Optimal Cubature Formulas Research Articles

Function recovery on manifolds using scattered data

We consider the task of recovering a Sobolev function on a connected compact Riemannian manifold M when given a sample on a finite point set. We prove that the quality of the sample is given by the Lγ(M)-average of the geodesic distance to the point set and determine the value of γ∈(0,∞]. This extends our findings on bounded convex domains [IMA J. Numer. Anal., 2024]. As a byproduct, we prove the optimal rate of convergence of the nth minimal worst case error for Lq(M)-approximation for all 1≤q≤∞.Further, a limit theorem for moments of the average distance to a set consisting of i.i.d. uniform points is proven. This yields that a random sample is asymptotically as good as an optimal sample in precisely those cases with γ<∞. In particular, we obtain that cubature formulas with random nodes are asymptotically as good as optimal cubature formulas if the weights are chosen correctly. This closes a logarithmic gap left open by Ehler, Gräf and Oates [Stat. Comput., 29:1203-1214, 2019].

Optimal cubature formulas for approximate integrals of functions defined on a sphere in three-dimensional space

The development of effective methods of approximate calculation of integrals using optimal cubature formulas and optimal quadrature formulas with trigonometric weights for defined functions on a sphere, the creation of new algorithms for approximate calculation of trigonometric weighted integrals in different classes of functions, as well as the assessment of their errors, is the goal of this work.

Compound cubature formulas on a lattice

In the present paper lattice optimal cubature formulas are constructed by the variational method in the Sobolev space. In addition, the square of the norm of the error functional of the constructed lattice optimal cubature formulas in the conjugate Sobolev space is explicitly calculated.

Optimal by the Order of Accuracy Cubature Formula for the Approximate Calculation of Triple Integrals from Fast Oscillating Functions in General View

Introduction. The rapid development of digital technologies encourages scientists to create new or improve existing mathematical models of technical processes. It is time to develop mathematical models with different types of data. In the tasks of digital signal and image processing, the approximate calculation of integrals from rapidly oscillating functions using new information operators makes it possible to build cubature formulas using different types of information: the values of functions on planes, lines and points can be used as data. The purpose is to present and investigate the optimal cubature formula for the approximate calculation of the triple integral from rapidly oscillating functions in the general form on the class of differential functions. Information about functions are traces on systems of mutually perpendicular planes. Results. The study of the problems of digital signal and image processing continued using the example of numerical integration of triple integrals from rapidly oscillating functions in the general form. The values of functions on systems of mutually perpendicular planes are using for constructed cubature formula. The main attention in the research focuses on obtaining the estimations of errors. Proposed cubature formula for the approximate calculation of the triple integral from rapidly oscillating functions in general view is optimal in order of accuracy on the class of differential functions. The conducted numerical experiment confirmed the theoretical results. Conclusions. The obtained results make it possible to build new and improve existing mathematical models of processes with different types of input information. New information operators are a powerful tool in the development of such models. Cubature formulas for the approximate calculation of integrals from rapidly oscillating functions of many variables have been created. Іn the construction of the formulas traces of the function on planes, lines, and points are used. Formulas in their construction use function traces on planes, lines, and points. In this work, a cubature formula for the approximate calculation of the triple integral from a rapidly oscillating function in the general form, which is optimal in order of accuracy, is constructed and investigated on the class of differentiable functions. A feature of the proposed formula is the use of values of functions on systems of mutually perpendicular planes as data. Keywords: integrals of rapidly oscillating functions of many variables, cubature formulas, new information operators, digital signal and image processing.

Asymptotically optimal cubature formulas on manifolds for prefixed weights

Given a Riemannian manifold X with Riemannian measure μX and positive weights {ωj}j=1N, we study the conditions under which there exist points {xj}j=1N⊂X so that a cubature formula of the form (1)∫XPdμX=∑j=1NωjP(xj)holds for all polynomials P of order less than or equal to L. The problem is studied for diffusion polynomials (linear combinations of eigenfunctions of the Laplace–Beltrami operator) in the context of abstract Riemannian manifolds and for algebraic polynomials in the context of algebraic manifolds in Rn.

Optimal Cubature Formulas on Classes of Periodic Functions in Several Variables

Optimal Cubature Formulas on Classes of Periodic Functions in Several Variables

Spherical cubature formulas in Sobolev spaces

We study sequences of cubature formulas on the unit sphere in a multidimensional Euclidean space. The grids for the cubature formulas under consideration embed in each other consecutively, forming in the limit a dense subset on the initial sphere. As the domain of cubature formulas, i.e. as the class of integrands, we take spherical Sobolev spaces. These spaces may have fractional smoothness. We prove that, among all possible spherical cubature formulas with given grid, there exists and is unique a formula with the least norm of the error, an optimal formula. The weights of the optimal cubature formula are shown to be solutions to a special nondegenerate system of linear equations. We prove that the errors of cubature formulas tend to zero as the number of nodes grows indefinitely.

Optimal cubature formulas for calculation of multidimensional integrals in weighted Sobolev spaces

Optimal cubature formulas are constructed for calculations of multidimensional integrals in weighted Sobolev spaces. We consider some classes of functions defined in the cube Ω = [-1, 1]l, l = 1, 2,..., and having bounded partial derivatives up to the order r in Ω and the derivatives of jth order (r < j ≤ s) whose modulus tends to infinity as power functions of the form (d(x, Г))-(j-r), where x ∈ Ω Г, x = (x1,..., xl), Г = ∂Ω, and d(x, Г) is the distance from x to Г.

Sampling and Cubature on Sparse Grids Based on a B-spline Quasi-Interpolation

Let $$X_n = \{x^j\}_{j=1}^n$$Xn={xj}j=1n be a set of n points in the d-cube $${\mathbb {I}}^d:=[0,1]^d$$Id:=[0,1]d, and $$\Phi _n = \{\varphi _j\}_{j =1}^n$$źn={źj}j=1n a family of n functions on $${\mathbb {I}}^d$$Id. We consider the approximate recovery of functions f on $${{\mathbb {I}}}^d$$Id from the sampled values $$f(x^1), \ldots , f(x^n)$$f(x1),ź,f(xn), by the linear sampling algorithm $$ L_n(X_n,\Phi _n,f) := \sum _{j=1}^n f(x^j)\varphi _j. $$Ln(Xn,źn,f):=źj=1nf(xj)źj. The error of sampling recovery is measured in the norm of the space $$L_q({\mathbb {I}}^d)$$Lq(Id)-norm or the energy quasi-norm of the isotropic Sobolev space $$W^\gamma _q({\mathbb {I}}^d)$$Wqź(Id) for $$1 0. Functions f to be recovered are from the unit ball in Besov-type spaces of an anisotropic smoothness, in particular, spaces $$B^{\alpha ,\beta }_{p,\theta }$$Bp,źź,β of a hybrid of mixed smoothness $$\alpha > 0$$ź>0 and isotropic smoothness $$\beta \in {\mathbb {R}}$$βźR, and spaces $$B^a_{p,\theta }$$Bp,źa of a nonuniform mixed smoothness $$a \in {\mathbb {R}}^d_+$$aźR+d. We constructed asymptotically optimal linear sampling algorithms $$L_n(X_n^*,\Phi _n^*,\cdot )$$Ln(Xnź,źnź,·) on special sparse grids $$X_n^*$$Xnź and a family $$\Phi _n^*$$źnź of linear combinations of integer or half integer translated dilations of tensor products of B-splines. We computed the asymptotic order of the error of the optimal recovery. This construction is based on B-spline quasi-interpolation representations of functions in $$B^{\alpha ,\beta }_{p,\theta }$$Bp,źź,β and $$B^a_{p,\theta }$$Bp,źa. As consequences, we obtained the asymptotic order of optimal cubature formulas for numerical integration of functions from the unit ball of these Besov-type spaces.

Constructive sparse trigonometric approximation and other problems for functions with mixed smoothness

Our main interest in this paper is to study some approximation problems for classes of functions with mixed smoothness. We use a technique based on a combination of results from hyperbolic cross approximation, which were obtained in 1980s–1990s, and recent results on greedy approximation to obtain sharp estimates for best -term approximation with respect to the trigonometric system. We give some observations on the numerical integration and approximate recovery of functions with mixed smoothness. We prove lower bounds, which show that one cannot improve the accuracy of sparse grids methods with points in the grid by adding arbitrary points. In the case of numerical integration these lower bounds provide the best available lower bounds for optimal cubature formulae and for sparse grids based cubature formulae. Bibliography: 31 titles.

Lower bounds for the integration error for multivariate functions with mixed smoothness and optimal Fibonacci cubature for functions on the square

We prove lower bounds for the error of optimal cubature formulae for d-variate functions from Besov spaces of mixed smoothness in the case , and , where is either the d-dimensional torus or the d-dimensional unit cube . In addition, we prove upper bounds for QMC integration on the Fibonacci-lattice for bivariate periodic functions from in the case , and . A non-periodic modification of this classical formula yields upper bounds for if . In combination these results yield the correct asymptotic error of optimal cubature formulae for functions from and indicate that a corresponding result is most likely also true in case . This is compared to the correct asymptotic of optimal cubature formulae on Smolyak grids which results in the observation that any cubature formula on Smolyak grids can never achieve the optimal worst-case error.

Optimal cubature formulas for tensor products of certain classes of functions

We study the problem of constructing an optimal formula of approximate integration along a d -dimensional parallelepiped. Our construction utilizes mean values along intersections of the integration domain with n hyperplanes of dimension ( d − 1 ) , each of which is perpendicular to some coordinate axis. We find an optimal cubature formula of this type for two classes of functions. The first class controls the moduli of continuity with respect to all variables, whereas the second class is the intersection of certain periodic multivariate Sobolev classes. We prove that all node hyperplanes of the optimal formula in each case are perpendicular to a certain coordinate axis and are equally spaced and the weights are equal. For specific moduli of continuity and for sufficiently large n , the formula remains optimal for the first class among cubature formulas with arbitrary positions of hyperplanes.

Construction of Optimal Cubature Formulas Related to Computer Tomography

We study the problem of the optimization of approximate integration on the class of functions defined on the parallelepiped Πd=[0,a1]×⋅⋅⋅×[0,ad], a1,…,ad>0, having a given majorant for the modulus of continuity (relative to the l1-metric in ℝd). An optimal cubature formula, which uses as information integrals of f along intersections of Πd with n arbitrary (d−1)-dimensional hyperplanes in ℝd (d>1) is obtained. We also find an asymptotically optimal sequence of cubature formulas, whose information functionals are integrals of f along intersections of Πd with shifts of (d−2)-dimensional coordinate subspaces of ℝd (d>2).

Optimal cubature formulas on compact homogeneous manifolds

We find lower bounds for the rate of convergence of optimal cubature formulas on sets of differentiable functions on compact homogeneous manifolds of rank I or two-point homogeneous spaces. It is shown that these lower bounds are sharp in the power scale in the case of S 2 , the unit sphere in R 3 .

Cubature formulas on a sphere invariant under the icosahedral rotation group

An algorithm is proposed for constructing cubature formulas on a sphere that are invariant under the icosahedral rotation group. The algorithm is applied to construct new cubature formulas of algebraic order of accuracy n = 19, 20, 21, 23, 24, 25. Parameters of these cubature formulas are given to 16 significant digits. A table, which containsmain characteristics of all optimal cubature formulas for the icosahedral rotation group exact to order 35, is included. A real version of F. Klein’s formula, which establishes a connection between basic invariant forms of the icosahedral rotation group, is presented.

Optimal quadrature and cubature formulas for computing Fourier transform of finite functions of one class. Case of strong oscillation

The paper deals with quadrature and cubature formulas, optimal and nearly optimal in accuracy, used to evaluate integrals of quickly oscillating functions in a class of bounded functions with piecewise continuous first derivatives bounded by a constant in one-and multidimensional cases. Optimal estimates for the numerical integration error and upper bounds are obtained for these integrals in case of strong oscillation of subintegral functions.

On the construction of optimal cubature formulae which use integrals over hyperspheres

We consider formulae of approximate integration over a d -dimensional ball which use n surface integrals along ( d - 1 ) -dimensional spheres centered at the origin. For a class of functions defined on the ball with gradients satisfying an integral restriction, optimal formulae of this type are obtained.

Cubature formulas, discrepancy, and nonlinear approximation

The main goal of this paper is to demonstrate connections between the following three big areas of research: the theory of cubature formulas (numerical integration), the discrepancy theory, and nonlinear approximation. In Section 1, we discuss a relation between results on cubature formulas and on discrepancy. In particular, we show how standard in the theory of cubature formulas settings can be translated into the discrepancy problem and into a natural generalization of the discrepancy problem. This leads to a concept of the r-discrepancy. In Section 2, we present results on a relation between construction of an optimal cubature formula with m knots for a given function class and best nonlinear m-term approximation of a special function determined by the function class. The nonlinear m-term approximation is taken with regard to a redundant dictionary also determined by the function class. Sections 3 and 4 contain some known results on the lower and the upper estimates of errors of optimal cubature formulas for the class of functions with bounded mixed derivative. One of the important messages (well known in approximation theory) of this paper is that the theory of discrepancy is closely connected with the theory of cubature formulas for the classes of functions with bounded mixed derivative. We have included in the paper both new results (Section 2) and known results. We included some known results with their proofs for the following two reasons. First of all we want to make the paper self-contained (within reasonable limits). Secondly, we selected the proofs which demonstrate different methods and are not very much technically involved. Section 5 contains historical notes on discrepancy and cubature formulas, some further comments and remarks. Historical remarks on nonlinear approximation are included in Section 2. We want to point out that this paper is not a complete survey in any of the above-mentioned areas. We did not even try to provide a complete list of results in those areas. We rather wanted to highlight the most typical results in cubature formulas (Sections 3 and 4) and show their relation to the discrepancy theory.

Optimal Cubature Formulae and Recovery of Fast-Oscillating Functions from an Interpolational Class

The method of limit functions is used to construct optimal-by-accuracy and optimal-by-order (with constant not exceeding two) cubature formulae for the integration of fast oscillatory functions given by their values at a finite number of fixed nodes in a square region. The construction is based on explicit forms of the majorant and minorant in the given interpolational class C1,L,N2 and the solution of the problem of optimal-by-accuracy recovery of functions from this class. It is shown that an appropriate choice of the grid in this interpolational class leads to a substantial reduction in a priori information required for the application of the proposed approach.

Cubature formulae for Besov classes

This paper is devoted to an investigation of optimal cubature formulae for Besov classes of functions with restrictions on the mixed difference. This study is a continuation of papers [1] and [2]. In this paper we find the exact order of the error of the cubature formula using the values of the function at particular points.