10:00am - 12:00pmMultivariate 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)
Algebraic Approaches to Spline Theory
Michael DiPasquale
Colorado State University
In this talk we will give a brief overview of some of the algebraic methods which are used in spline theory. We will give particular attention to the pioneering work of Billera, in which homological methods were introduced for the calculation of dimension formulas. These methods have proved very fruitful for splines on all types of subdivisions; we will attempt to give a flavor for the various results that have been obtained this way, the questions that remain open, and the connections to algebraic geometry that result from these methods. Computing dimension formulas is only a beginning in spline theory - time permitting we will address how the gluing data for splines may be solved in some cases to give basis functions.
Polynomial splines of non-uniform degree: Combinatorial bounds on the dimension
Deepesh Toshniwal1, Bernard Mourrain2, Thomas Hughes1
1The University of Texas at Austin, 2Inria
Polynomial splines on triangulations and quadrangulations have myriad applications and are ubiquitous, especially in the fields of computer aided design, computer graphics and computational analysis. Meaningful use of splines for these purposes requires the construction and analysis of a suitable set of basis functions for the spline spaces. In turn, the computation or estimation of their dimensions is useful which, following the definition of smooth splines, depends on an interplay between algebra and geometry.
We consider the general case of splines with polynomial pieces of differing degrees. The flexibility of such splines would allow design of complex shapes with fewer control points, i.e., cleaner and simpler designs; while the same would also lead to more efficient engineering analysis. Using homological techniques, introduced by Billera (1988), we analyze the dimension of splines on triangulations and T-meshes. Specifically, we generalize the frameworks presented in Mourrain and Villamizar (2013) and Mourrain (2014) to the setting of both mixed polynomial degrees and mixed smoothness. Combinatorial bounds on the dimension are presented. Several examples are provided to illustrate application of the theory developed.
Approximation power of C1-smooth isogeometric functions on trivariate two-patch domains
Katharina Birner1, Bert Jüttler1, Angelos Mantzaflaris2
1Johannes Kepler University Linz, 2Inria
Bases and dimensions of trivariate spline functions possessing first order geometric continuity on two-patch domains were studied in (Birner, Jüttler, Mantzaflaris, Graph. Mod. 2018). It was shown that the properties of the spline space depend strongly on the type of the gluing data that is used to specify the relation between the partial derivatives along the interface between the patches. Locally supported bases were shown to exist for trilinear geometric gluing data (that corresponds to piecewise trilinear domain parameterizations) and sufficiently high degree.
In this talk we discuss the approximation properties of these spline functions. In particular, we perform numerical experiments with L2 projection in order to explore the approximation power. Despite the existence of locally supported bases, we observe a reduction of the approximation order for low degrees, and we provide a theoretical explanation for this phenomenon. This is joint work with Bert Jüttler and Angelos Mantzaflaris.
Splines, representations, and the Stanley-Stembridge conjecture
Julianna Tymoczko
Smith College
Splines can be used to construct the (equivariant) cohomology of certain algebraic varieties. We describe one such construction, and how the action of certain permutations on the splines relates to a longstanding open question in combinatorics called the Stanley-Stembridge conjecture. We also discuss certain steps towards resolving the conjecture in special cases.