Cubature Formulas Research Articles

Numerical Integration over Pyramids

Pyramidal elements are often used to connect tetrahedral and hexahedral elements in the finite element method. In this paper we derive three new higher order numerical cubature formulae for pyramidal elements.

Higher-order interpolated lattice schemes for multidimensional option pricing problems

This article presents a new class of higher-order space–time discretization schemes for multidimensional diffusions via lattice systems which involve space interpolation techniques. Through the explanation of the methodology, the author shows that the algorithm can be simply represented by multiplications of sparse matrices. The key idea is to combine the weak approximation approach for stochastic differential equations and some techniques on high-dimensional spaces to break the curse of dimensionality (cubature formulas and sparse grid interpolation techniques). The first objective is to investigate the error estimates derived from short time asymptotics of certain semigroup operators, together with the discussion of numerical stability. As the second objective, several computational experiments for some derivative pricing models are presented in one and three dimensional settings.

A meshless like method for the approximate solution of integral equations over polygons

In this paper, a numerical scheme based on a Gauss-like cubature formula from Sammariva and Vianello (2007) [1] is introduced for approximate solution of integral equations over a polygonal domain with a piecewise straight lines boundary in R2. The proposed technique is a meshless like method with sufficient precision, which does not require any discretization of the polygon domain or any preprocessing such as mesh refinement. The error analysis of the method is provided and some numerical experiments are also presented to evaluate the performance of the proposed algorithm.

Polynomial Spline Approach for Double Integrals with Algebraic Singularity

In this note, cubature formulas are constructed to evaluate the double integrals on the rectangle with algebraic singularity by replacing the density function f(x,y) with the SΔ(P) modified spline function interpolation of type (0,2). Rate of convergence are obtained in the classes of function f(x,y) ∊ C2,α(D).

The algebra of interpolatory cubature formulæ for generic nodes

We consider the classical problem of computing the expected value of a real function f of the d-variate random variable X using cubature formulae. We use in synergy tools from Commutative Algebra for cubature rulae, from elementary orthogonal polynomial theory and from Probability.

Computation of volume potentials over bounded domains via approximate approximations

We obtain cubature formulas of volume potentials over bounded domains combining the basis functions introduced in the theory of approximate approximations with their integration over the tangential-halfspace. Then the computation is reduced to the quadrature of one dimensional integrals over the halfline. We conclude the paper providing numerical tests which show that these formulas give very accurate approximations and confirm the predicted order of convergence.

Weighted spherical semidesigns and cubature formulae for calculating integrals on a sphere

In this paper we study weighted spherical semidesigns, i.e., systems of points on a sphere of a specific type. We propose a new proof of the necessary and sufficient condition for a system of points on a sphere to be a weighted spherical semidesign. This criterion gives new approaches to the construction of cubature formulae for calculating integrals over a sphere with the degree of accuracy of 5 and 9.

Cubature formulas on a sphere that are invariant with respect to a group of dihedron rotations with inversion D 6h

An algorithm of searching for the best (in a sense) cubature formulas on a sphere that are invariant with respect to a group of dihedron rotations with inversion D6h is developed. This algorithm is applied to find parameters of all the best cubature formulas of this group of symmetry up to the 23rd order of accuracy n. In the course of the study performed, exact values of parameters of the corresponding cubature formulas are found for n ≤ 11, and approximate values are obtained by numerical solving systems of nonlinear algebraic equations by a Newton-type method for other values of n. For the first time, ways of obtaining the best cubature formulas for a sphere are systematically investigated for the case of a group that is not a subgroup of the groups of symmetry of regular polyhedrons.

Fourier expansions for a logarithmic fundamental solution of the polyharmonic equation

In even-dimensional Euclidean space for integer powers of the Laplacian greater than or equal to the dimension divided by two, a fundamental solution for the polyharmonic equation has logarithmic behavior. We give two approaches for developing an azimuthal Fourier expan- sion of this logarithmic fundamental solution. The first approach is algebraic and relies upon the construction of two-parameter polynomials which we call logarithmic polynomials. The second approach depends on the computation of parameter derivatives of Fourier series expressions for a power-law fundamental solution of the polyharmonic equation. We conclude by comparing the two approaches and giving the azimuthal Fourier series for a logarithmic fundamental solution of the polyharmonic equation in rotationally-invariant coordinate systems. Solutions of the polyharmonic equation (powers of the Laplacian operator) are ubiquitous in many areas of computational, pure, applied mathematics, physics and en- gineering. We concern ourselves, in this paper, with a fundamentalsolution of the poly- harmonic equation (Laplace, biharmonic, etc.), which by convolution yields a solution to the inhomogeneous polyharmonic equation. Solutions to inhomogeneous polyhar- monic equations are useful in many physical applications including those areas related to Poisson's equation such as Newtonian gravity, electrostatics, magnetostatics, quan- tum direct and exchange interactions (cf. Section 1 in (3)), etc. Furthermore, applica- tions of higher-powers of the Laplacian include such varied areas as minimal surfaces (12), Continuum Mechanics (8), Mesh deformation (6), Elasticity (9), Stokes Flow (7), Geometric Design (20), Cubature formulae (17), mean value theorems (cf. Pizzetti's formula) (13), and Hartree-Fock calculations of nuclei (21). Closed-form expressions for the Fourier expansions of a logarithmic fundamental solution for the polyharmonic equation are extremely useful when solving inhomoge- neous polyharmonic problems in even-dimensionalEuclidean space, especially when a degreeof rotationalsymmetry is involved. A fundamentalsolution of the polyharmonic equation on d-dimensionalEuclidean space R d has two argumentsand therefore maps from a 2d-dimensional space to the reals. Solutions to the inhomogeneous polyhar- monic equation can be obtained by convolution of a fundamental solution with an in- tegrable source distribution. Eigenfunction decompositions of a fundamental solution

Asymptotically optimal unsaturated lattice cubature formulae with bounded boundary layer

This paper describes a new algorithm for constructing lattice cubature formulae with bounded boundary layerwhereThese formulae are unsaturated (in the sense of Babenko) both with respect to the order and in regard to the property of asymptotic optimality on -spaces, . Most of the results obtained apply also to -spaces with a hypoelliptic multiplier of smoothness .Bibliography: 6 titles.

Note on Cubature Formulae and Designs Obtained from Group Orbits

Abstract In 1960, Sobolev proved that for a finite reflection group G, a G-invariant cubature formula is of degree t if and only if it is exact for all G-invariant polynomials of degree at most t . In this paper, we make some observations on invariant cubature formulas and Euclidean designs in connection with the Sobolev theorem. First, we give an alternative proof of theorems by Xu (1998) on necessary and sufficient conditions for the existence of cubature formulas with some strong symmetry. The new proof is shorter and simpler compared to the original one by Xu, and, moreover, gives a general interpretation of the analytically-written conditions of Xu's theorems. Second, we extend a theorem by Neumaier and Seidel (1988) on Euclidean designs to invariant Euclidean designs, and thereby classify tight Euclidean designs obtained from unions of the orbits of the corner vectors. This result generalizes a theorem of Bajnok (2007), which classifies tight Euclidean designs invariant under the Weyl group of type B, to other finite reflection groups.

Errors, condition numbers, and guaranteed accuracy of higher-dimensional spherical cubatures

We give upper bounds for the deviation of the norm of a perturbed error functional from the norm of the original error of a higher-dimensional spherical cubature formula. The deviation arises as a result of the combined influence on the computation of small variations of the weights of the cubature formula and rounding for the subsequent calculation of the cubature sum in the given standards of approximation to real numbers. We estimate the practical error of the cubature formula for its action on an arbitrary function in the unit ball of the normed space of integrands. The resulting estimates are applied to studying the practical error of spherical cubature formulas in the case of integrands in Sobolev-type spaces on the higher-dimensional unit sphere. We represent the norm of the error functional in the dual space of the Sobolev class as a positive definite quadratic form in the weights of the cubature formula. We estimate the practical error for spherical cubature formulas, each of which is constructed as the direct product of Gauss’s quadrature formula along the meridian of the sphere and of the rectangle quadrature formula along the equator. The weights of this direct product with 2m2 nodes are positive. The formula itself is exact at all spherical harmonics up to order 2m − 1.

Cubature formula for spherical basis function networks

Some mathematical models in geophysics and graphic processing need to compute integrals with scattered data on the sphere. Thus cubature formula plays an important role in computing these spherical integrals. This paper is devoted to establishing an exact positive cubature formula for spherical basis function networks. The authors give an existence proof of the exact positive cubature formula for spherical basis function networks, and prove that the cubature points needed in the cubature formula are not larger than the number of the scattered data.

Polyharmonic Hardy Spaces on the Complexified Annulus and Error Estimates of Cubature Formulas

The present paper has a twofold contribution: first, we introduce a new concept of Hardy spaces on a multidimensional complexified annular domain which is closely related to the annulus of the Klein–Dirac quadric important in Conformal Quantum Field Theory. Secondly, for functions in these Hardy spaces, we provide error estimate for the polyharmonic Gaus–Jacobi cubature formulas, which have been introduced in previous papers.

On Trigonometric Blending Interpolation and Cubature Formulae

We discuss error representations for Hermite-Lagrange trigono- metric interpolation introduced in Dryanov and Petrov (Interpolation and L1-approximation by trigonometric polynomials and blending func- tions, J. Approx. Theory 164:1049-1064, 2012) and obtain one-sided trigonometric quadratures for approximate integration of one-dimensional integrals. Next, we study error representations of multivariate Hermite- Lagrange transfinite trigonometric interpolation and derive one-sided trigonometric blending interpolants to multivariate functions, under some restrictions. Then, we construct one-sided transfinite cubature formulae for approximate integration of multivariate integrals. We construct also cubature formulae with positive coefficients, based on line integrals and exact in a vector space of trigonometric blending functions with pre- scribed order.

Estimation of the remainder of a cubature formula on a Chebyshev grid

Let C(Q) denote the space of continuous functions f(x, y) in the square Q = [−1, 1] × [−1, 1] with the norm $$\begin{gathered} \left\| f \right\| = \max \left| {f(x,y)} \right|, \hfill \\ (x,y) \in Q. \hfill \\ \end{gathered} $$ On a Chebyshev grid, a cubature formula of the form $$\int\limits_{ - 1}^1 {\int\limits_{ - 1}^1 {\frac{1} {{\sqrt {(1 - x^2 )(1 - y^2 )} }}f(x,y)dxdy = \frac{{\pi ^2 }} {{mn}}\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^m {f\left( {\cos \frac{{2i - 1}} {{2n}}\pi ,\cos \frac{{2j - 1}} {{2m}}\pi } \right)} + R_{m,n} (f)} } } $$ is considered in some class H(r1, r2) of functions f ∈ C(Q) defined by a generalized shift operator. The remainder Rm, n(f) is proved to satisfy the estimate $$\mathop {\sup }\limits_{f \in H(r_1 ,r_2 )} \left| {R_{m,n} (f)} \right| = O(n^{ - r_1 + 1} + m^{ - r_2 + 1} ), $$ where r1, r2 > 1; λ−1 ≤ n/m ≤ λ with λ > 0; and the constant in O(1) depends on λ.

Tight p-fusion frames

Fusion frames enable signal decompositions into weighted linear subspace components. For positive integers p, we introduce p-fusion frames, a sharpening of the notion of fusion frames. Tight p-fusion frames are closely related to the classical notions of designs and cubature formulas in Grassmann spaces and are analyzed with methods from harmonic analysis in the Grassmannians. We define the p-fusion frame potential, derive bounds for its value, and discuss the connections to tight p-fusion frames.

Numerical integration based on bivariate quadratic spline quasi-interpolants on Powell-Sabin partitions

In this paper we generate and study new cubature formulas based on spline quasi-interpolants in the space of quadratic Powell-Sabin splines on nonuniform triangulations of a polygonal domain in ℝ2. By using a specific refinement of a generic triangulation, optimal convergence orders are obtained for some of these rules. Numerical tests are presented for illustrating the theoretical results.

Extremal functions of cubature formulas on a multidimensional sphere and spherical splines

We establish the general form of extremal cubature formulas on multidimensional spheres. The domains of definition for the cubature formulas under consideration are Sobolev-type spaces on the sphere. The smoothness of the class function under study may be fractional. We prove that, for a given set of nodes, there exists a one-to-one correspondence between the set of extremal functions of cubature formulas on the sphere and the set of natural spherical splines with zero spherical mean.

Estimation of the remainder of a cubature formula on a Chebyshev grid for two-variable functions

For a cubature formula of the form $$\int\limits_0^{2\pi } {\int\limits_0^{2\pi } {f(x,y)dxdy = \frac{{4\pi ^2 }} {{mn}}\sum\limits_{i = 0}^{n - 1} {\sum\limits_{j = 0}^{m - 1} {f\left( {\frac{{2\pi i}} {n},\frac{{2\pi j}} {m}} \right) + R_{n,m} (f)} } } }$$ on a Chebyshev grid, the remainder Rn,m(f) is proved to satisfy the sharp estimate $$\mathop {\sup }\limits_{f \in H\left( {r_1 ,r_2 } \right)} \left| {R_{n,m} (f)} \right| = O\left( {n^{ - r_1 + 1} + m^{ - r_1 + 1} } \right)$$ in some class of functions H(r1, r2) defined by a generalized shift operator. Here, r1, r2 > 1; λ−1 ≤ n/m ≤ λ with λ > 0; and the constant in the O-term depends only on λ.