to reduce the smoothing problem with obstacles to an equivalent problem with weights. We prove that for problems with obstacles in Hilbert spaces and also in classical case the associated Lagrangian has a saddle point. This implies the existence of equivalent problems with weights. We present the equation connecting deviations of initial problem with obstacles and weights of equivalent problem with weights. The equation contains an arbitrary symmetric regular matrix as a free parameter.
September 24, Peter Nørtoft (Sintef)
Title: Isogeometric Analysis, Shape Optimization and Locally Refinable B-splines in Fluid Mechanics
Abstract: Isogeometric analysis is a recently proposed numerical method for solving partial differential equations (PDEs). The method aims at bridging the gap between finite element methods and computer aided design (CAD), thus facilitating exact representations of complex geometries, as well as smooth approximations of state variables. A key ingredient of the isogeometric method is to represent the geometry, and to approximate the state variables using smooth basis functions often of similar types. Tensor product B-splines and Non-Uniform Rational B-splines (NURBS) are simple first choices, since these are standard tools within the CAD community. Both of these, however, inhibit efficient local refinement, which is a crucial aspect in most practical applications. The recently proposed Locally Refinable (LR) B-splines represent a new approach to local refinement based on B-splines. This work focuses on three different aspects of the isogeometric paradigm within fluid mechanics, where the underlying PDE is the so-called Navier-Stokes equation. The first part of the talk discusses how the state variables may be discretized using B-splines, the second part how the method may be used for designing optimal shapes, and the third part how LR B-splines may be used for adaptive mesh refinements, all demonstrated through representative examples. This is joint work with T. Dokken (SINTEF, Norway), J. Gravesen (DTU, Denmark), D.M. Nguyen (SINTEF, Norway).
September 17, Maria Charina (Universität Dortmund)
Title: Polynomial Reproduction of Multivariate Scalar Subdivision Schemes
Abstract: In this talk we study a polynomial reproduction property of multivariate scalar subdivision scheme with dilation matrix mI, |m| ≥ 2. Namely, we derive simple algebraic conditions on the subdivision symbol that guarantee the reproduction of exactly the same polynomials from which the starting data for the subdivision recursion is being sampled. Our motivation for this work is the result by Adi Levin that states that such a reproduction property of degree k of a subdivision scheme is sufficient for its limits to have the approximation order k + 1. We illustrate our results with several examples. This is a joint work with Costanza Conti, Università di Firenze, Italy.
September 3, Maria Charina (Universität Dortmund)
Title: Real algebra and optimization for signal and image processing
Abstract: Recent results from real algebraic geometry and the theory of polynomial optimization are related in a new framework to the existence question of multivariate tight wavelet frames whose generators have at least one vanishing moment. Namely, several equivalent formulations of the so-called Unitary Extension Principle (UEP) by Ron and Shen are interpreted in terms of hermitian sums of squares of certain nonnegative trigonometric polynomials and in terms of semi- definite programming. The latter together with the results by Lai, Stöckler and Scheiderer answer affirmatively the long standing open question of the existence of such tight wavelet frames in dimension d = 2; we also provide numerically efficient methods for checking their existence and actual construction in any dimension. We exhibit a class of counterexamples in dimension d = 3 showing that, in general, the UEP property is not sufficient for the existence of tight wavelet frames. On the other hand we provide stronger sufficient conditions for the existence of tight wavelet frames in dimension d ≥ 3 and illustrate our results by several examples. This is a joint work with Mihai Putinar University of California at Santa Barbara, USA, Claus Scheiderer, Universität Konstanz, Germany and Joachim Stöckler, Dortmund University of Technology, Germany.
June 18, Bart Siwek (CMA)
Title: A discrete Chebyshev spline finding algorithms
Abstract: We discuss a family of discrete algorithms for finding a piecewise polynomial analogue of a Chebyshev polynomial, the Chebyshev spline. This particular equioscillating function has numerous imporant properties - for example locations of its extrema are the best points for interpolation. We begin our discussion by a review of classical Remez type algorithm then we move to discrete, control polygon based, methods and present their structure as well as numerical results.
June 11, Dang Manh Nguyen (SINTEF)
Title: Shape Optimization and Isogeometric Analysis in Electromagnetism
Abstract: Isogeometric analysis (IGA), a recently proposed numerical method for solving PDEs, has brought a new turn for FEM-based shape optimization due to its capability of representing a geometry exactly with relatively few design control points. Our main focus is to investigate the incorporation the methodology into shape optimization with a very tight connection to electromagnetic problems. In this talk, an iterative procedure for shape optimization using IGA and the following applications of the investigation will be presented
Hearing the shape of a drum.
Shape optimization of magnetic energy concentrators.
Remarkably, we have obtained concentrators that perform one million times better than an earlier topology optimization result. This shows a great potential of shape optimization using IGA in electromagnetism.
June 4, Juan Gerardo Alcázar Arribas (Universidad de Alcalá, Madrid)
Title: Detecting Features in the Shape of Rational Curves
Abstract: In this talk we will discuss a novel, efficient method by Alcázar and Hermoso to deterministically detect whether a curve defined by a rational parametrization exhibits some kind of symmetry (central, mirror, rotation). This is a question of interest not only from a theoretical point of view, but also for applications in pattern recognition or CAGD. The method is based on the existing relationship between two proper parametrizations of a same curve.
May 21, Tom Lyche (CMA)
Title: Locally Refinable Splines on Box Partitions
Abstract: This will be a talk about splines, but the motivation for this work comes from isogeometric analysis. The central idea is to replace traditional Finite Element spaces by Non Uniform Rational B-Splines (NURBS) to provide accurate shape description and elements with higher degrees and smoothness. Since the introduction of the idea in 2005 by Tom Hughes and co-workers, promising results have been obtained documenting its potential. However, it has also been demonstrated that NURBS do not support the local refinement needed in efficient finite element analysis. To overcome this deficiency the use of T-splines is a promising alternative. T- splines use tensor product B-splines on a quadrilateral mesh with T-joins, called a T-mesh. There are some unresolved problems with T-splines. For example, T- splines are not always linearly independent, it has been observed that refinement along a diagonal in a T-mesh can lead to non-local refinement, and it leads to rational basis functions. To overcome some of these problems we present a general theory for an alternative called Locally Refinable splines or LR-splines. In two dimensions we obtain a special case of a T-mesh, here named an LR-mesh. However, the concept works in any space dimension, local refinement is guaranteed, the tensor product basis functions are piecewise polynomials, they span the full piecewise polynomial space on the underlying partition, and simple strategies guarantee linear independence. This is joint work with Tor Dokken and Kjell Fredrik Pettersen at SINTEF, Oslo.
May 4, Kai Hormann (University of Lugano)
Title: Polynomial Reproduction for Univariate Subdivision Schemes of any Arity
Abstract: This talk is about the ability of convergent subdivision schemes to reproduce polynomials in the sense that for initial data, which is sampled from some polynomial function, the scheme yields the same polynomial in the limit. This property is desirable because the reproduction of polynomials up to some degree d implies that a scheme has approximation order d+1. We first show that any convergent, linear, uniform, and stationary subdivision scheme reproduces linear functions with respect to an appropriately chosen parameterization. We then present a simple algebraic condition for polynomial reproduction of higher order. All results are given for subdivision schemes of any arity m>1 and we use them to derive a unified definition of general m-ary pseudo-splines. Our framework also covers non-symmetric schemes and we give an example where the smoothness of the limit functions can be increased by giving up symmetry.
March 12, Bruno Simões (University of Trento)
Title: Gröbner basis computation revised
Abstract: Algebraic geometry is the mathematical field that studies geometric objects by means of algebra. Its origins go back to Descartes together with his introduction to the coordinate geometry. In the twentieth century, algebraic geometry became much broader and in many ways much more abstract due to the emergence of commutative algebra and homological algebra as the foundational language of the subject.
As the abstract theory of algebraic geometry was being developed in the middle of the twentieth century, a parallel development was taking place concerning the algorithmic aspects of the subject. The groebner bases theory emerged from that branch with the goal of providing a way to manipulate systems of equations systematically. Recent engineering applications of this theory include computer graphics, computer vision, geometric modeling, geometric theorem proving, optimization, control theory, statistics, communications, biology, robotics, coding theory, and cryptography.
In this talk we will provide an elementary introduction to Groebner bases, comparing as well some of the best-known algorithms in this area (e.g. F5 and G2V). We will also describe a combination of ideas that can be used to improve the performance of signature-based Groebner basis algorithms (e.g. using the information about syzygy module and Hilbert series).
February 27, Georg Muntingh (CMA)
Title: A Classification of the Generalized Principal Lattices in Space
Abstract: In multivariate polynomial interpolation theory, the properties of polynomial interpolants depend very much on the configuration of the interpolation points in space. An important class is made up by the generalized principal lattices, which form a corner stone in the classification of the meshes with simple Lagrange formula and can be viewed as a generalization of the triangular meshes.
While generalized principal lattices are defined by an abstract combinatorial definition, all generalized principal lattices in the projective plane arise from a real cubic curve in the dual projective plane. As all such curves are of arithmetic genus 1, one can ask the question: Which space curves of arithmetic genus 1 and degree 4 give rise to generalized principal lattices in dual projective space?
In this talk we show how complete intersections of quadric surfaces can be used to define generalized principal lattices in space.
January 30, Gianpaolo Orioli (Università di Roma)
Title: A quick journey through algorithms and polytopes for the stable set problem on claw graphs
Abstract: In his seminal paper, Edmonds provided a polynomial-time algorithm for solving the maximum weighted matching problem and a complete linear description of the matching polytope. Later, Padberg and Rao provided a fast combinatorial algorithm for the separation problem over the latter polytope.
In the talk we will survey a few recents results on the maximum weighted stable set problem for a claw-free graph, that is a non-trivial generalization of the weighted matching problem. Namely, we will illustrate a polynomial-time algorithm for the problem, that is based on some graph decomposition results, and a linear description of the stable set polytope (in a slightly extended space) that comes together with an efficient separation routine.
January 23, Michael Floater (IFI/CMA)
Title: A smoothness criterion for monotonicity-preserving subdivision
Abstract: In this paper we study subdivision schemes that both interpolate and preserve the monotonicity of the data, and we derive a simple `ratio' condition that guarantees the continuous differentiability of the limit function. We then show that the condition holds for three specific non-linear, four-point schemes of this type: Kuijt and van Damme's scheme, based on rational functions, Sabin and Dodgson's scheme, based on square roots; and a new scheme based on fourth roots. This is joint work with C. Beccari, T. Cashman, and L. Romani.
January 9, Heidi Dahl (SINTEF/CMA/Vilnius University)
Title: Rational rolling ball blends between natural quadrics
Abstract: Natural quadrics are the simplest primitive shapes used in Computer Aided Design (CAD): spheres, right circular cones and right circular cylinders. By combining natural quadrics with rolling ball blends between them, we are able to model the majority of mechanical parts exactly. However, though the natural quadrics are rational surfaces, a rolling ball blend between two natural quadrics needs not be.
In this talk I will give a complete classification of the configurations of natural quadrics that can be blended with rational fixed radius rolling ball blends. Their parametrizations are constructed by applying results from canal surfaces. For variable radius blends, I will present a general algorithm for constructing minimal bi- degree rational parametrizations of patches on canal surfaces. Finally, we consider how to blend corners, where the blends of the adjoining edges intersect.
December 12, Karoline Moe (CMA)
Disclaimer: This seminar will contain a fairytale. It will necessarily involve some slides with boring math, but also blowups of exceptional power as well as intriguing space travels.
Title: Rational cuspidal curves
Abstract: Once upon a time, in a space far, far away, there was a curve and a problem... How many and what kind of cusps can a rational cuspidal curve have? This is a fairly easily posed question, which yet has no answer - even for rational cuspidal curves on the projective plane. In this talk I will first try to explain the mentioned problem and the partial results in the projective plane. Then I will try to show what happens if we move to another surface - namely P1 x P1...
December 5, Bart Siwek (IFI/CMA)
Title: A simple Hermite interpolatory subdivision scheme
Abstract: In this talk a simple Hermite interpolatory subdivision scheme will be presented. Problems with natural Hermite extension of the 4-point Dubuc's scheme will be discussed first, and a simplified version will be presented. The talk will continue with a more in-depth look at that particular version and conclude with a presentation of its most important properties: smoothness, approximation order and Holder regularity.
November 28, Elisa Postinghel (CMA), Slides
Title: Algebraic geometry, secant varieties and multivariate interpolation
Abstract: The problem of determining the dimension of higher secant varieties of an algebraic variety X is a very hard one to be solved in general. It has an equivalent reformulation, via Terracini's Lemma, in the framework of multivariate Hermite interpolation. Indeed calculating the dimension of the n-secant variety of X is equivalent to calculating the dimension of the linear system of the hypersurfaces of X with n prescribed double points in general position. In this talk I will discuss an inductive approach, inspired by works of C. Ciliberto and R. Miranda, to compute the dimension of all secant varieties of a class of Segre-Veronese varieties. This is joint work with A. Laface (Concepción, Chile).
October 31, Oliver Barrowclough (SINTEF)
Title: Implicitization using interpolation
Abstract: In this talk we will look at how methods from interpolation theory can be used for implicitization of parametric curves and surfaces. We will present two different approaches and discuss the computational benefits of these methods over traditional algorithms for implicitization. We will also show how the methods can be extended to produce low degree implicit approximations, thereby obtaining even more computational efficiency.
October 17, Tom Lyche (IFI/CMA)
Title: B-Splines: A Fundamental Tool for Analysis and Computation
Abstract: Starting from their inception in approximation theory 60 years ago, the use of B-splines has blossomed in diverse areas in science, engineering and electronic arts. These include animation, computational geometry, computer-aided design, computer-aided manufacturing, computer graphics, control theory, geometric design, image analysis, medical visualization, optimization, partial differential equations, robotics, and statistics. More recently B-splines have been assuming a fundamental role in the isogeometric analysis, an ambitious effort to unify shape representation and engineering analysis. Honoring its interesting historical development, this talk gives a mathematical introduction to splines and B-splines in one and several space dimensions and also considers various generalizations.
October 3, Georg Muntingh (CMA), Slides
Title: Divided Differences of Multivariate Implicit Functions
Abstract: Under general conditions, the equation g(x1, ..., xq, y) = 0 implicitly defines y locally as a function of x1, ..., xq. In this talk we present a formula that expresses divided differences of y in terms of divided differences of g, generalizing a recent formula for the case where y is univariate. The formula involves a sum over a combinatorial structure whose elements can be viewed either as polygonal partitions or as planar trees. Through this connection we indicate how to arrive at a simpler formula for derivatives.
June 10, Nada Sissouno (University of Darmstadt, Germany), Slides
Title: Approximation with Tensor Product Splines on Domains
Abstract: Tensor product splines obviously work well on domains that are rectangular boxes. However, in applications general domains naturally arise and in that case the approximation with tensor product splines faces various problems. To resolve them for the bivariate case we suggest a new variation of the tensor product B- splines called condensed B-splines.
June 6, Michael Floater (CMA), Slides
Title: Interpolatory Subdivision of Irregularly Spaced Data
Abstract: The talk is about recent work on establishing the Holder regularity and approximation properties of interpolatory subdivision schemes in the case that the data are irregularly spaced. The focus will be on the generalization of the family of schemes based on polynomials of odd degree proposed by Dubuc and Deslauriers.
May 30, Richard F. Riesenfeld (University of Utah, USA)
Title: Dynamic Geometric Computation of Interacting Models
Abstract: In many diverse applications involving design and manipulation of free-form geometric modeling, it is important to compute continuously and robustly the precise interactions among objects undergoing dynamic deformations over a period of time. This problem is transformed into a computation on a higher dimensional manifold where singularity theory is introduced to develop a rigorous framework for solving it robustly, and in a manner that guarantees identifying all points of abrupt topological transition.
Keywords: Deformations, Spline Intersection, B-splines
Joint work with Xianming Chen (Utah), Elaine Cohen (Utah), James Damon (University of North Carolina, USA)
May 9, Nikolay Qviller (CMA), Slides
Title: Fundamental Polynomials of Nodal Curve Counting
Abstract: Nodal curves in a fixed complete linear system |L| on a projective surface S are enumerated by Bell polynomials in certain universal, linear combinations of the four Chern numbers of (S,L). These linear polynomials can be interpreted through the contribution of certain diagonals to certain intersection products, but have proven difficult to describe explicitly. It turns out that part of the reason is the failure of the Segre class of a closed subscheme to satisfy an inclusion-exclusion principle. However, as was pointed out by Aluffi, this failure can be corrected by slightly modifying the definition of the Segre class. After recalling some essential points in the enumerative geometry of nodal curves, I will discuss the application of Aluffi's ideas to this particular problem.
May 2, Tom Lyche (CMA), Slides
Title: Simplex Splines for the Powell-Sabin 12-Split
Abstract: A Simplex Spline is the natural generalization of a B-spline to several dimensions. We give a short introduction to bivariate simplex splines and then introduce a simplex spline basis for a space of C1 quadratics on the well-known Powell-Sabin 12-split triangular region. Among its many important desirable properties, we show that it has an associated recurrence relations for evaluation and differentiation. Also discussed briefly are a Marsden-like identity, quasi- interpolants, approximation methods exhibiting unconditional stability, a subdivision scheme, and smoothness conditions across macro-element edges.
March 28, Tomas Sauer (Justus Liebig University Giessen, Germany), Slides
Title: Divided Differences in Several Variables – an Interpolating Approach
Abstract: There are many generalizations of divided differences to the multivariate case, depending on which of the many properties of the univariate differences one wishes to extend. One possibility, pursued by de Boor, is to extend simplex spline integrals, another one is to define divided differences as the leading coefficients or leading forms of an interpolation polynomial. This latter approach, however, requires to clarify which concept of multivariate polynomial interpolation one wishes to pursue. The talk considers divided differences based on monomial minimal degree interpolation spaces which indeed generalize a lot of properties of the univariate divided difference. A recurrence relation, connections to simplex splines and even a product formula will be given. Partially, this is joint work with Jesus Carnicer from Zaragoza.
March 7, Nelly Villamizar (CMA), Slides
Title: Bounds on the Dimension of Triangular Splines
Abstract: We consider the space of spline functions defined on a triangular subdivision of a polygonal domain. We will show how homological techniques can be used to give bounds to the dimension of this spline space. The lower and upper bound that we present are, in many or perhaps most of the cases, more general and give better approximations to the exact value of the dimension than the already existing ones. These results can also be extended to find bounds on the dimension of spline spaces on 3-dimensional (simplicial) complexes embedded in R3.
February 14, Solveig Bruvoll (CMA)
Title: Uniformly Stable Wavelets on General Triangulations
Abstract: In this talk we construct wavelet-like functions on arbitrary triangulations. Nested triangulations are obtained through refinement by two standard strategies, in which no regularity is required. One strategy inserts a new node inside a triangle and splits it into three smaller triangles. The other strategy splits two neighboring triangles into four smaller triangles by inserting a new node on the edge between the triangles. The refinement results in nested spaces of piecewise linear functions. These functions are made to satisfy certain orthogonality conditions, which locally correspond to vanishing linear moments. It turns out that this construction is uniformly stable in the L∞-norm independently of the geometry of the original triangulation and the refinements.
February 7, Tor Dokken (SINTEF/CMA)
Title: Locally Refined B-Splines and Isogeometric Analysis
Abstract: It is well known that tensor product B-splines do not support local refinement needed in many applications. In this talk we discuss Local Refinement (LR) B- splines on meshes with T-joins. The central idea of isogeometric analysis is to replace traditional Finite Elements by NonUniform Rational B-splines (NURBS) to provide accurate shape description and higher order elements in analysis. Since the introduction of the idea by T. Hughes in 2005 excellent results have been obtained documenting the potential of isogeometric analysis.
January 17, Georg Muntingh (CMA), Slides
Title:Generalized Principal Lattices in Space
Abstract: In multivariate polynomial interpolation theory, the properties of polynomial interpolants depend very much on the configuration of the interpolation points in space. An important class is made up by the generalized principal lattices, which form a corner stone in the classification of the meshes with simple Lagrange formula and can be viewed as a generalization of the triangular meshes.
While generalized principal lattices are defined by an abstract combinatorial definition, all generalized principal lattices in the projective plane arise from a real cubic curve in the dual projective plane. As all such curves are of arithmetic genus 1, one can ask the question: Which space curves of arithmetic genus 1 and degree 4 give rise to generalized principal lattices in dual projective space?
In this talk we show how generalized principal lattices arise naturally from the notion of a triangular mesh and try to give an answer to this question.