Hyperbolic Cross Approximation Research Articles

Estimates of efficiency for two methods of stable numerical summation of smooth functions

We consider a classical ill-posed problem of reconstruction of continuous functions from their noisy Fourier coefficients. We study the case of functions of two variables that has been much less investigated. The smoothness of reconstructed functions is measured in terms of the Sobolev classes as well as the classes of functions with dominated mixed derivatives. We investigate two summation methods, that are based on ideas of the rectangle and the hyperbolic cross, respectively. For both of these methods we establish the estimates of the accuracy on the classes that are considered as well the estimates of computational costs. Moreover, we made the comparison of their efficiency based on obtained estimates. A somehow surprising outcome of our study is that for both types of the considered smoothness classes one should employ hyperbolic cross approximation that is not typical for the functions under consideration.

Numerical meshless solution of high-dimensional sine-Gordon equations via Fourier HDMR-HC approximation

In this paper, an implicit time stepping meshless scheme is proposed to find the numerical solution of high-dimensional sine-Gordon equations (SGEs) by combining the high dimensional model representation (HDMR) and the Fourier hyperbolic cross (HC) approximation. To ensure the sparseness of the relevant coefficient matrices of the implicit time stepping scheme, the whole domain is first divided into a set of subdomains, and the relevant derivatives in high-dimension can be separately approximated by the Fourier HDMR-HC approximation in each subdomain. The proposed method allows for stable large time-steps and a relatively small number of nodes with satisfactory accuracy. The numerical examples show that the proposed method is very attractive for simulating the high-dimensional SGEs.

Error analysis of the Wiener–Askey polynomial chaos with hyperbolic cross approximation and its application to differential equations with random input

It is well-known that sparse grid algorithm has been widely accepted as an efficient tool to overcome the “curse of dimensionality” in some degree. In this note, we give the error estimate of hyperbolic cross (HC) approximations with all sorts of Askey polynomials. These polynomials are useful in generalized polynomial chaos (gPC) in the field of uncertainty quantification. The exponential convergences in both regular and optimized HC approximations have been shown under the condition that the random variable depends on the random inputs smoothly in some degree. Moreover, we apply gPC to numerically solve the ordinary differential equations with slightly higher dimensional random inputs. Both regular and optimized HC have been investigated with Laguerre-chaos, Charlier-chaos and Hermite-chaos in the numerical experiment. The discussion of the connection between the standard ANOVA approximation and Galerkin approximation is in the appendix.

Formula omitted]-dimension in infinite dimensional hyperbolic cross approximation and application to parametric elliptic PDEs

In this article, we present a cost–benefit analysis of the approximation in tensor products of Hilbert spaces of Sobolev-analytic type. The Sobolev part is defined on a finite dimensional domain, whereas the analytical space is defined on an infinite dimensional domain. As main mathematical tool, we use the ε-dimension in Hilbert spaces which gives the lowest number of linear information that is needed to approximate an element from the unit ball W in a Hilbert space Y up to an accuracy ε>0 with respect to the norm of a Hilbert space X. From a practical point of view this means that we a priori fix an accuracy and ask for the amount of information to achieve this accuracy. Such an analysis usually requires sharp estimates on the cardinality of certain index sets which are in our case infinite-dimensional hyperbolic crosses. As main result, we obtain sharp bounds of the ε-dimension of the Sobolev-analytic-type function classes which depend only on the smoothness differences in the Sobolev spaces and the dimension of the finite dimensional domain where these spaces are defined. This implies in particular that, up to constants, the costs of the infinite dimensional (analytical) approximation problem is dominated by the finite-variate Sobolev approximation problem. We demonstrate this procedure with examples of functions spaces stemming from the regularity theory of parametric partial differential equations.

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.

Hyperbolic cross approximation in infinite dimensions

We give tight upper and lower bounds of the cardinality of the index sets of certain hyperbolic crosses which reflect mixed Sobolev–Korobov-type smoothness and mixed Sobolev-analytic-type smoothness in the infinite-dimensional case where specific summability properties of the smoothness indices are fulfilled. These estimates are then applied to the linear approximation of functions from the associated spaces in terms of the ε-dimension of their unit balls. Here, the approximation is based on linear information. Such function spaces appear for example for the solution of parametric and stochastic PDEs. The obtained upper and lower bounds of the approximation error as well as of the associated ε-complexities are completely independent of any parametric or stochastic dimension. Moreover, the rates are independent of the parameters which define the smoothness properties of the infinite-variate parametric or stochastic part of the solution. These parameters are only contained in the order constants. This way, linear approximation theory becomes possible in the infinite-dimensional case and corresponding infinite-dimensional problems get tractable.

Hyperbolic cross approximation for the spatially homogeneous Boltzmann equation

We consider the nonlinear spatially homogeneous Boltzmann equation and its Fourier spectral discretization in velocity space involving periodic continuation of the density and a truncation of the collision operator. We allow discretization based on arbitrary sets of active Fourier modes with particular emphasis on the family of so-called hyperbolic cross approximations. We also discuss an offset method that takes advantage of the known equilibrium solutions. Extending the analysis in Filbet & Mouhot (2011, Analysis of spectral methods for the homogeneous Boltzmann equation. Trans. Amer. Math. Soc., 363, 1947–1980), we establish consistency estimates for the discrete collision operators and stability of the semidiscrete evolution. Under an assumption of Gaussian-like decay of the discrete solution, we give a detailed bound for |$H^{s}$|-Sobolev norms of the error due to Fourier spectral discretization.

A sparse grid method for the Navier-Stokes equations based on hyperbolic cross

A sparse grid method for the time-dependent Navier–Stokes equations based on hyperbolic cross approximation is considered in this article. Subsequent truncation of the associated series expansion results in a sparse grid discretization. Stability and convergence of the fully discrete sparse grid method are established. Finally, the numerical experiment is presented to show the effectiveness of this sparse grid method. Copyright © 2013 John Wiley & Sons, Ltd.

N-Widths and ε-Dimensions for High-Dimensional Approximations

In this paper, we study linear trigonometric hyperbolic cross approximations, Kolmogorov n-widths d n (W,H γ ), and e-dimensions n e (W,H γ ) of periodic d-variate function classes W with anisotropic smoothness, where d may be large. We are interested in finding the accurate dependence of d n (W,H γ ) and n e (W,H γ ) as a function of two variables n, d and e, d, respectively. Recall that n, the dimension of the approximating subspace, is the main parameter in the study of convergence rates with respect to n going to infinity. However, the parameter d may seriously affect this rate when d is large. We construct linear approximations of functions from W by trigonometric polynomials with frequencies from hyperbolic crosses and prove upper bounds for the error measured in isotropic Sobolev spaces H γ . Furthermore, in order to show the optimality of the proposed approximation, we prove upper and lower bounds of the corresponding n-widths d n (W,H γ ) and e-dimensions n e (W,H γ ). Some of the received results imply that the curse of dimensionality can be broken in some relevant situations.

Hermite Spectral Method with Hyperbolic Cross Approximations to High-Dimensional Parabolic PDEs

It is well-known that sparse grid algorithm has been widely accepted as an efficient tool to overcome the "curse of dimensionality" in some degree. In this note, we first give the error estimate of hyperbolic cross (HC) approximations with generalized Hermite functions. The exponential convergence in both regular and optimized hyperbolic cross approximations has been shown. Moreover, the error estimate of Hermite spectral method to high-dimensional linear parabolic PDEs with HC approximations has been investigated in the properly weighted Korobov spaces. The numerical result verifies the exponential convergence of this approach.

Sparse Spectral Approximations of High-Dimensional Problems Based on Hyperbolic Cross

Hyperbolic cross approximations by some classical orthogonal polynomials/functions in both bounded and unbounded domains are considered in this paper. Optimal error estimates in proper anisotropic weighted Korobov spaces for both regular hyperbolic cross approximations and optimized hyperbolic cross approximations are established. These fundamental approximation results indicate that spectral methods based on hyperbolic cross approximations can be effective for treating certain high-dimensional problems and will serve as basic tools for analyzing sparse spectral methods in high dimensions.

Nonequispaced Hyperbolic Cross Fast Fourier Transform

A straightforward discretization of problems in $d$ spatial dimensions often leads to an exponential growth in the number of degrees of freedom. Thus, even efficient algorithms like the fast Fourier transform (FFT) have high computational costs. Hyperbolic cross approximations allow for a severe decrease in the number of used Fourier coefficients to represent functions with bounded mixed derivatives. We propose a nonequispaced hyperbolic cross FFT based on one hyperbolic cross FFT and a dedicated interpolation by splines on sparse grids. Analogously to the nonequispaced FFT for trigonometric polynomials with Fourier coefficients supported on the full grid, this allows for the efficient evaluation of trigonometric polynomials with Fourier coefficients supported on the hyperbolic cross at arbitrary spatial sampling nodes.

Tensor products of Sobolev–Besov spaces and applications to approximation from the hyperbolic cross

Besov as well as Sobolev spaces of dominating mixed smoothness are shown to be tensor products of Besov and Sobolev spaces defined on R . Using this we derive several useful characterizations from the one-dimensional case to the d -dimensional situation. Finally, consequences for hyperbolic cross approximations, in particular for tensor product splines, are discussed.

Computing Semiclassical Quantum Dynamics with Hagedorn Wavepackets

We consider the approximation of multiparticle quantum dynamics in the semiclassical regime by Hagedorn wavepackets, which are products of complex Gaussians with polynomials that form an orthonormal $L^2$ basis and preserve their type under propagation in Schrödinger equations with quadratic potentials. We build a fully explicit, time-reversible time-stepping algorithm to approximate the solution of the Hagedorn wavepacket dynamics. The algorithm is based on a splitting between the kinetic and potential part of the Hamiltonian operator, as well as on a splitting of the potential into its local quadratic approximation and the remainder. The algorithm is robust in the semiclassical limit. It reduces to the Strang splitting of the Schrödinger equation in the limit of the full basis set, and it advances positions and momenta by the Störmer–Verlet method for the classical equations of motion. The algorithm allows for the treatment of multiparticle problems by thinning out the basis according to a hyperbolic cross approximation and of high-dimensional problems by Hartree-type approximations in a moving coordinate frame.

A periodic wavelet method for the second kind of the logarithmic integral equation

A periodic wavelet Galerkin method is presented in this paper to solve a weakly singular integral equations with emphasis on the second kind of Fredholm integral equations. The kernel function, which includes of a smooth part and a log weakly singular part, is discretised by the periodic Daubechies wavelets. The wavelet compression strategy and the hyperbolic cross approximation technique are used to approximate the weakly singular and smooth kernel functions. Meanwhile, the sparse matrix of systems can be correspondingly obtained. The bi-conjugate gradient iterative method is used to solve the resulting algebraic equation systems. Especially, the analytical computational formulae are presented for the log weakly singular kernel. The computational error for the representative matrix is also evaluated. The convergence rate of this algorithm is O (N-p log(N)), where p is the vanishing moment of the periodic Daubechies wavelets. Numerical experiments are provided to illustrate the correctness of the theory presented here.

Integral Operators on Sparse Grids

In this paper we are concerned with the construction and use of wavelet approximation spaces for the fast evaluation of integral expressions. The spaces are based on biorthogonal anisotropic tensor product wavelets. We introduce sparse grid (hyperbolic cross) approximation spaces which are adapted not only to the smoothness of the kernel but also to the norm in which the error is measured. Furthermore, we introduce compression schemes for the corresponding discretizations. Numerical examples for the Laplace equation with Dirichlet boundary conditions and an additional integral term with a smooth kernel demonstrate the validity of our theoretical results.

Hyperbolic Wavelet Approximation

We study the multivariate approximation by certain partial sums (hyperbolic wavelet sums) of wavelet bases formed by tensor products of univariate wavelets. We characterize spaces of functions which have a prescribed approximation error by hyperbolic wavelet sums in terms of a K -functional and interpolation spaces. The results parallel those for hyperbolic trigonometric cross approximation of periodic functions [DPT].

Information Complexity of Multivariate Fredholm Integral Equations in Sobolev Classes

In this paper, the complexity of full solution of Fredholm integral equations of the second kind with data from the Sobolev classWr2is studied. The exact order of information complexity is derived. The lower bound is proved using a Gelfand number technique. The upper bound is shown by providing a concrete algorithm of optimal order, based on a specific hyperbolic cross approximation of the kernel function. Numerical experiments are included, comparing the optimal algorithm with the standard Galerkin method.

Complexity of Local Solution of Multivariate Integral Equations

In this paper the complexity of the local solution of Fredholm integral equations is studied. For certain Sobolev classes of multivariate periodic functions with dominating mixed derivative we prove matching lower and upper hounds. The lower bound is shown using relations to s-numbers. The upper hound is proved in a constructive way providing an implementable algorithm of optimal order based on Fourier coefficients and a hyperbolic cross approximation.

Lower and Upper Bounds for the Number of Lattice Points in a Simplex

Previous article Next article Lower and Upper Bounds for the Number of Lattice Points in a SimplexAharon Gavriel Beged-dovAharon Gavriel Beged-dovhttps://doi.org/10.1137/0122012PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] A. G. Beged-Dov, Some computational aspects of the P printers M paper mills trim problem, Business Administration, 2 (1970), 15–34 Google Scholar[2] P. C. Gilmore and , R. E. Gomory, A linear programming approach to the cutting-stock problem, Operations Res., 9 (1961), 849–859 MR0137589 0096.35501 CrossrefISIGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Homotopy techniques for solving sparse column support determinantal polynomial systemsJournal of Complexity, Vol. 66 Cross Ref Percolating sets in bootstrap percolation on the Hamming graphs and triangular graphsEuropean Journal of Combinatorics, Vol. 92 Cross Ref Polynomial Approximation of Anisotropic Analytic Functions of Several Variables16 July 2020 | Constructive Approximation, Vol. 51 Cross Ref Greedy approximations by signed harmonic sums and the Thue–Morse sequenceAdvances in Mathematics, Vol. 366 Cross Ref Sampling inequalities for anisotropic tensor product grids8 January 2019 | IMA Journal of Numerical Analysis, Vol. 107 Cross Ref Novel results for the anisotropic sparse grid quadratureJournal of Complexity, Vol. 47 Cross Ref On tensor product approximation of analytic functionsJournal of Approximation Theory, Vol. 207 Cross Ref Hyperbolic cross approximation in infinite dimensionsJournal of Complexity, Vol. 33 Cross Ref Rethinking the encounter probability for direct-to-target nuclear attacks for aviation security5 May 2011 | Journal of Transportation Security, Vol. 4, No. 3 Cross Ref Эффективные оценки производных обратной функцииИзвестия Российской академии наук. Серия математическая, Vol. 72, No. 4 Cross Ref Improved lower and upper bounds for the number of feasible solutions to a knapsaek problemOptimization, Vol. 19, No. 6 Cross Ref APPLICATION OF OPERATIONS RESEARCH IN PAPER PROCUREMENT Cross Ref The Number of Feasible Solutions to a Knapsack ProblemO. Achou12 July 2006 | SIAM Journal on Applied Mathematics, Vol. 27, No. 4AbstractPDF (209 KB)Bounds on the Number of Feasible Solutions to a Knapsack ProblemThomas A. Lambe12 July 2006 | SIAM Journal on Applied Mathematics, Vol. 26, No. 2AbstractPDF (229 KB)The Paper Trim Problem: A Variable Demand AnalysisA I I E Transactions, Vol. 6, No. 1 Cross Ref A Remark on “an Inequality for the Number of Lattice Points in a Simplex”Manfred W. Padberg12 July 2006 | SIAM Journal on Applied Mathematics, Vol. 20, No. 4AbstractPDF (253 KB) Volume 22, Issue 1| 1972SIAM Journal on Applied Mathematics History Submitted:09 December 1969Published online:12 July 2006 InformationCopyright © 1972 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0122012Article page range:pp. 106-108ISSN (print):0036-1399ISSN (online):1095-712XPublisher:Society for Industrial and Applied Mathematics