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Approximation Theory XV: San Antonio 2016

herausgegeben von: Gregory E. Fasshauer, Larry L. Schumaker

Verlag: Springer International Publishing

Buchreihe : Springer Proceedings in Mathematics & Statistics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

These proceedings are based on papers presented at the international conference Approximation Theory XV, which was held May 22–25, 2016 in San Antonio, Texas. The conference was the fifteenth in a series of meetings in Approximation Theory held at various locations in the United States, and was attended by 146 participants. The book contains longer survey papers by some of the invited speakers covering topics such as compressive sensing, isogeometric analysis, and scaling limits of polynomials and entire functions of exponential type.

The book also includes papers on a variety of current topics in Approximation Theory drawn from areas such as advances in kernel approximation with applications, approximation theory and algebraic geometry, multivariate splines for applications, practical function approximation, approximation of PDEs, wavelets and framelets with applications, approximation theory in signal processing, compressive sensing, rational interpolation, spline approximation in isogeometric analysis, approximation of fractional differential equations, numerical integration formulas, and trigonometric polynomial approximation.

Inhaltsverzeichnis

Frontmatter

Linear Barycentric Rational Interpolation with Guaranteed Degree of Exactness

Abstract

In recent years, linear barycentric rational interpolants, introduced in 1988 and improved in 2007 by Floater and Hormann, have turned out to be among the most efficient infinitely smooth interpolants, in particular with equispaced points. In the present contribution, we introduce a new way of obtaining linear barycentric rational interpolants with relatively high orders of convergence. The basic idea is to modify the interpolant with equal weights of 1988 to force it to interpolate exactly the monomials up to a certain degree. This is obtained by modifying a few weights at each extremity of the interval of interpolation. Numerical experience demonstrates that the method is indeed able to interpolate with much higher orders than the original 1988 interpolant and in a very stable way.

Jean-Paul Berrut

Approximation by Splines on Piecewise Conic Domains

Abstract

We develop a Hermite interpolation scheme and prove error bounds for $C^1$ bivariate piecewise polynomial spaces of Argyris type vanishing on the boundary of curved domains enclosed by piecewise conics.

Oleg Davydov, Wee Ping Yeo

A Rescaled Method for RBF Approximation

Abstract

In the recent paper [1], a new method to compute stable kernel-based interpolants has been presented. This rescaled interpolation method combines the standard kernel interpolation with a properly defined rescaling operation, which smooths the oscillations of the interpolant. Although promising, this procedure lacks a systematic theoretical investigation. Through our analysis, this novel method can be understood as standard kernel interpolation by means of a properly rescaled kernel. This point of view allows us to consider its error and stability properties.

Stefano De Marchi, Andrea Idda, Gabriele Santin

Flavors of Compressive Sensing

Abstract

About a decade ago, a couple of groundbreaking articles revealed the possibility of faithfully recovering high-dimensional signals from some seemingly incomplete information about them. Perhaps more importantly, practical procedures to perform the recovery were also provided. These realizations had a tremendous impact in science and engineering. They gave rise to a field called ‘compressive sensing,’ which is now in a mature state and whose foundations rely on an elegant mathematical theory. This survey presents an overview of the field, accentuating elements from approximation theory, but also highlighting connections with other disciplines that have enriched the theory, e.g., statistics, sampling theory, probability, optimization, metagenomics, graph theory, frame theory, and Banach space geometry.

Simon Foucart

Computing with Functions on Domains with Arbitrary Shapes

Abstract

We describe an approximation scheme and an implementation technique that enables numerical computations with functions defined on domains with an arbitrary shape. The scheme is spectrally accurate for smooth functions. The main advantage of the technique is that, unlike most spectral approximation schemes in higher dimensions, it is not limited to domains with tensor-product structure. The scheme itself is a discrete least squares approximation in a redundant set (a frame) that originates from a basis on a bounding box. The implementation technique consists of representing a domain by its characteristic function, i.e., the function that indicates whether or not a point belongs to the set. We show in a separate paper that the least squares approximation with N degrees of freedom can be solved in ${\mathscr {O}}(N^2\log ^2 N)$ operations for any domain that has non-trivial volume. The computational cost improves to ${\mathscr {O}}(N \log ^2 N)$ operations for domains that do have tensor-product structure. The scheme applies to domains even with fractal shapes, such as the Mandelbrot set, since such domains are defined precisely by their characteristic function.

Daan Huybrechs, Roel Matthysen

A Polygonal Spline Method for General Second-Order Elliptic Equations and Its Applications

Abstract

We explain how to use polygonal splines to numerically solve second-order elliptic partial differential equations. The convergence of the polygonal spline method will be studied. Also, we will use this approach to numerically study the solution of some mixed parabolic and hyperbolic partial differential equations. Comparison with standard bivariate spline method will be given to demonstrate that our polygonal splines have some better numerical performance.

Ming-Jun Lai, James Lanterman

An Adaptive Triangulation Method for Bivariate Spline Solutions of PDEs

Abstract

We report numerical performance of our adaptive triangulation algorithms to improve numerical solutions of PDEs using bivariate spline functions. Our ultimate goal is to find a PDE-solution-dependent triangulation which improves both the accuracy and computational efficiency of the spline solution. We present little theory to guide our search for such a triangulation, but instead approach the problem numerically. Starting with some initial triangulation $\triangle $, we use the gradient values of the spline solution based on $\triangle $ to generate an updated triangulation $\triangle '$ and compute a new spline solution. We consider both refining and coarsening the initial triangulation $\triangle $ in order to make the spline solution more effective. As we add vertices to and remove vertices from the vertex set, we use a global retriangulation instead of local refinement techniques. We introduce a new concept of mesh efficiency to measure the effectiveness of a spline solution over a given triangulation. Extensive numerical experiments have been conducted and are summarized in this paper. In addition, we report a heuristic for generating an initial solution-dependent triangulation and show numerical evidence that this algorithm produces an initial triangulation which yields a better spline solution than one based on a uniform initial mesh.

Ming-Jun Lai, Clayton Mersmann

Refinable Functions with PV Dilations

Abstract

A PV number is an algebraic integer $\alpha $ of degree $d \ge 2$ all of whose Galois conjugates other than itself have modulus less than 1. Erdös [8] proved that the Fourier transform $\widehat{\varphi },$ of a nonzero compactly supported scalar-valued function satisfying the refinement equation $\varphi (x) = \frac{|\alpha |}{2}\varphi (\alpha x) + \frac{|\alpha |}{2}\varphi (\alpha x-1)$ with PV dilation $\alpha ,$ does not vanish at infinity so by the Riemann–Lebesgue lemma $\varphi $ is not integrable. Dai, Feng, and Wang [5] extended his result to scalar-valued solutions of $\varphi (x) = \sum _k a(k) \varphi (\alpha x - \tau (k))$ where $\tau (k)$ are integers and a has finite support and sums to $|\alpha |$. In ([22], Conjecture 4.2), we conjectured that their result holds under the weaker assumption that $\tau $ has values in the ring of polynomials in $\alpha $ with integer coefficients. This paper formulates a stronger conjecture and provides support for it based on a solenoidal representation of $\widehat{\varphi },$ and deep results of Erdös and Mahler [9]; Odoni [26] that gives lower bounds for the asymptotic density of integers represented by integral binary forms of degree $> 2;$ degree $=2$, respectively. We also construct an integrable vector-valued refinable function with PV dilation.

Wayne Lawton

Polyhyperbolic Cardinal Splines

Abstract

In this note, we discuss solutions of differential equation $(D^2-\alpha ^2)^{k}u=0$ on $\mathbb {R}\setminus \mathbb {Z}$, which we call polyhyperbolic splines. We develop the fundamental function of interpolation and prove various properties related to these splines.

Jeff Ledford

Adaptive Computation with Splines on Triangulations with Hanging Vertices

Abstract

It is shown how computational methods based on Bernstein–Bézier methods for polynomial splines on triangulations can be carried over to compute with splines on triangulations with hanging vertices. Allowing triangulations with hanging vertices provides much more flexibility than using ordinary triangulations and allows for simple adaptive algorithms based on local refinements. The use of these techniques is illustrated for two application areas of splines—namely, function fitting and the solution of boundary value problems.

Shiying Li, Larry L. Schumaker

Scaling Limits of Polynomials and Entire Functions of Exponential Type

Abstract

The connection between polynomials and entire functions of exponential type is an old one, in some ways harking back to the simple limit

$$\begin{aligned} \lim _{n\rightarrow \infty }\left( 1+\frac{z}{n}\right) ^{n}=e^{z}. \end{aligned}$$

On the left-hand side, we have $P_{n}\left( \frac{z}{n}\right) $, where $P_{n}$ is a polynomial of degree n, and on the right, an entire function of exponential type. We discuss the role of this type of scaling limit in a number of topics: Bernstein’s constant for approximation of $\left| x\right| $; universality limits for random matrices; asymptotics of $L_{p}$ Christoffel functions and Nikolskii inequalities; and Marcinkiewicz–Zygmund inequalities. Along the way, we mention a number of unsolved problems.

D. S. Lubinsky

Generalized B-Splines in Isogeometric Analysis

Abstract

In this paper, we survey the use of generalized B-splines in isogeometric Galerkin and collocation methods. Generalized B-splines are a special class of Tchebycheffian B-splines and form an attractive alternative to standard polynomial B-splines and NURBS in both modeling and simulation. We summarize their definition and main properties, and we illustrate their use in a selection of numerical examples in the context of isogeometric analysis. For practical applications, we mainly focus on trigonometric and hyperbolic generalized B-splines.

Carla Manni, Fabio Roman, Hendrik Speleers

On Polynomials with Vanishing Hessians and Some Density Problems

Abstract

We propose a conjecture regarding homogeneous polynomials with vanishing Hessian and indicate some evidence for its validity. The conjecture is related to a question of Allan Pinkus and Bronislaw Wajnryb regarding density of certain classes of polynomials.

Tom McKinley, Boris Shekhtman

Batched Stochastic Gradient Descent with Weighted Sampling

Abstract

We analyze a batched variant of Stochastic gradient descent (SGD) with weighted sampling distribution for smooth and non-smooth objective functions. We show that by distributing the batches computationally, a significant speedup in the convergence rate is provably possible compared to either batched sampling or weighted sampling alone. We propose several computationally efficient schemes to approximate the optimal weights and compute proposed sampling distributions explicitly for the least squares and hinge loss problems. We show both analytically and experimentally that substantial gains can be obtained.

Deanna Needell, Rachel Ward

A Fractional Spline Collocation Method for the Fractional-order Logistic Equation

Abstract

We construct a collocation method based on the fractional B-splines to solve a nonlinear differential problem that involves fractional derivatives, i.e., the fractional-order logistic equation. The use of the fractional B-splines allows us to express the fractional derivatives of the approximating function in an analytical form. Thus, the fractional collocation method is easy to implement, accurate, and efficient. Several numerical tests illustrate the efficiency of the proposed collocation method.

Francesca Pitolli, Laura Pezza

The Complete Length Twelve Parametrized Wavelets

Abstract

In this paper, a complete parametrization of the length twelve wavelets is given for the dilation coefficients of the trigonometric polynomials, $m(\omega )$, that satisfy the necessary conditions for orthogonality, that is $m(0)=\sqrt{2}$ and $|m(\omega )|^2+|m(\omega +\pi )|^2=2$. This parametrization has five free parameters and has a simple compatibility with the shorter length parametrizations for some specific choices of the free parameters. These wavelets have varying numbers of vanishing moments and regularity, but continuously transform from one to the other with the perturbation of the free parameters. Finally, we graph some example scaling functions from the parametrization which includes the standard Daubechies wavelets and some new wavelets that perform better than the CDF biorthogonal 9/7 wavelet in an image compression experiment on some fingerprint images.

David W. Roach

Potential Theoretic Approach to Design of Accurate Numerical Integration Formulas in Weighted Hardy Spaces

Abstract

We propose a method for designing accurate numerical integration formulas on weighted Hardy spaces, which are regarded as spaces of transformed integrands by some useful variable transformations such as the double-exponential transformation. We begin with formulating an optimality of numerical integration formulas in the space by using the norms of the error operators corresponding to those formulas. Then, we derive an expression of the minimum value of the norms, which gives a criterion for an optimal sequence of sampling points for numerical integration. Based on the expression, we propose an algorithm designing accurate numerical integration formulas on the space by a potential theoretic approach. The effectiveness of the designed formulas is supported by some numerical examples.

Ken’ichiro Tanaka, Tomoaki Okayama, Masaaki Sugihara

A Class of Intrinsic Trigonometric Mode Polynomials

Abstract

In this paper, we study a class of trigonometric polynomials that exhibit properties expected from intrinsic mode functions. In a series of lemmas, we provide sufficient conditions for a positiveness of the instantaneous frequency, number of zeros and extrema, and the proximity of upper and lower envelopes. The question of necessity of each of the conditions is discussed in numerical examples. We also introduce an orthonormal basis in $L_2$ of weak intrinsic mode functions.

Vesselin Vatchev

Kernel-Based Approximation Methods for Partial Differential Equations: Deterministic or Stochastic Problems?

Abstract

In this article, we present the kernel-based approximation methods to solve the partial differential equations using the Gaussian process regressions defined on the kernel-based probability spaces induced by the positive definite kernels. We focus on the kernel-based regression solutions of the multiple Poisson equations. Under the kernel-based probability measures, we show many properties of the kernel-based regression solutions including approximate formulas, convergence, acceptable errors, and optimal initialization. The numerical experiments show good results for the kernel-based regression solutions for the large-scale data.

Qi Ye

Titel: Approximation Theory XV: San Antonio 2016
herausgegeben von: Gregory E. Fasshauer
Larry L. Schumaker
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-59912-0
Print ISBN: 978-3-319-59911-3
DOI: https://doi.org/10.1007/978-3-319-59912-0