A fast and reliable algorithm for the optimal interpolation of scattered data on the torus by multivariate trigonometric polynomials is presented. The algorithm is based on a variant of the conjugate gradient method in combination with the fast Fourier transforms for nonequispaced nodes. The main result is that under mild assumptions the total complexity for solving the interpolation problem at M arbitrary nodes is of order O(M logM). This result is obtained by the use of localised trigonometric kernels where the localisation is chosen in accordance to the spatial dimension d. Numerical examples show the efficiency of the new algorithm.
The class $Ts(r,f)$ the trigonometric interpolation splines depending on the parameter vectors, selected convergence factors and interpolation factors is considered. The main properties of simple interpolation trigonometric splines are given, which are also transferred to periodic simple interpolation polynomial splines. These results lead to the possibility of combining the theory of simple polynomial interpolation splines and the basics of the theory of simple trigonometric splines into a single theory - the theory of interpolation splines
The paper deals with two fundamental types of trigonometric polynomials and splines on uniform grids, which allow us to construct interpolation approximations that depend linearly on the values of the interpolated function. Fundamental on the same grids are trigonometric LS polynomials and LS splines, which allow us to construct the approximation of functions by the least squares method and also depend linearly on the values of the approximate function. Trigonometric splines were considered as splines with polynomial analogues in some cases. The material is illustrated in graphs. The considere…
A class of trigonometric interpolation splines depending on parameter vectors, selected convergence factors and interpolation factors is considered. The concept of crosslink grids and interpolation grids is introduced; these grids can match or y not match. Parseval equality is given and some properties of splines are discussed. The material is illustrated by a numerical example.
One of the possible variants of the classification of trigonometric interpolation splines is considered, depending on the chosen convergence factors, the distribution of signs of the basis functions and the interpolation factors. The concept of crosslinking and interpolation grids is introduced; these grids can either match or not match. The proposed classification is illustrated by an example.
Interpolation of classes of differentiated functions given on a finite interval by trigonometric splines using the phantom node method is considered. This method consists in supplementing a given sequence of values of an approximate function with an even number of values of a phantom function, which is constructed in such a way as to eliminate gaps in both the function itself and its derivatives up to and including a certain order; in the General case, these gaps occur with the periodic continuation of the function given at a finite interval. The results of calculations on test examples for tr…
Complex signed measures of finite total variation are a powerful signal model in many applications. Restricting to the $d$-dimensional torus, finitely supported measures allow for exact recovery if the trigonometric moments up to some order are known. Here, we consider the approximation of general measures, e.g., supported on a curve, by trigonometric polynomials of fixed degree with respect to the Wasserstein-1 distance. We prove sharp lower bounds for their best approximation and (almost) matching upper bounds for effectively computable approximations when the trigonometric moments of the me…
Under consideration methods of constructing trigonometric interpolation splines of two variables on rectangular areas. These methods are easily generalized to the case of trigonometric interpolation splines of several variables on such domains. A numerical example illustrating the main theoretical propositions is considered. The given methods of constructing such splines can be widely used in practice.
We present a method for dimensionally adaptive sparse trigonometric interpolation of multidimensional periodic functions belonging to a smoothness class of finite order. This method targets applications where periodicity must be preserved and the precise anisotropy is not known a priori. To the authors' knowledge, this is the first instance of a dimensionally adaptive sparse interpolation algorithm that uses a trigonometric interpolation basis. The motivating application behind this work is the adaptive approximation of a multi-input model for a molecular potential energy surface (PES) where e…
In this article, we use the knowledge of positive definite tensors to develop a concept of positive definite multi-kernels to construct the kernel-based interpolants of scattered data. By the techniques of reproducing kernel Banach spaces, the optimal recoveries and error analysis of the kernel-based interpolants are shown for a special class of strictly positive definite multi-kernels.
We discuss a pointwise numerical differentiation formula on multivariate scattered data, based on the coefficients of local polynomial interpolation at Discrete Leja Points, written in Taylor's formula monomial basis. Error bounds for the approximation of partial derivatives of any order compatible with the function regularity are provided, as well as sensitivity estimates to functional perturbations, in terms of the inverse Vandermonde coefficients that are active in the differentiation process. Several numerical tests are presented showing the accuracy of the approximation.
Hybrid Gaussian-cubic radial basis functions for scattered data interpolation
Indian Institute of Technology Kharagpur（印度理工学院卡拉格普尔分校）；Hong Kong Baptist University（香港浸会大学）；University of Texas at Austin（德克萨斯大学奥斯汀分校）；Colorado School of Mines（科罗拉多矿业学院）
Spherical Gauss-Laguerre (SGL) basis functions, i. e., normalized functions of the type $L_{n-l-1}^{(l + 1/2)}(r^2) r^l Y_{lm}(\vartheta,φ)$, $|m| \leq l < n \in \mathbb{N}$, $L_{n-l-1}^{(l + 1/2)}$ being a generalized Laguerre polynomial, $Y_{lm}$ a spherical harmonic, constitute an orthonormal polynomial basis of the space $L^2$ on $\mathbb{R}^3$ with radial Gaussian (multivariate Hermite) weight $\exp(-r^2)$. We have recently described fast Fourier transforms for the SGL basis functions based on an exact quadrature formula with certain grid points in $\mathbb{R}^3$. In this paper, we presen…
Data sites selected from modeling high-dimensional problems often appear scattered in non-paternalistic ways. Except for sporadic clustering at some spots, they become relatively far apart as the dimension of the ambient space grows. These features defy any theoretical treatment that requires local or global quasi-uniformity of distribution of data sites. Incorporating a recently-developed application of integral operator theory in machine learning, we propose and study in the current article a new framework to analyze kernel interpolation of high dimensional data, which features bounding stoc…
The method of constructing trigonometric Hermite splines, which interpolate the values of some periodic function and its derivatives in the nodes of a uniform grid, is considered. The proposed method is based on the periodicity properties of trigonometric functions and is reduced to solving only systems of linear algebraic equations of the second order; solutions of these systems can be obtained in advance. When implementing this method, it is necessary to calculate the coefficients of interpolation trigonometric polynomials that interpolate the values of the function itself and the values of …
With the rapid growth of data, how to extract effective information from data is one of the most fundamental problems. In this paper, based on Tikhonov regularization, we propose an effective method for reconstructing the function and its derivative from scattered data with random noise. Since the noise level is not assumed small, we will use the amount of data for reducing the random error, and use a relatively small number of knots for interpolation. An indicator function for our algorithm is constructed. It indicates where the numerical results are good or may not be good. The corresponding…
In this paper we describe a parallel implicit method based on radial basis functions (RBF) for surface reconstruction. The applicability of RBF methods is hindered by its computational demand, that requires the solution of linear systems of size equal to the number of data points. Our reconstruction implementation relies on parallel scientific libraries and is supported for massively multi-core architectures, namely Graphic Processor Units (GPUs). The performance of the proposed method in terms of accuracy of the reconstruction and computing time shows that the RBF interpolant can be very effe…
In this present paper, I propose a derivation of unified interpolation and extrapolation function that predicts new values inside and outside the given range by expanding direct Taylor series on the middle point of given data set. Mathematical construction of experimental model derived in general form. Trigonometric and Power functions adopted as test functions in the development of the vital aspects in numerical experiments. Experimental model was interpolated and extrapolated on data set that generated by test functions. The results of the numerical experiments which predicted by derived mod…
The paper is devoted to the construction of an optimal interpolation formula in $K_2(P_2)$ Hilbert space. Here the interpolation formula consists of a linear combination $\sum_{β=0}^NC_β(z)φ(x_β)$ of given values of a function $φ$ from the space $K_2(P_2)$. The difference between functions and the interpolation formula is considered as a linear functional called the error functional. The error of the interpolation formula is estimated by the norm of the error functional. We obtain the optimal interpolation formula by minimizing the norm of the error functional by coefficients $C_β(z)$ of the i…
Institut für Numerische Simulation, Universität Bonn（波恩大学数值模拟研究所）；Fraunhofer Institute for Algorithms and Scientific Computing (SCAI)（弗劳恩霍夫算法与科学计算研究所）；Departement Mathematik und Informatik, Universität Basel（巴塞尔大学数学与信息学系）；Istituto Dalle Molle di studi sull’intelligenza artificiale, Università della Svizzera italiana（意大利瑞士大学达莱莫勒人工智能研究学院）
We study approximation properties of general multivariate periodic quasi-interpolation operators, which are generated by distributions/functions $\widetildeφ_j$ and trigonometric polynomials $φ_j$. The class of such operators includes classical interpolation polynomials ($\widetildeφ_j$ is the Dirac delta function), Kantorovich-type operators ($\widetildeφ_j$ is a characteristic function), scaling expansions associated with wavelet constructions, and others. Under different compatibility conditions on $\widetildeφ_j$ and $φ_j$, we obtain upper and lower bound estimates for the $L_p$-error of a…
