Hyperbolic Cross Research Articles

Adaptive hyperbolic-cross-space mapped Jacobi method on unbounded domains with applications to solving multidimensional spatiotemporal integrodifferential equations

In this paper, we develop a new adaptive hyperbolic-cross-space mapped Jacobi (AHMJ) method for solving multidimensional spatiotemporal integrodifferential equations in unbounded domains. By devising adaptive techniques for sparse mapped Jacobi spectral expansions defined in a hyperbolic cross space, our proposed AHMJ method can efficiently solve various spatiotemporal integrodifferential equations such as the anomalous diffusion model with reduced numbers of basis functions. Our analysis of the AHMJ method gives a uniform upper error bound for solving a class of spatiotemporal integrodifferential equations, leading to effective error control.

Exploiting relation of video segments for temporal action detection

With the development of urban intelligence, intelligent identification of human activities has become imminent. Temporal Action Detection (TAD) is designed to identify real human social activities, which is a challenging task in video understanding. The current methods mainly use global features for boundary matching or predefine all possible proposals, while ignoring the interference of background and the causal relevance of the temporal action, resulting in the generation of more redundant proposals and the decline of detection accuracy. To fill this gap, we propose a novel Dilated Convolution Locate and Action Relevant Score model called DCAR. Specifically, DCAR includes a Dilated Location Network (DL-Net) and Action Relevance Calculation (ARC) block. For the DL-Net, we design a Boundary Feature Enhancement (BFE) block, which enhances the boundary feature of actions and fuses the similar features of different channels by pooling and channel squeeze to reduce the interference of the background. Meanwhile, we also design multiple dilated convolutional structures to aggregate long contextual information in time point/interval after boundary enhancement. For the ARC block, we use the hyperbolic space distance and cross attention to calculate the causal correlation of action proposals classification, which removes the misclassification of action proposals. We conduct extensive experiments on Thumos14 and ActivityNet-1.3, which shows our method significantly improves the performance and achieves state-of-the-art results. On Thumos14, at tIoU 0.3, 0.4 and 0.5, it achieves mAP of 63.2%, 57.0% and 48.5%, respectively; On ActivityNet-1.3 it reaches 9.42% at tIoU 0.95.

Using a hyperbolic cross to solve non-linear macroeconomic models

The paper presents a sparse grid approximation method based on the hyperbolic cross and applies it to solve non-linear macroeconomic models. We show how the standard hyperbolic cross can be extended to give greater control over the approximating grid and we discuss how to implement an anisotropic hyperbolic cross. Applying the approximation method to four macroeconomic models, we establish that it delivers a level of accuracy on par or better than Smolyak's method and that it can produce accurate approximations using fewer points than Smolyak's method.

On the mixed regularity of N-body Coulombic wavefunctions

In this paper, we prove a new mixed regularity of Coulombic wavefunction taking into account the Pauli exclusion principle. We also study the hyperbolic cross space approximation of eigenfunctions associated with this new regularity, and deduce the corresponding error estimates in L2-norm and H1-semi-norm. The proofs are based on the study of extended Hardy-type inequalities for Coulomb-type potentials.

Fast hyperbolic wavelet regression meets ANOVA

We use hyperbolic wavelet regression for the fast reconstruction of high-dimensional functions having only low dimensional variable interactions. Compactly supported periodic Chui–Wang wavelets are used for the tensorized hyperbolic wavelet basis. In a first step we give a self-contained characterization of tensor product Sobolev–Besov spaces on the d-torus with arbitrary smoothness in terms of the decay of such wavelet coefficients. In the second part we perform and analyze scattered-data approximation using a hyperbolic cross type truncation of the basis expansion for the associated least squares method. The corresponding system matrix is sparse due to the compact support of the wavelets, which leads to a significant acceleration of the matrix vector multiplication. In case of i.i.d. samples we can even bound the approximation error with high probability by loosing only log -terms that do not depend on d compared to the best approximation. In addition, if the function has low effective dimension (i.e. only interactions of few variables), we qualitatively determine the variable interactions and omit ANOVA terms with low variance in a second step in order to increase the accuracy. This allows us to suggest an adapted model for the approximation. Numerical results show the efficiency of the proposed method.

Numerical weighted integration of functions having mixed smoothness

We investigate the approximation of weighted integrals over Rd for integrands from weighted Sobolev spaces of mixed smoothness. We prove upper and lower bounds of the convergence rate of optimal quadratures with respect to n integration nodes for functions from these spaces. In the one-dimensional case (d=1), we obtain the right convergence rate of optimal quadratures. For d≥2, the upper bound is performed by sparse-grid quadratures with integration nodes on step hyperbolic crosses in the function domain Rd.

Transformer's frequency response analysis results interpretation using a novel cross entropy based methodology

Transformer defects can be identified by the FRA (frequency response analysis) that is a promising diagnostic technique. Despite the standardization in FRA measuring technique, its results interpretation is yet a research area. Because different faults types can be identified in various frequency bounds of the FRA signatures, it is necessary to identify the possible relationships between specific failures and frequency ranges in this contribution. For this purpose, a real transformer is used to conduct the essential tests, which include both healthy and faulted circumstances (axial displacement (AD), radial deformation (RD), and short-circuits (SC)). To identify efficient characteristics from the produced frequency response traces and improve interpretation accuracy of such traces, a new hyperbolic fuzzy cross entropy (FCE) measure is demonstrated and then utilized for the aim of discrimination and classification of transformer winding defects in pre-defined frequency ranges. After normalizing FRA results of the transformer under healthy and various fault circumstances the lower bounds from such responses have been extracted and then utilized to construct the desired form of the fuzzy sets of healthy and faulted circumstances. Then, a new hyperbolic FCE measure-based discrimination and classification of winding faults methodology is offered on the basis of highest and lowest FCE measure values. The highest FCE measure value between the fuzzy sets of healthy and faulted circumstances such as AD, RD and SC is designated to confirm the occurrence of winding faults in a suitable frequency range. The suggested methodology ensures smart interpretation of FRA signature and accurate classification of winding faults as it can effectively discriminate both healthy and faulted circumstances in the desired frequency ranges. The proposed approaches' performance is tested and compared by applying the experimental data after feature extraction.

On computing high-dimensional Riemann theta functions

Riemann theta functions play a crucial role in the field of nonlinear Fourier analysis, where they are used to realize inverse nonlinear Fourier transforms for periodic signals. The practical applicability of this approach has however been limited since Riemann theta functions are multi-dimensional Fourier series whose computation suffers from the curse of dimensionality. In this paper, we investigate several new approaches to compute Riemann theta functions with the goal of unlocking their practical potential. Our first contributions are novel theoretical lower and upper bounds on the series truncation error. These bounds allow us to rule out several of the existing approaches for the high-dimension regime. We then propose to consider low-rank tensor and hyperbolic cross based techniques. We first examine a tensor-train based algorithm which utilizes the popular scaling and squaring approach. We show theoretically that this approach cannot break the curse of dimensionality. Finally, we investigate two other tensor-train based methods numerically and compare them to hyperbolic cross based methods. Using finite-genus solutions of the Korteweg–de Vries (KdV) and nonlinear Schrödinger equation (NLS) equations, we demonstrate the accuracy of the proposed algorithms. The tensor-train based algorithms are shown to work well for low genus solutions with real arguments but are limited by memory for higher genera. The hyperbolic cross based algorithm also achieves high accuracy for low genus solutions. Its novelty is the ability to feasibly compute moderately accurate solutions (a relative error of magnitude 0.01) for high dimensions (up to 60). It therefore enables the computation of complex inverse nonlinear Fourier transforms that were so far out of reach.

Effect of magnetic induction on EG-water-based composite nanofluid flow across an elongated region: a KB approach

This study investigates the result of magnetic induction on Ethylene Glycol-water (EG-water) based tangent hyperbolic cross nanoliquid flow above a porous stretched region in the existence of thermic energy. Due to effective thermal conductivity, we considered Zirconium dioxide (ZrO2) and Graphene oxide (GO) nanoparticles and embedded them in the EG-water combination (50%–50%). A theoretical model was established and resolved numerically using the Keller-Box approach. The outcomes are executed in the existence and absence of magnetic induction and exemplified graphically to understand the behavior of flow and thermal areas. The magnetic induction Prandtl number inclines to enlarge the heat diffusion rate of hyperbolic-tangent cross nano liquid in the occurrence of the magnetic induction. The heat transmission rate of tangent hyperbolic hybrid nanofluids is high in the existence of magnetic induction when related to its absence. Weissenberg numbers and permeability are controlling factors for wall friction. The electromagnetic field assists in regulating the thermal and momentum fields.

Approximation of functions with small mixed smoothness in the uniform norm

In this paper we present results on asymptotic characteristics of multivariate function classes in the uniform norm. Our main interest is the approximation of functions with mixed smoothness parameter not larger than 1/2. Our focus will be on the behavior of the best m-term trigonometric approximation as well as the decay of Kolmogorov and entropy numbers in the uniform norm. It turns out that these quantities share a few fundamental abstract properties like their behavior under real interpolation, such that they can be treated simultaneously. We start with proving estimates on finite rank convolution operators with range in a step hyperbolic cross. These results imply bounds for the corresponding function space embeddings by a well-known decomposition technique. The decay of Kolmogorov numbers have direct implications for the problem of sampling recovery in L2 in situations where recent results in the literature are not applicable since the corresponding approximation numbers are not square summable.

Using a Hyperbolic Cross to Solve Non-Linear Macroeconomic Models

Using a Hyperbolic Cross to Solve Non-Linear Macroeconomic Models

Bounds on Kolmogorov widths and sampling recovery for classes with small mixed smoothness

Results on asymptotic characteristics of classes of functions with mixed smoothness are obtained in the paper. Our main interest is in estimating the Kolmogorov widths in the uniform norm of classes with small mixed smoothness. We prove the corresponding bounds for the unit balls of the trigonometric polynomials with frequencies from a hyperbolic cross. We demonstrate how our results on the Kolmogorov widths imply new upper bounds for the optimal sampling recovery in the L2 norm of functions with small mixed smoothness.

Sampling Discretization of Integral Norms

The paper is devoted to discretization of integral norms of functions from a given finite dimensional subspace. Even though this problem is extremely important in applications, its systematic study has begun recently. In this paper we obtain a conditional theorem for all integral norms $L_q$, $1\le q<\infty$, which is an extension of known results for $q=1$. To discretize the integral norms successfully, we introduce a new technique, which is a combination of probabilistic technique with results on the entropy numbers in the uniform norm. As an application of the general conditional theorem, we derive a new Marcinkiewicz type discretization for the multivariate trigonometric polynomials with frequencies from the hyperbolic crosses.

Static and free vibration analyses of moderately thick hyperbolic paraboloidal cross ply laminated composite shell structure

The main objective of this paper is to investigate the response behaviour of moderately thick hyperbolic paraboloidal laminated shells using C0 finite element formulations based on Higher order Shear Deformation Theory (HSDT). The mathematical model of HSDT includes the thickness co-ordinate to radius of curvature ratio (z/R) as well as zero shear conditions at free surfaces of the shell. The finite element equations have been developed using eight noded isoparametric shell element with seven degrees of freedom per node. The Jacobian matrix used in the finite element procedure is formulated by considering the curvilinear coordinates of the shell element. The accuracy of the present HSDT is validated by comparing the obtained results with those available published 2D and 3D elasticity solutions. An extensive parametric study on static and free vibration analyses of moderately thick hyperbolic paraboloidal laminated shells has been carried out to study the performance of shell for various geometric configurations.

Sampling Discretization of Integral Norms of the Hyperbolic Cross Polynomials

The paper is devoted to discretization of integral norms of functions from a given finite-dimensional subspace. We use recent general results on sampling discretization to derive a new Marcinkiewicz type discretization theorem for the multivariate trigonometric polynomials with frequencies from the hyperbolic crosses. It is shown that recently developed techniques allow us to improve the known results in this direction.

Using a hyperbolic cross to solve non-linear macroeconomic models

Using a hyperbolic cross to solve non-linear macroeconomic models

Approximative characteristics and properties of operators of the best approximation of classes of functions from the Sobolev and Nikol’skii–Besov spaces

We have obtained the exact-order estimates for some approximative characteristics of the Sobolev classes $$ {\mathbbm{W}}_{p,\alpha}^r $$ and Nikоl’skii–Besov classes $$ {\mathbbm{B}}_{p,\theta}^r $$ of periodic functions of one and several variables in the norm of the space B∞,1. Properties of the linear operators realizing the orders of the best approximation for the classes $$ {\mathbbm{B}}_{\infty, \theta}^r $$ in this space by trigonometric polynomials generated by a set of harmonics with “numbers” from step hyperbolic crosses are investigated.

A Fourier‐based machine learning technique with application in engineering

AbstractThe generic problem in supervised machine learning is to learn a function f from a collection of samples, with the objective of predicting the value taken by f for any given input. In effect, the learning procedure consists in constructing an explicit function that approximates f in some sense. In this article is introduced a Fourier‐based machine learning method which could be an alternative or a complement to neural networks for applications in engineering. The basic idea is to extend f into a periodic function so as to use partial sums of the Fourier series as approximations. For this approach to be effective in high dimension, it proved necessary to use several ideas and concepts such as regularization, Sobol sequences and hyperbolic crosses. An attractive feature of the proposed method is that the training stage reduces to a quadratic programming problem. The presented method is first applied to some examples of high‐dimensional analytical functions, which allows some comparisons with neural networks to be made. An application to a homogenization problem in nonlinear conduction is discussed in detail. Various examples related to global sensitivity analysis, assessing effective energies of microstructures, and solving boundary value problems are presented.

Approximative characteristics and properties of operators of the best approximation of classes of functions from the Sobolev and Nikol'skii-Besov spaces

We have obtained the exact-order estimates for some approximative characteristics of the Sobolev classes $\mathbb{W}^{\boldsymbol{r}}_{p,\boldsymbol{\alpha}}$ and Nikоl'skii--Besov classes $\mathbb{B}^{\boldsymbol{r}}_{p,\theta}\ $ of periodic functions of one and several variables in the norm of the space $B_{\infty, 1}$. Properties of the linear operators realizing the orders of the best approximation for the classes $\mathbb{B}^{\boldsymbol{r}}_{\infty, \theta}$ in this space by trigonometric polynomials generated by a set of harmonics with $``$numbers$"$ from step hyperbolic crosses are investigated.

Fourier restriction to a hyperbolic cone

Using a bilinear restriction theorem of Lee and a bilinear-to-linear argument of Stovall, we obtain the conjectured range of Fourier restriction estimates for a conical hypersurface in R4 with hyperbolic cross sections.