Conference Agenda

Overview and details of the sessions of this conference. Please select a date or location to show only sessions at that day or location. Please select a single session for detailed view (with abstracts and downloads if available).

 
Session Overview
Session
MS140, part 3: Multivariate spline approximation and algebraic geometry
Time:
Thursday, 11/Jul/2019:
10:00am - 12:00pm

Location: Unitobler, F-107
30 seats, 56m^2

Presentations
10:00am - 12:00pm

Multivariate spline approximation and algebraic geometry

Chair(s): Michael DiPasquale (Colorado State University, United States of America), Nelly Villamizar (Swansea University)

The focus of the proposed minisymposium is on problems in approximation theory that may be studied using techniques from commutative algebra and algebraic geometry. Research interests of the participants relevant to the minisymposium fall broadly under multivariate spline theory, interpolation, and geometric modeling. For instance, a main problem of interest is to study the dimension of the vector space of splines of a bounded degree on a simplicial complex; recently there have been several advances on this front using notions from algebraic geometry. Nevertheless this problem remains elusive in low degree; the dimension of the space of piecewise cubics on a planar triangulation (especially relevant for applications) is still unknown in general.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Bivariate Semialgebraic Splines

Frank Sottile1, Michael DiPasquale2
1Texas A&M University, 2Colorado State University

We consider bivariate splines over partitions defined by arcs of irreducible algebraic curves. We compute the dimension of the space of semialgebraic splines in two extreme cases. If the forms defining the edges span a three-dimensional space of forms of degree $n$, then we show that the dimensions can be reduced to the linear case. If the partition is sufficiently generic, we give a formula for the dimension of the spline space in large degree and bound how large the degree must be for the formula to be correct. We also study the dimension of the spline space in some examples where the curves do not satisfy either extreme. The results are derived using commutative and homological algebra. This is joint work with Michael DiPasquale.

 

Geometrically smooth spline bases for geometric modeling

Ahmed Blidia, Bernard Mourrain
Inria

Given a topological complex M with glueing data along edges shared by adjacent faces, we study the associated space of geometrically smooth splines that satisfy differentiability properties across shared edges. We present new constructions of basis functions of the space of $G^1$- spline functions on quadrangular meshes, which are tensor product bspline functions on each quadrangle and with b-spline transition maps across the shared edges. By analysing the syzygy equation induced by the $G^1$ constraints over a single edge, we show that the separability of the space of $G^1$ splines across an edge allows to determine the dimension and a bases of the space of $G^1$ splines on M. This leads to new explicit construction of basis functions attached to the vertices, edges and faces of M.

The construction of smooth basis functions attached to a topological structure has important applications in geometric modeling. We illustrate it on the fitting of point clouds by $G^1$ splines on quadrangular meshes of complex topology. The ingredients are detailled and experimentation results showing the behavior of the method are presented.

 

Splines, Stable Bundles, and PDE’s

Peter Stiller
Texas A&M University

We will explain a number of connections between certain local and global problems in approximation theory related to spaces of splines and certain stable or semi-stable vector bundles/reflexive sheaves on complex projective spaces. These connections lead to an interesting relationship between the spaces of solutions of certain systems of constant coefficient partial differential equations and the first cohomology group of those vector bundles/reflexive sheaves. Using results of Grothendieck and Shatz, the case of two variables and the projective plane is analyzed. We will also discuss extensions to vector bundles on higher-dimensional projective spaces as they relate to splines and PDE’s in three or more variables.

 

Computing the dimension of spline spaces using homological techniques

Andrea Bressan
University of Oslo

Homological techniques have been successfully employed for computing the dimensions of piecewise polynomial spaces on triangulations and quad meshes in the plane. Examples of applications to other spline spaces will be presented.