Polynomial Exactness Research Articles

TetraFreeQ: Tetrahedra-free quadrature on polyhedral elements

In this paper we provide a tetrahedra-free algorithm to compute low-cardinality quadrature rules with a given degree of polynomial exactness, positive weights and interior nodes on a polyhedral element with arbitrary shape. The key tools are the notion of Tchakaloff discretization set and the solution of moment-matching equations by Lawson-Hanson iterations for NonNegative Least-Squares. Several numerical tests are presented. The method is implemented in Matlab as open-source software.

On the computation of Patterson-type quadrature rules

We describe a stable and efficient algorithm for computing positive suboptimal extensions of the Gaussian quadrature rule with one or two degrees less of polynomial exactness than the corresponding Kronrod extension. These rules constitute a particular case of those first considered by Begumisa and Robinson (1991) and then by Patterson (1993) and have been proven to verify asymptotically good properties for a large class of weight functions. In particular, they may exist when the Gauss–Kronrod rule does not. The proposed algorithm is a nontrivial modification of the one introduced by Laurie (1997) for the Gauss–Kronrod quadrature, and it is based on the determination of an associated Jacobi matrix. The nodes and weights of the rule are then given as the eigenvalues and eigenvectors of the matrix, as in the classical Golub–Welsch algorithm (1969).

On the Unification of Schemes and Software for Wavelets on the Interval

We revisit the construction of wavelets on the interval with various degrees of polynomial exactness, and explain how existing schemes for orthogonal- and Spline wavelets can be extended to compactly supported delay-normalized wavelets. The contribution differs substantially from previous ones in how results are stated and deduced: linear algebra notation is exploited more heavily, and the use of sums and complicated index notation is reduced. This extended use of linear algebra eases translation to software, and a general open source implementation, which uses the deductions in this paper as a reference, has been developed. Key features of this implementation is its flexibility w.r.t. the length of the input, as well as its generality regarding the wavelet transform.

Spectral Mesh-Free Quadrature for Planar Regions Bounded by Rational Parametric Curves

This work presents spectral, mesh-free, Green’s theorem-based numerical quadrature schemes for integrating functions over planar regions bounded by rational parametric curves. Our algorithm proceeds in two steps: (1) We first find intermediate quadrature rules for line integrals along the region’s boundary curves corresponding to Green’s theorem. (2) We then use a high-order quadrature rule to compute the numerical antiderivative of the integrand along a coordinate axis, which is used to evaluate the Green’s theorem line integral. We present two methods to compute the intermediate quadrature rule. The first is spectrally accurate (it converges faster than any algebraic order with respect to number of quadrature points) and is relatively easy to implement, but has no guarantee of polynomial exactness. The second guarantees exactness for polynomial integrands up to a pre-specified degree k with an a priori-known number of quadrature points and retains the convergence properties of the first, but is slightly more complicated. The quadrature schemes have applications to computation of geometric moments, immersogeometric analysis, conservative field transfer between high-order meshes, and initialization of multi-material simulations with rational geometry. We compare the quadrature schemes produced using our method to other methods in the literature and show that they are much more efficient both in terms of number of quadrature points and computational time. We provide an open-source implementation of the algorithm in MATLAB.

Stable and efficient cubature rules by metaheuristic optimization with application to Kalman filtering

The numerical approximation of Gaussian integrals is of great relevance for a wide variety of scientific applications. In recent decades, numerous efficient cubature rules of high polynomial exactness for the case of high-dimensional Gaussian integrals have been proposed. In addition to the question of efficiency, however, the question of stability, which depends on the influence of negative weights, is also of crucial importance. For certain degrees of polynomial exactness no stable rules are known. Therefore, a new approach based on metaheuristic optimization is presented which allows the computation of cubature rules which exceed comparable rules known from the literature with respect to stability and efficiency. Especially with regard to iterative numerical calculations, the use of unstable cubature rules over time can lead to a distortion and thus a significant deterioration of the results. To illustrate this fact, the cubature-based Kalman filter is utilized. Based on the results of two simulation studies, the necessity for the use of stable cubature rules is motivated.

Optimal stencils in Sobolev spaces

This paper proves that the approximation of pointwise derivatives of order $s$ of functions in Sobolev space $W_2^m(\R^d)$ by linear combinations of function values cannot have a convergence rate better than $m-s-d/2$, no matter how many nodes are used for approximation and where they are placed. These convergence rates are attained by {\em scalable} approximations that are exact on polynomials of order at least $\lfloor m-d/2\rfloor +1$, proving that the rates are optimal for given $m,\,s,$ and $d$. And, for a fixed node set $X\subset\R^d$, the convergence rate in any Sobolev space $W_2^m(\Omega)$ cannot be better than $q-s$ where $q$ is the maximal possible order of polynomial exactness of approximations based on $X$, no matter how large $m$ is. In particular,scalable stencil constructions via polyharmonic kernels are shown to realize the optimal convergence rates, and good approximations of their error in Sobolev space can be calculated via their error in Beppo-Levi spaces. This allows to construct near-optimal stencils in Sobolev spaces stably and efficiently, for use in meshless methods to solve partial differential equations via generalized finite differences (RBF-FD). Numerical examples are included for illustration.

Modified Stieltjes polynomials and Gauss–Kronrod quadrature rules

Modified Stieltjes polynomials are defined and used to construct suboptimal extensions of Gaussian rules with one or two degrees less of polynomial exactness than the corresponding Kronrod extension. We prove that, for wide classes of weight functions and a sufficiently large number of nodes, the extended quadratures have positive weights and simple nodes on the interval $$[-1,1]$$ . The classes of weight functions considered complement those for which the Gauss–Kronrod rule is known to exist. Also, strong asymptotic representations on the whole interval $$[-1,1]$$ are given for the modified Stieltjes polynomials, which prove that they behave asymptotically as orthogonal polynomials. Finally, we provide some numerical examples.

Overcoming the sign problem in one-dimensional QCD by new integration rules with polynomial exactness

In this paper we describe a new integration method for the groups $U(N)$ and $SU(N)$, for which we verified numerically that it is polynomially exact for $N\le 3$. The method is applied to the example of 1-dimensional QCD with a chemical potential. We explore, in particular, regions of the parameter space in which the sign problem appears due the presence of the chemical potential. While Markov Chain Monte Carlo fails in this region, our new integration method still provides results for the chiral condensate on arbitrary precision, demonstrating clearly that it overcomes the sign problem. Furthermore, we demonstrate that our new method leads to orders of magnitude reduced errors also in other regions of parameter space.

Local interpolation with optimal polynomial exactness in refinement spaces

Local interpolation with optimal polynomial exactness in refinement spaces

On weak tractability of the Smolyak algorithm for approximation problems

We consider the problems of Lp-approximation of d-variate analytic functions defined on the cube with directional derivatives of all orders bounded by 1. For 1≤p<∞, it is shown that the Smolyak algorithm based on polynomial interpolation at the extrema of the Chebyshev polynomials leads to weak tractability of these problems. This gives an affirmative answer to one of the open problems raised recently by Hinrichs et al. (2014). Our proof uses the polynomial exactness of the algorithm and an explicit bound on the operator norm of the algorithm.

On weak tractability of the Clenshaw–Curtis Smolyak algorithm

We consider the problem of integration of d-variate analytic functions defined on the unit cube with directional derivatives of all orders bounded by 1. We prove that the Clenshaw–Curtis Smolyak algorithm leads to weak tractability of the problem. This seems to be the first positive tractability result for the Smolyak algorithm for a normalized and unweighted problem. The space of integrands is not a tensor product space and therefore we have to develop a different proof technique. We use the polynomial exactness of the algorithm as well as an explicit bound on the operator norm of the algorithm.

Cubic spline wavelets with complementary boundary conditions

We propose a new construction of a stable cubic spline-wavelet basis on the interval satisfying complementary boundary conditions of the second order. It means that the primal wavelet basis is adapted to homogeneous Dirichlet boundary conditions of the second order, while the dual wavelet basis preserves the full degree of polynomial exactness. We present quantitative properties of the constructed bases and we show superiority of our construction in comparison to some other known spline wavelet bases in an adaptive wavelet method for the partial differential equation with the biharmonic operator.

On the construction of sparse tensor product spaces

Let Ω1 ⊂ Rn1 and Ω2 ⊂ Rn2 be two given domains and consider on each domain a multiscale sequence of ansatz spaces of polynomial exactness r1 and r2, respectively. In this paper, we study the optimal construction of sparse tensor products made from these spaces. In particular, we derive the resulting cost complexities to approximate functions with anisotropic and isotropic smoothness on the tensor product domain Ω1×Ω2. Numerical results validate our theoretical findings.

Numerical integration with polynomial exactness over a spherical cap

This paper presents rules for numerical integration over spherical caps and discusses their properties. For a spherical cap on the unit sphere $\mathbb{S}^2$ , we discuss tensor product rules with n 2/2?+?O(n) nodes in the cap, positive weights, which are exact for all spherical polynomials of degree ??n, and can be easily and inexpensively implemented. Numerical tests illustrate the performance of these rules. A similar derivation establishes the existence of equal weight rules with degree of polynomial exactness n and O(n 3) nodes for numerical integration over spherical caps on $\mathbb{S}^2$ . For arbitrary d???2, this strategy is extended to provide rules for numerical integration over spherical caps on $\mathbb{S}^d$ that have O(n d ) nodes in the cap, positive weights, and are exact for all spherical polynomials of degree ??n. We also show that positive weight rules for numerical integration over spherical caps on $\mathbb{S}^d$ that are exact for all spherical polynomials of degree ??n have at least O(n d ) nodes and possess a certain regularity property.

Cubature formulas for symmetric measures in higher dimensions with few points

We study cubature formulas for $d$-dimensional integrals with an arbitrary symmetric weight function of product form. We present a construction that yields a high polynomial exactness: for fixed degree $\ell =5$ or $\ell =7$ and large dimension $d$ the number of knots is only slightly larger than the lower bound of Möller and much smaller compared to the known constructions. We also show, for any odd degree $\ell = 2k+1$, that the minimal number of points is almost independent of the weight function. This is also true for the integration over the (Euclidean) sphere.

Smolyak cubature of given polynomial degree with few nodes for increasing dimension

Some recent investigations (see e.g., Gerstner and Griebel [5], Novak and Ritter [9] and [10], Novak, Ritter and Steinbauer [11], Wasilkowski and Woźniakowski [18] or Petras [13]) show that the so-called Smolyak algorithm applied to a cubature problem on the d-dimensional cube seems to be particularly useful for smooth integrands. The problem is still that the numbers of nodes grow (polynomially but) fast for increasing dimensions. We therefore investigate how to obtain Smolyak cubature formulae with a given degree of polynomial exactness and the asymptotically minimal number of nodes for increasing dimension d and obtain their characterization for a subset of Smolyak formulae. Error bounds and numerical examples show their good behaviour for smooth integrands. A modification can be applied successfully to problems of mathematical finance as indicated by a further numerical example.

Biorthogonal Multiwavelets on the Interval: Cubic Hermite Splines

Starting with Hermite cubic splines as the primal multigenerator, first a dual multigenerator onR is constructed that consists of continuous functions, has small support, and is exact of order 2. We then derive multiresolution sequences on the interval while retaining the polynomial exactness on the primal and dual sides. This guarantees moment conditions of the corresponding wavelets. The concept of stable completions (CDP) is then used to construct the corresponding primal and dual multiwavelets on the interval as follows. An appropriate variation of what is known as a hierarchical basis in finite element methods is shown to be an initial completion. This is then, in a second step, projected into the desired complements spanned by compactly supported biorthogonal multiwavelets. The masks of all multigenerators and multiwavelets are finite so that decomposition and reconstruction algorithms are simple and efficient. Furthermore, in addition to the Jackson estimates which follow from the exactness, one can also show Bernstein inequalities for the primal and dual multiresolutions. Consequently, sequence norms for the coefficients based on such multiwavelet expansions characterize Sobolev normsk¢k Hs.(0;1)/ for s 2 .i0:824926; 2:5/. In particular, the multiwavelets form Riesz bases for L2.(0; 1)/.

High dimensional polynomial interpolation on sparse grids

We study polynomial interpolation on a d-dimensional cube, where d is large. We suggest to use the least solution at sparse grids with the extrema of the Chebyshev polynomials. The polynomial exactness of this method is almost optimal. Our error bounds show that the method is universal, i.e., almost optimal for many different function spaces. We report on numerical experiments for d = 10 using up to 652 065 interpolation points.

Simple Cubature Formulas with High Polynomial Exactness

We study cubature formulas for d -dimensional integrals with arbitrary weight function of tensor product form. We present a construction that yields a high polynomial exactness: for fixed degree, the number of knots depends on the dimension in an order-optimal way. The cubature formulas are universal: the order of convergence is almost optimal for two different scales of function spaces. The construction is simple: a small number of arithmetical operations is sufficient to compute the knots and the weights of the formulas.

Biorthogonal Spline Wavelets on the Interval—Stability and Moment Conditions

This paper is concerned with the construction of biorthogonal multiresolution analyses on [0, 1] such that the corresponding wavelets realize any desired order of moment conditions throughout the interval. Our starting point is the family of biorthogonal pairs consisting of cardinal B-splines and compactly supported dual generators on R developed by Cohen, Daubechies, and Feauveau. In contrast to previous investigations we preserve the full degree of polynomial reproduction also for the dual multiresolution and prove in general that the corresponding modifications of dual generators near the end points of the interval still permit the biorthogonalization of the resulting bases. The subsequent construction of compactly supported biorthogonal wavelets is based on the concept of stable completions. As a first step we derive an initial decomposition of the spline spaces where the complement spaces between two successive levels are spanned by compactly supported splines which form uniformly stable bases on each level. As a second step these initial complements are then projected into the desired complements spanned by compactly supported biorthogonal wavelets. Since all generators and wavelets on the primal and the dual sides have finitely supported masks, the corresponding decomposition and reconstruction algorithms are simple and efficient. The desired number of vanishing moments is implied by the polynomial exactness of the dual multiresolution. Again due to the polynomial exactness the primal and dual spaces satisfy corresponding Jackson estimates. In addition, Bernstein inequalities can be shown to hold for a range of Sobolev norms depending on the regularity of the primal and dual wavelets. Then it follows from general principles that the wavelets form Riesz bases forL2([0, 1]) and that weighted sequence norms for the coefficients of such wavelet expansions characterize Sobolev spaces and their duals on [0, 1] within a range depending on the parameters in the Jackson and Bernstein estimates. We conclude by addressing several issues concerning numerical implementation. In particular, we test the quantitative stability properties of corresponding multiscale transformations, indicate strategies for improving them, and present some numerical experiments.