Convergence Rates for Logspline Tomography

We consider bivariate logspline density estimation for tomography data. In the usual logspline density estimation for bivariate data, the logarithm of the unknown density function is estimated by tensor product splines, the unknown parameters of which are given by maximum likelihood. In this paper we use tensor product B-splines and the projection-slice theorem to construct the logspline density estimators for tomography data. Rates of convergence are established for log-density functions assumed to belong to a Besov space.     

Introduction to the Web-method and its applications

The Web-method is a meshless finite element technique which uses weighted extended B-splines (Web-splines) on a tensor product grid as basis functions. It combines the computational advantages of B-splines and standard mesh-based elements. In particular, degree and smoothness can be chosen arbitrarily without substantially increasing the dimension. Hence, accurate approximations are obtained with relatively few parameters. Moreover, the regular grid is well suited for hierarchical refinement and multigrid techniques. This article should serve as an introduction to finite element approximation with B-splines. We first review the construction of Web-bases and discuss their basic properties. Then we illustrate the performance of Ritz–Galerkin schemes for a model problem and applications in linear elasticity. Finally, we discuss several implementation aspects. AMS subject classification 65N30, 41A15, 74S05Anja Streit: Present address: Fraunhofer ITWM, 67663 Kaiserslautern, Germany.     

An orthogonal basis for non-uniform algebraic-trigonometric spline space

Non-uniform algebraic-trigonometric B-splines shares most of the properties as those of the usual polynomial B-splines. But they are not orthogonal. We construct an orthogonal basis for the n-order(n ≥ 3) algebraic-trigonometric spline space in order to resolve the theoretical problem that there is not an explicit orthogonal basis in the space by now. Motivated by the Legendre polynomials, we present a novel approach to define a set of auxiliary functions,which have simple and explicit expressions. Then the proposed orthogonal splines are given as the derivatives of these auxiliary functions.     

Schoenberg’s polynomial B-splines of odd degrees: A brief review of application

A brief overview of applications of Schoenberg’s polynomial B-splines of odd degrees in mathematical statistics, computational mathematics, and statistical radio engineering is provided. Exact formulas for the found Schoenberg B-spline of 15th degree are presented. High-quality approximations of smooth functions with an infinite Fourier transform by functions with a finite Fourier transform are found.     

A planar cubic Bézier spiral

A planar cubic Bézier curve segment that is a spiral, i.e., its curvature varies monotonically with arc-length, is discussed. Since this curve segment does not have cusps, loops, and inflection points (except for a single inflection point at its beginning), it is suitable for applications such as highway design, in which the clothoid has been traditionally used. Since it is polynomial, it can be conveniently incorporated in CAD systems that are based on B-splines, Bézier curves, or NURBS (nonuniform rational B-splines) and is thus suitable for general curve design applications in which fair curves are important.     

Recurrence relations for Tchebycheffian B-splines

Four-term recurrence relations with constant coefficients are derived for a wide class of T chebycheffian B-splines, LB-splines and complex B-splines. Such a relation exists whenever the differential operator defining the underlying “polynomial” space can be factored in two essentially different ways. The four lower order B-splines in the recurrence relation appear in two pairs, each pair corresponding to one of these factorization. It is shown that the two-term recurrence relations for polynomial, trigonometric and hyperbolic B-splines as well as other known two-term recurrence relations are obtained directly from the four-term recurrence relations in a unified and systematic way. The above derivation also yields two different two-term recurrence relations for Green’s functions of these “polynomial” spaces In this context the special examples of exponential functions and rational functions are analyzed in detail.     

Algorithms for surface fitting using Powell-Sabin splines

Algorithms are presented for fitting a Powell-Sabin spline toa set of scattered data. Both the detemination of least-squaresand smoothing splines are considered. For the latter we adoptthe philosophy of an existing tensor product spline algorithm.The triangulation is determined in an automatic and adaptiveway. The algorithm employs a single parameter to control thetradeoff between closeness of fit and smoothness of fit. The Powell-Sabin splines are represented in terms of locallysupported basis functions. The use of the Bernstein-Bzier ordinatesof these B-splines results in efficient calculations. Numericalexamples illustrate the usefulness of the given algorithms.     

Average case tractability of non-homogeneous tensor product problems with the absolute error criterion

We study average case tractability of non-homogeneous tensor product problems with the absolute error criterion. We consider algorithms that use finitely many evaluations of arbitrary linear functionals. For general non-homogeneous tensor product problems, we obtain the matching necessary and sufficient conditions for strong polynomial tractability in terms of the one-dimensional eigenvalues. We give some examples to show that strong polynomial tractability is not equivalent to polynomial tractability, and polynomial tractability is not equivalent to quasi-polynomial tractability. But for non-homogeneous tensor product problems with decreasing eigenvalues, we prove that strong polynomial tractability is always equivalent to polynomial tractability, and strong polynomial tractability is even equivalent to quasi-polynomial tractability when the one-dimensional largest eigenvalues are less than one. In particular, we find an example that quasi-polynomial tractability with the absolute error criterion is not equivalent to that with the normalized error criterion even if all the one-dimensional largest eigenvalues are one. Finally we consider a special class of non-homogeneous tensor product problems with improved monotonicity condition of the eigenvalues.     

A General Multiresolution Method for Fitting Functions on the Sphere

In [7], Lyche and Schumaker have described a method for fitting functions of class C ¹ on the sphere which is based on tensor products of quadratic polynomial splines and trigonometric splines of order three associated with uniform knots. In this paper, we present a multiresolution method leading to C ²-functions on the sphere, using tensor products of polynomial and trigonometric splines of odd order with arbitrary simple knot sequences. We determine the decomposition and reconstruction matrices corresponding to the polynomial and trigonometric spline spaces. We describe the general tensor product decomposition and reconstruction algorithms in matrix form which are convenient for the compression of surfaces. We give the different steps of the computer implementation of these algorithms and, finally, we present a test example.     

Rational B-splines with prescribed poles

A recursion formula for rational B-splines with prescribed poles is given that reduces to DeBoor's recursion when all poles are at infinity. Some properties of polynomial B-splines generalize to these rational B-splines: partition of unity, a knot inserting algorithm, numerical stability. It can be proved that the rational B-splines are identical with the Chebyshevian B-splines constructed by T. Lyche. The recursions are not identical and the one for the rational B-splines is more convenient. Furthermore, the rational B-splines are identified as special NURBS. The weights can be chosen depending on the poles.     

(3,3,1,1)层次B样条与分级T网格上多项式样条的相互转换

1引言 B样条在计算机图形学和几何建模等领域有着广泛的应用[3,8].在应用过程中,通常都需要对得到的模型进行修改以到达更好的效果.对于B样条曲线,利用节点插入算法可以有效地进行局部修改.     

The construction of plane elastomechanics and Mindlin plate elements of B-spline wavelet on the interval

Based on two-dimensional tensor product B-spline wavelet on the interval (BSWI), a class of C0 type plate elements is constructed to solve plane elastomechanics and moderately thick plate problems. Instead of traditional polynomial interpolation, the scaling functions of two-dimensional tensor product BSWI are employed to form the shape functions and construct BSWI elements. Unlike the process of direct wavelets adding in the previous work, the elemental displacement field represented by the coefficients of wavelets expansions is transformed into edges and internal modes via the constructed transformation matrix in this paper. The method combines the versatility of the conventional finite element method (FEM) with the accuracy of B-spline functions approximation and various basis functions for structural analysis. Some numerical examples are studied to demonstrate the proposed method and the numerical results presented are in good agreement with the closed-form or traditional FEM solutions.     

Delaunay configurations and multivariate splines: A generalization of a result of B. N. Delaunay

In the 1920s, B. N. Delaunay proved that the dual graph of the Voronoi diagram of a discrete set of points in a Euclidean space gives rise to a collection of simplices, whose circumspheres contain no points from this set in their interior. Such Delaunay simplices tessellate the convex hull of these points. An equivalent formulation of this property is that the characteristic functions of the Delaunay simplices form a partition of unity. In the paper this result is generalized to the so-called Delaunay configurations. These are defined by considering all simplices for which the interiors of their circumspheres contain a fixed number of points from the given set, in contrast to the Delaunay simplices, whose circumspheres are empty. It is proved that every family of Delaunay configurations generates a partition of unity, formed by the so-called simplex splines. These are compactly supported piecewise polynomial functions which are multivariate analogs of the well-known univariate B-splines. It is also shown that the linear span of the simplex splines contains all algebraic polynomials of degree not exceeding the degree of the splines.

     

Tractability through increasing smoothness

We prove that some multivariate linear tensor product problems are tractable in the worst case setting if they are defined as tensor products of univariate problems with logarithmically increasing smoothness. This is demonstrated for the approximation problem defined over Korobov spaces and for the approximation problem of certain diagonal operators. For these two problems we show necessary and sufficient conditions on the smoothness parameters of the univariate problems to obtain strong polynomial tractability. We prove that polynomial tractability is equivalent to strong polynomial tractability, and that weak tractability always holds for these problems. Under a mild assumption, the Korobov space consists of periodic functions. Periodicity is crucial since the approximation problem defined over Sobolev spaces of non-periodic functions with a special choice of the norm is not polynomially tractable for all smoothness parameters no matter how fast they go to infinity. Furthermore, depending on the choice of the norm we can even lose weak tractability.     

Multivariate spline functions. I. Construction, properties and computation

A construction is given which allows the Hilbert space treatment of spline functions to be applied to the case of more than one variable, when the basic operator is a linear partial differential one. The particular case of the tensor product polynomial spline in two variables is then studied using a reproducing kernel, and its main properties, including the minimization ones, are deduced. A stable computational method is then given for this spline function, with certain point evaluation functionals. Finally, extensions are discussed, for more general linear functionals, for more general differential operators, and for more than two variables.     

Invariants of Composite Networks Arising as a Tensor Product

We generalize Brylawski’s formula of the Tutte polynomial of a tensor product of matroids to colored connected graphs, matroids, and disconnected graphs. Unlike the non-colored tensor product where all edges have to be replaced by the same graph, our colored generalization of the tensor product operation allows individual edge replacement. The colored Tutte polynomials we compute exists by the results of Bollobás and Riordan. The proof depends on finding the correct generalization of the two components of the pointed Tutte polynomial, first studied by Brylawski and Oxley, and on careful enumeration of the connected components in a tensor product. Our results make the calculation of certain invariants of many composite networks easier, provided that the invariants are obtained from the colored Tutte polynomials via substitution and the composite networks are represented as tensor products of colored graphs. In particular, our method can be used to calculate (with relative ease) the expected number of connected components after an accident hits a composite network in which some major links are identical subnetworks in themselves.      

A sinusoidal polynomial spline and its Bezier blended interpolant

Functional polynomials composed of sinusoidal functions are introduced as basis functions to construct an interpolatory spline. An interpolant constructed in this way does not require solving a system of linear equations as many approaches do. However there are vanishing tangent vectors at the interpolating points. By blending with a Bezier curve using the data points as the control points, the blended curve is a proper smooth interpolant. The blending factor has the effect similar to the “tension” control of tension splines. Piecewise interpolants can be constructed in an analogous way as a connection of Bezier curve segments to achieve C¹ continuity at the connecting points. Smooth interpolating surface patches can also be defined by blending sinusoidal polynomial tensor surfaces and Bezier tensor surfaces. The interpolant can very efficiently be evaluated by tabulating the sinusoidal function.     

Orthonormality of Cardinal Chebyshev B-Spline Bases in Weighted Sobolev Spaces

Extending a recent result of Ulrich Reif on cardinal polynomial B-splines, we show that the cardinal Chebyshev B-spline basis associated with a linear differential operator with constant real coefficients is orthonormal with respect to a unique weighted Sobolev-type inner product.     

Ritz-Galerkin approximations in blending function spaces

Summary This paper considers the theoretical development of finite dimensional bivariate blending function spaces and the problem of implementing the Ritz-Galerkin method in these approximation spaces. More specifically, the approximation theoretic methods of polynomial blending function interpolation and approximation developed in [2, 11–13] are extended to the general setting of L-splines, and these methods are then contrasted with familiar tensor product techniques in application of the Ritz-Galerkin method for approximately solving elliptic boundary value problems. The key to the application of blending function spaces in the Ritz-Galerkin method is the development of criteria which enable one to judiciously select from a nondenumerably infinite dimensional linear space of functions, certain finite dimensional subspaces which do not degrade the asymptotically high order approximation precision of the entire space. With these criteria for the selection of subspaces, we are able to derive a virtually unlimited number of new Ritz spaces which offer viable alternatives to the conventional tensor product piecewise polynomial spaces often employed. In fact, we shall see that tensor product spaces themselves are subspaces of blending function spaces; but these subspaces do not preserve the high order precision of the infinite dimensional parent space.Considerable attention is devoted to the analysis of several specific finite dimensional blending function spaces, solution of the discretized problems, choice of bases, ordering of unknowns, and concrete numerical examples. In addition, we extend these notations to boundary value problems defined on planar regions with curved boundaries.