Rational bases for convex polyhedra

In Wachspress (1975) [2] rational bases were constructed for convex polyhedra whose vertices were all of order three. The restriction to order three was first removed by Warren (1996) [3] and his analysis was refined subsequently by Warren and Schaefer (2004) [4]. A new algorithm (GADJ) for finding the denominator polynomial common to all the basis functions was exposed in Dasgupta and Wachspress (2007) [1] for convex polyhedra with all vertices of order three. This algorithm is applied here for generating bases for general convex polyhedra.     

Construction of a hermite rational “Wachspress type” finite element

This paper concerns the construction of a quadrilateral finite element whose interpolation space admits of rational fractions for basis functions of “Wachspress type” [1, 2]. The construction of this finite element, which is in a way the “rational” equivalent of the ADINI finite element[3, 4], is founded on a method analogous to the one used for Serendip degree-two finite element construction in[2]. The study of interpolation error is dealt with in a paper by Apprato, Arcangeli and Gout in this journal “Rational interpolation of Wachspress error estimates”.     

Recent advances in the construction of polygonal finite element interpolants

Summary This paper is an overview of recent developments in the construction of finite element interpolants, which areC ⁰-conforming on polygonal domains. In 1975, Wachspress proposed a general method for constructing finite element shape functions on convex polygons. Only recently has renewed interest in such interpolants surfaced in various disciplines including: geometric modeling, computer graphics, and finite element computations. This survey focuses specifically on polygonal shape functions that satisfy the properties of barycentric coordinates: (a) form a partition of unity, and are non-negative; (b) interpolate nodal data (Kronecker-delta property), (c) are linearly complete or satisfy linear precision, and (d) are smooth within the domain. We compare and contrast the construction and properties of various polygonal interpolants—Wachspress basis functions, mean value coordinates, metric coordinate method, natural neighbor-based coordinates, and maximum entropy shape functions. Numerical integration of the Galerkin weak form on polygonal domains is discussed, and the performance of these polygonal interpolants on the patch test is studied.     

On Wachspress quadrilateral patches

Wachspress initiated the study of rational basis functions for finite element construction over quadrilaterals and more general polygonal and curved elements. Later Apprato et al. (1979) and Gout (1979, 1985) studied the interpolatory and convergence properties of lower degree rational finite elements and their applications in solving second order boundary value problems. In the present paper we introduce higher degree Wachspress functions by an iterative technique and study their properties from the point of view of applications to surface fitting problems. It is indeed remarkable to note that these functions possess properties similar to tensor product Bernstein polynomials and hence could be effectively used to generate quadrilateral patches.     

A Complex View of Barycentric Mappings

Barycentric coordinates are very popular for interpolating data values on polyhedral domains. It has been recently shown that expressing them as complex functions has various advantages when interpolating two‐dimensional data in the plane, and in particular for holomorphic maps. We extend and generalize these results by investigating the complex representation of real‐valued barycentric coordinates, when applied to planar domains. We show how the construction for generating real‐valued barycentric coordinates from a given weight function can be applied to generating complex‐valued coordinates, thus deriving complex expressions for the classical barycentric coordinates: Wachspress, mean value, and discrete harmonic. Furthermore, we show that a complex barycentric map admits the intuitive interpretation as a complex‐weighted combination of edge‐to‐edge similarity transformations, allowing the design of “home‐made” barycentric maps with desirable properties. Thus, using the tools of complex analysis, we provide a methodology for analyzing existing barycentric mappings, as well as designing new ones.     

Complex Transfinite Barycentric Mappings with Similarity Kernels

Transfinite barycentric kernels are the continuous version of traditional barycentric coordinates and are used to define interpolants of values given on a smooth planar contour. When the data is two‐dimensional, i.e. the boundary of a planar map, these kernels may be conveniently expressed using complex number algebra, simplifying much of the notation and results. In this paper we develop some of the basic complex‐valued algebra needed to describe these planar maps, and use it to define similarity kernels, a natural alternative to the usual barycentric kernels. We develop the theory behind similarity kernels, explore their properties, and show that the transfinite versions of the popular three‐point barycentric coordinates (Laplace, mean value and Wachspress) have surprisingly simple similarity kernels. We furthermore show how similarity kernels may be used to invert injective transfinite barycentric mappings using an iterative algorithm which converges quite rapidly. This is useful for rendering images deformed by planar barycentric mappings.     

A Maple package to find first order differential invariants of 2ODEs via a Darboux approach

Here we present an implementation of a semi-algorithm to find elementary first order differential invariants (elementary first integrals) of a class of rational second order ordinary differential equations (rational 2ODEs). The algorithm was developed in Duarte and da Mota (2009)  [18]; it is based on a Darboux-type procedure, and it is an attempt to construct an analog (generalization) of the method built by Prelle and Singer (1983)  [6] for rational first order ordinary differential equations (rational 1ODEs). to deal, this time, with 2ODEs. The FiOrDi package presents a set of software routines in Maple for dealing with rational 2ODEs. The package presents commands permitting research investigations of some algebraic properties of the ODE that is being studied.     

Interpolation error estimates on hermite rational “Wachspress type” third degree finite element

This paper is about the study of interpolation error for the Hermite rational “Wachspress type” third degree finite element that is constructed in[1]. We obtain results analogous with those of the “corresponding” ADINI (polynomial) finite element.     

A unified, integral construction for coordinates over closed curves

We propose a simple generalization of Shephard's interpolation to piecewise smooth, convex closed curves that yields a family of boundary interpolants with linear precision. Two instances of this family reduce to previously known interpolants: one based on a generalization of Wachspress coordinates to smooth curves and the other an integral version of mean value coordinates for smooth curves. A third instance of this family yields a previously unknown generalization of discrete harmonic coordinates to smooth curves. For closed, piecewise linear curves, we prove that our interpolant reproduces a general family of barycentric coordinates considered by Floater, Hormann and Kós that includes Wachspress coordinates, mean value coordinates and discrete harmonic coordinates.     

Hermite interpolation: The barycentric approach

The barycentric formulas for polynomial and rational Hermite interpolation are derived; an efficient algorithm for the computation of these interpolants is developed. Some new interpolation principles based on rational interpolation are discussed.     

On the stable exact model matching problem

The stable exact model matching problem (SEMMP) is investigated. We state and prove a number of equivalent necessary and sufficient conditions for the existence of proper solutions to the exact model matching problem that are also Ω-stable, i.e. have no poles inside a symmetric ‘forbidden’ subset Ω of the finite complex plane . These results can be viewed as the counterpart of the results in [3] and [9] for the case of the ring of proper and Ω-stable rational functions.     

Towards a geometry of recursion

Any mathematical theory of algorithms striving to offer a foundation for programming needs to provide a rigorous definition for an abstract algorithm. The works reported by Girard (1988) in [10] and by Moschovakis (1989, 1995) in [29], [30] and [31] are among the best examples of such attempts. They both try to offer a mathematically precise and rigorous formulation of an abstract algorithm, intend to keep the algorithmic flavour present and take the notion of recursion as primary and central. The present work is motivated by Girard’s GoI 2 paper (Girard (1988) [10], which offers a treatment of recursion in terms of fixed points of linear functions. It is situated in the context of the geometry of interaction (GoI) program and is carried out in the concrete setting of the space of bounded linear maps on a Hilbert space. In this paper, we extend the work in Girard (1988) [10] to the context of traced unique decomposition categories, once again emphasizing the role of abstract trace in the theory of computing. We show that traced unique decomposition categories enriched over partially additive monoids or their variations suffice to axiomatize and hence extend the work in Girard’s GoI 2 paper. The theory developed here allows us to formulate an abstract notion of an algorithm as a pair of morphisms in a traced unique decomposition category, an abstract notion of computation as the execution formula (defined using the trace operator) applied to an algorithm, and finally a notion of deadlock-freeness for algorithms. In addition, we can treat recursive definitions, fixed points and fixed point operators in a uniform way in terms of traced unique decomposition categories.     

Honeycomb Wachspress finite elements for structural topology optimization

Traditionally, standard Lagrangian-type finite elements, such as linear quads and triangles, have been the elements of choice in the field of topology optimization. However, finite element meshes with these conventional elements exhibit the well-known “checkerboard” pathology in the iterative solution of topology optimization problems. A feasible alternative to eliminate such long-standing problem consists of using hexagonal (honeycomb) elements with Wachspress-type shape functions. The features of the hexagonal mesh include two-node connections (i.e. two elements are either not connected or connected by two nodes), and three edge-based symmetry lines per element. In contrast, quads can display one-node connections, which can lead to checkerboard; and only have two edge-based symmetry lines. In addition, Wachspress rational shape functions satisfy the partition of unity condition and lead to conforming finite element approximations. We explore the Wachspress-type hexagonal elements and present their implementation using three approaches for topology optimization: element-based, continuous approximation of material distribution, and minimum length-scale through projection functions. Examples are presented that demonstrate the advantages of the proposed element in achieving checkerboard-free solutions and avoiding spurious fine-scale patterns from the design optimization process.     

Positive Gordon–Wixom coordinates

We introduce a new construction of transfinite barycentric coordinates for arbitrary closed sets in two dimensions. Our method extends weighted Gordon–Wixom interpolation to non-convex shapes and produces coordinates that are positive everywhere in the interior of the domain and that are smooth for shapes with smooth boundaries. We achieve these properties by using the distance to lines tangent to the boundary curve to define a weight function that is positive and smooth. We derive closed-form expressions for arbitrary polygons in two dimensions and compare the basis functions of our coordinates with several other types of barycentric coordinates.     

Near-optimal parameterization of the intersection of quadrics: I. The generic algorithm

We present an exact and efficient algorithm for computing a proper parametric representation of the intersection of two quadrics in three-dimensional real space given by implicit equations with rational coefficients. The output functions parameterizing the intersection in projective space are polynomial, whenever it is possible, which is the case when the intersection is not a smooth quartic (for example, a singular quartic, a cubic and a line, and two conics). Furthermore, the parameterization is near-optimal in the sense that the number of distinct square roots appearing in the coefficients of these functions is minimal, except in a small number of well-identified cases where there may be an extra square root. In addition, the algorithm is practical: a complete and efficient C++ implementation is described in Lazard et al. [Lazard, S., Peñaranda, L.M., Petitjean, S., 2006. Intersecting quadrics: An efficient and exact implementation. In: 20th ACM Symposium on Computational Geometry, 2004. Computational Geometry: Theory and Applications 35 (1–2), 74–99 (special issue)].     

A Prolate-Element Method for Nonlinear PDEs on the Sphere

A p-type spectral-element method using prolate spheroidal wave functions (PSWFs) as basis functions, termed as the prolate-element method, is developed for solving partial differential equations (PDEs) on the sphere. The gridding on the sphere is based on a projection of the prolate-Gauss-Lobatto points by using the cube-sphere transform, which is free of singularity and leads to quasi-uniform grids. Various numerical results demonstrate that the proposed prolate-element method enjoys some remarkable advantages over the polynomial-based element method: (i) it can significantly relax the time step size constraint of an explicit time-marching scheme, and (ii) it can increase the accuracy and enhance the resolution.     

Riskformer: Survival prediction from MR imaging in patients with IDH-wildtype glioblastoma

BackgroundGlioblastoma is the most common malignant brain tumor in adults. Despite aggressive treatment, its prognosis is still poor, with a median overall survival of less than 15 months.MethodsIn this study, we developed and validated a vision transformer-based model, named Riskformer, built from preoperative MRI for predicting OS in patients with newly diagnosed IDH-wildtype glioblastoma patients.ResultsRiskformer score was associated with OS as an independent prognostic factor in patients with IDH-wildtype glioblastoma. In the training dataset (hazard ratio [HR]: 5.930, 95% confidence interval [CI]: 3.933–8.942, P < 0.001), validation dataset (HR: 2.436, 95% CI: 1.334–4.447, P < 0.001) and testing dataset (HR: 4.651, 95% CI: 2.233–9.688, P < 0.001).ConclusionThe Riskformer score could provide independent and incremental prognostic value over existing clinical factors in OS prediction in patients with IDH-wildtype glioblastoma.     

构造有理插值函数的一种参数法

对设定有理分式函数次数类型的有理插值问题研究,已有许多很多的结论。有理插值问题是否有解,取决于被插函数一些给定的函数值[f(xi),i=0,1,?,m+n]。指出分子和分母多项式次数之和为[N]的有理插值问题总有解,然后从设定的有理插值函数次数类型出发,引入正整参数[d],给出一种构造有理插值函数的方法。用该方法总可以构造出满足插值条件的有理分式函数,且有较大灵活性,计算量也不大。     

Optimized implementations of rational approximations—a case study on the Voigt and complex error function

Rational functions are frequently used as efficient yet accurate numerical approximations for real and complex valued special functions. For the complex error function w(x+iy), whose real part is the Voigt function K(x,y), the rational approximation developed by Hui, Armstrong, and Wray [Rapid computation of the Voigt and complex error functions, J. Quant. Spectrosc. Radiat. Transfer 19 (1978) 509-516] is investigated. Various optimizations for the algorithm are discussed. In many applications, where these functions have to be calculated for a large x grid with constant y, an implementation using real arithmetic and factorization of invariant terms is especially efficient.