Schedule for: 24w5313 - Homological Perspective on Splines and Finite Elements

Nechako residence

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Equivariant cohomology is an algebraic invariant associated to a group action on a space. GKM theory describes this graded ring in a very special set of commonly arising circumstances. In this talk, we will explain defining properties of equivariant cohomology for these spaces (with their group actions) and how to use GKM theory to calculate it. Along the way, we will explain how to associate labeled graphs, recently interpreted as splines, to these spaces.

Differential complexes are sequences of vector spaces and graded linear differential operators such that their composition vanishes. Examples of differential complexes include the de Rham complex and the Bernstein-Gelfand-Gelfand (BGG) sequences. These complexes play an important role in the analysis and numerical computation of electromagnetism and continuum mechanics. In the framework of Finite Element Exterior Calculus (FEEC), the key to obtaining stable and structure-preserving numerical methods is to use discrete (finite element, spline etc.) versions of differential complexes. The construction of such complexes has drawn increased attention and is challenging due to the tensor symmetries and continuity requirements. Finite element/spline differential complexes also provide a new perspective for some theoretical problems in spline theory, such as the dimension of spline spaces. Via exact sequences, dimension problems of a smooth scalar space can be equivalently formulated as problems of vector- or tensor-valued problems with less regularity. In this talk, we present some existing results and methods for constructing finite element differential complexes. This is part of an effort to vectorize or tensorize the results of scalar splines by taking differentials and completing them in a sequence.

Splines are piecewise functions consisting of polynomial pieces glued together in a certain smooth way. They find application in a wide range of contexts such as computer aided geometric design, data fitting, and finite element analysis, just to mention a few. The success of univariate splines is built on two main pillars. $\textit{Representation}$: spline spaces possess a special basis, called the B-spline basis, which enjoys several nice properties relevant for both theoretical and practical purposes. $\textit{Approximation}$: spline spaces have an optimal approximation order for smooth functions and their derivatives. Both pillars may benefit from the local representation of the polynomial pieces in terms of Bernstein polynomials, called the Bernstein-Bézier form, which allows for a stable and geometrically meaningful description of the single pieces and their smooth connections. When moving to the multivariate setting, splines on triangulations emerge as a natural and powerful extension of univariate splines. Dealing with highly smooth splines on triangulations is very appealing but requires additional efforts to obtain stable dimensions and stable local bases, to achieve local constructions, and to get full approximation power. Bernstein polynomials naturally extend to triangles and the Bernstein-Bézier form generalizes in an elegant way to the triangular setting. Here the simple and clear geometric interpretation of the inter-element smoothness conditions play an even more important role in the analysis of spline spaces. This talk is divided into two parts. In the first part, we review the properties of univariate Bernstein polynomials and the Bernstein-Bézier form, with a focus on their geometric meaning. We also discuss their role for univariate splines, both representation and approximation. In the second part, we extend this form to splines on triangulations. Again we describe the main issues concerning representation and approximation, emphasizing the analogies with the univariate case and highlighting the difficulties arising from the multivariate setting.

The first part of the talk presents a brief summary of the framework of Isogeometric Analysis (IGA), which was introduced by T.J.R. Hughes et al. in 2005 as a concept to bridge the gap between design (in particular Computer Aided Geometric Design) and analysis (i.e., numerical simulation via discretization methods for PDEs such as the Finite Element Method). IGA relies on spline-based discretizations of physical fields, in particular tensor-product constructions, and methods for adaptive refinement of the resulting spline spaces are therefore of vital interest. Consequently, the second part of the talk focuses on adaptive generalizations of tensor-product splines and their mathematical properties, in particular algebraic completeness, as this is one of the main topics of this BIRS workshop. More specifically, we discuss hierarchical splines, whose origins can be traced back to the seminal work of D.R. Forsey and R.H. Bartels in 1988, splines defined by control meshes with T-joints (T-splines) as introduced by T.W. Sederberg et al. (2003), the polynomial splines over hierarchical T-meshes of J. Deng et al. (2008), and the polynomial splines over locally refined box partitions (LR-splines) of Dokken et al. (2013). Finally, the talk concludes with some suggestions for further research.

In this talk, I would like to draw people’s attention to the dimension counting problem of spline spaces. This problem has an algebraic nature: it can be translated into the computation of Hilbert functions. Hilbert functions encode many important algebraic invariants. They are additive on short exact sequences, and that is why we can use homological methods, i.e. chain complexes, to obtain Hilbert functions of complicated objects from those of simple ones. A classical example is Billera’s homological approach to prove Strang’s conjecture. Later, his method was modified by Schenck and Stillman. This set of tools has been implemented in the computer algebra system Macaulay 2.

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Abstract: Starting with C^1 spaces on macro triangulations (Alfeld, Worsey-Farin splits), we show that they are part of a finite element complex. We then discuss how some of the spaces can be used for problems in fluid flow, electro-magnetics and solid mechanics.

Spline functions are a well-known and widely employed tool, with applications to CAD, CAE, CAGD, numerical solution of PDEs, etc.. The theory about spline spaces has been and still is continuously growing, and led to countless types of of spline spaces [1, 2]: univariate and multivariate, polynomial and generalized, defined on triangulations and on (locally) tensor-product meshes, etc.. All these splines are traditionally functions s : Rm −→ Rn with varying regularities. In this presentation we explore the possibility to extend the concept of spline to operators (functions) s : X −→ R where X is an infinite-dimensional Hilbert space, motivated by the fact that such a tool could be applied both to approximation methods and to the solution of functional differential equations [3, 4]. We will present a couple of constructions for piecewise k-linear operators, as a proof of concept that splines can be defined and used in this more general setting.

Given a scalar-valued discontinuous piecewise polynomial, a “potential reconstruction” is a piecewise polynomial that is trace-continuous, i.e., H1-conforming. It is best obtained via a conforming finite element solution of local homogeneous Dirichlet problems on patches of elements sharing a vertex. Similarly, given a vector-valued discontinuous piecewise polynomial not satisfying the target divergence, a “flux reconstruction” is a piecewise polynomial that is normal-trace-continuous, i.e., H(div)-conforming, and has the target divergence. It is best obtained via local homogeneous Neumann problems on patches of elements, using the mixed finite element method. These concepts are known to lead to guaranteed, locally efficient, and polynomial-degree-robust a posteriori error estimates. We show that they also allow to devise stable local commuting projectors that lead to p-robust equivalence of global-best approximation over the whole computational domain using a conforming finite element space with local- (elementwise-)best approximations without any continuity requirement along the interfaces and without any constraint on the divergence. Therefrom, optimal hp approximation / a priori error estimates under minimal elementwise Sobolev regularity follow.

Recently, Eric Brechner and I have developed an open source Python package for building spline models of points, curves, surfaces, solids, and n-dimensional manifolds called BSpy. This package, based primarily on a single object class and a handful of methods, offers a powerful capability for building and manipulating geometric models in many dimensions. This talk will explain the history and philosophy that have gone into BSpy and will demonstrate some of the surprisingly complex operations that can be performed with coding idioms that are often only a few lines of code long.

Additional breakout rooms: Art 106, Art 108 and Art 114(big room).

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We describe in more detail a dual perspective on splines, developed by Billera and Rose in the context of splines and by Goresky-Kottwitz-MacPherson in the context of equivariant cohomology. We then use this perspective to give two tricks that help calculations.

This talk will explain the tool of folded alcove walks, which enjoy a wide range of applications throughout combinatorics, representation theory, number theory, and algebraic geometry. We will survey the construction of flag varieties through this lens, focusing on the problem of understanding intersections of different kinds of Schubert cells. We then highlight a key application in GKM theory.

The talk concerns the degrees of freedom that can be used to determine univocally the fields in high order Whitney finite element spaces. There are indeed two different families of such degrees of freedom, the weights and the moments. Weights and moments coincide in the lower order case, but are rather different as soon as we consider the high order case. I will mainly fo- cus on weights. Thank to their natural geometrical localization on the mesh, weights allow, for instance, to generalize, to the polynomial interpolation of differential k-forms, some fundamental concepts of the polynomial interpo- lation of regular scalar functions or to extend to the high order case graph techniques used in the low order case. I will also discuss about the relation- ship between weights and moments through a particular isomorphism that preserves the matrix of the gradient operator. This is a long term collaboration with Francesca Rapetti, from the University Coˆte d’Azur, France.

The Powell-Sabin 6-refinement has proven to be a convenient splitting technique for constructing smooth splines over a general triangulation. In this talk we use it to define a C^1 spline space of arbitrary degree with optimal polynomial precision and prescribed super-smoothness at split points inside triangles. Instead of traditional interpolation, we use blossoming to establish a set of functionals that characterize the spline space. The associated basis functions have some favorable properties, namely, they form a convex partition of unity and can be naturally represented in the Bernstein-Bezier form.

Additional breakout rooms: Art 106, Art 108 and Art 114(big room).

Please meet up at the parking lot.

Please use the meal card and get the breakfast items from Comma(or any food merchants)

The Wang-Shi split is a cross-cut partition of a triangle and consequently the dimension formula for the space of C^{p-1} splines of degree p does not have an homology term and it only involves the number of crossing lines containing each inner point. The talk will present a proof that the number of cut-lines containing any inner point is less than or equal to p+1 allowing to simplifying the formula to dim spline = dim polynomials + number of cut-lines.

This talk will introduce a novel construction of adaptive and yet structure preserving finite element discretizations with function spaces induced by subdivision. In many applications, for example in geo- physical fluid dynamics, adaptive and structure-preserving methods can be highly beneficial to simulate the long-term evolution of a multi-scale system with several invariants of motion like the total energy. If the discretizations of such systems do not preserve these invariants, the simulation results can differ significantly from the true physical behaviour of the systems. Combining the benefits of structure preservation and adaptive finite elements is notoriously difficult. If no special care is taken, adaptive mesh refinement algorithms of standard finite element approaches usually lose the property of structure preservation. Alternatively, the refinement can be chosen to be conforming, which in turn leads to unnecessary propagation of the refinement because surrounding cells need to be refined as well. On the other hand, IGA tensor-product techniques suffer from mesh topology restrictions. For this reason, we chose to build our function spaces upon subdivision. We extend the work of [1], who introduced vector field subdivision schemes that commute with the standard vector calculus operators like the gradient or the curl. Translating their work to the finite ele- ments realm yields de-Rham-complex-preserving finite elements for scalar functions, vector fields, and density functions. We added adaptivity to their structure-preserving discretization by leveraging the hi- erarchy of the basis functions induced by the subdivision algorithm. By carefully keeping track of the introduced degrees of freedom across the refinement levels, we maintain a discrete de Rham complex and thus enable structure-preserving simulations. Our method was verified by simulating the Maxwell eigenvalue problem, a well-known test case that reproduces the analytical eigenvalues if the chosen finite element spaces constitute a discrete de Rham complex. We show that our discretization indeed yields the correct spectrum and investigate the compu-tational effort and accuracy gains of our method.

TBD

Through the seminal works of Buffa et al., the fruitful integration of the two discretization paradigms of Finite Element Exterior Calculus (FEEC) and Isogeometric Analysis (IGA) was demonstrated already in 2011. The latter evolved over the last nearly 20 years, stemming from the publications of Hughes et al., into a powerful concept for linking Finite Element Methods with Computer-aided design. In fact, the introduction of isogeometric discrete differential forms by Buffa et al. laid the foundation for discretizing de Rham complexes in a structure-preserving manner using B-splines. However, although the FEEC theory was derived in an abstract setting, and while Hilbert sequences play a role in various physical applications, connecting IGA and FEEC often proves challenging or is sometimes not directly clear. This is especially true for Hilbert complexes that also encompass differential operators of higher orders. We present two approaches to obtain well-posed discretizations of a whole class of Hodge–Laplace problems using IGA, while maintaining the inf-sup stability condition. We focus on mixed weak formulations of saddle-point structure and second-order Hilbert complexes. In particular, we go beyond the standard de Rham case and demonstrate that ideas from FEEC and IGA are useful for non-de Rham chains as well. A central tool for describing the underlying settings and for choosing the Finite Element spaces is the Bernstein–Gelfand–Gelfand (BGG) construction discussed by Arnold and Hu in 2021. Our approach allows us to incorporate geometries with curved boundaries, which is not directly possible with classical FEEC approaches, and also provides suitable discretizations in arbitrary dimensions. We show error estimates for both approximation methods and explain their applicability in the field of linear elasticity theory. The theoretical discussions and estimates are further illustrated with various numerical examples perfomed utilizing the GeoPDEs software package.

Kelowna https://www.tourismkelowna.com/blog/post/how-to-use-transit-to-explore-kelownas-hiking-trails/

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The construction of spline spaces for the isogeometric analysis of higher order PDEs is partially understood for bivariate problems, but offers great challenges for trivariate problems. In this talk, we discuss the current situation and identify tasks to be addressed in the future. In particular, we consider volumetric subdivision as a possible candidate for the construction of function spaces with sufficient Sobolev regularity. While many analytic questions still remain unsolved, we can report on progress concerning the construction of algorithms with favorable properties.