Padua Points Research Articles

Padua points and fake nodes for polynomial approximation: old, new and open problems

Padua points, discovered in 2005 at the University of Padua, are the first set of points on the square [−1,1]2[−1,1]2 that are explicitly known, unisolvent for total degree polynomial interpolation and with Lebesgue constant increasing like log2(n)log2⁡(n) of the degree. One of the key features of the Padua points is that they lie on a particular Lissajous curve. Other important properties of Padua points arein two dimensions, Padua points are a WAM for interpolation and for extracting approximate Fekete points and discrete Leja sequences.in three dimensions, Padua points can be used for constructing tensor product WAMs on different compacts.Unfortunately, their extension to higher dimensions is still the biggest open problem. The concept of mapped bases has been widely studied (cf. e.g. [35] and references therein), which turns out to be equivalent to map the interpolating nodes and then construct the approximant in the classical form without the need of resampling. The mapping technique is general, in the sense that works with any basis and can be applied to continuous, piecewise or discontinuous functions or even images. All the proposed methods show convergence to the interpolant provided that the function is resampled at the mapped nodes. In applications, this is often physically unfeasible. An effective method for interpolating via mapped bases in the multivariate setting, referred as Fake Nodes Approach (FNA), has been presented in [37]. In this paper, some interesting connection of the FNA with Padua points and families of relatives nodes, that can be used as fake nodes for multivariate approximation, are presented and we conclude with some open problems.

Evaluating RBF methods for solving PDEs using Padua points distribution

In Radial Basis Functions (RBFs) methods, the distribution and the number of points have a significant effect on the accuracy, stability, and computational costs. Padua distribution is known as optimal points in the interpolation of Lagrange polynomials in a square-shaped computational domain and two-dimensional space. This point distribution was applied in rectangular and circular domains using mapping. In this research, for the first time, Padua points were used in three-dimensional space. The results of this distribution were compared with other distributions such as Fibonacci, points on vertices of a triangular mesh, points on regular grid plus extra points on boundaries, Halton, and Sobol and their combinations. Padua points were used to solve Partial Differential Equations (PDEs) using the RBF methods and Compactly Supported-Radial Basis Functions (CS-RBFs) and their combinations. Numerical results showed that Padua distribution in most of the cases with the lowest number of center points obtained the highest accuracy. Furthermore, Padua points in combination with other distributions increase the accuracy. The remarkable result of this paper is that these points can be used in higher dimensions and got accurate results.

Quantitative Tomography for Continuous Variable Quantum Systems.

We present a continuous variable tomography scheme that reconstructs the Husimi Q function (Wigner function) by Lagrange interpolation, using measurements of the Q function (Wigner function) at the Padua points, conjectured to be optimal sampling points for two dimensional reconstruction. Our approach drastically reduces the number of measurements required compared to using equidistant points on a regular grid, although reanalysis of such experiments is possible. The reconstruction algorithm produces a reconstructed function with exponentially decreasing error and quasilinear runtime in the number of Padua points. Moreover, using the interpolating polynomial of the Q function, we present a technique to directly estimate the density matrix elements of the continuous variable state, with only a linear propagation of input measurement error. Furthermore, we derive a state-independent analytical bound on this error, such that our estimate of the density matrix is accompanied by a measure of its uncertainty.

Polynomial interpolation on interlacingrectangular grids

In this paper we establish the unisolvence of any interlacing pair of rectangular grids of points with respect to a large class of associated polynomial spaces. This includes interpolation on Padua points and the schemes of Morrow and Patterson (1978), Xu (1996), and Erb et al. (2016). We use Newton polynomials and divided differences and an apparently new formula for determinants of certain divided difference matrices.

Multivariate polynomial interpolation on Lissajous–Chebyshev nodes

In this article, we study multivariate polynomial interpolation and quadrature rules on non-tensor product node sets related to Lissajous curves and Chebyshev varieties. After classifying multivariate Lissajous curves and the interpolation nodes linked to these curves, we derive a discrete orthogonality structure on these node sets. Using this orthogonality structure, we obtain unique polynomial interpolation in appropriately defined spaces of multivariate Chebyshev polynomials. Our results generalize corresponding interpolation and quadrature results for the Chebyshev–Gauß–Lobatto points in dimension one and the Padua points in dimension two.

Bivariate Lagrange interpolation at the node points of Lissajous curves – the degenerate case

In this article, we study bivariate polynomial interpolation on the node points of degenerate Lissajous figures. These node points form Chebyshev lattices of rank 1 and are generalizations of the well-known Padua points. We show that these node points allow unique interpolation in appropriately defined spaces of polynomials and give explicit formulas for the Lagrange basis polynomials. Further, we prove mean and uniform convergence of the interpolating schemes. For the uniform convergence the growth of the Lebesgue constant has to be taken into consideration. It turns out that this growth is of logarithmic nature.

Bivariate Lagrange interpolation at the node points of non-degenerate Lissajous curves

Motivated by an application in Magnetic Particle Imaging, we study bivariate Lagrange interpolation at the node points of Lissajous curves. The resulting theory is a generalization of the polynomial interpolation theory developed for a node set known as Padua points. With appropriately defined polynomial spaces, we will show that the node points of non-degenerate Lissajous curves allow unique interpolation and can be used for quadrature rules in the bivariate setting. An explicit formula for the Lagrange polynomials allows to compute the interpolating polynomial with a simple algorithmic scheme. Compared to the already established schemes of the Padua and Xu points, the numerical results for the proposed scheme show similar approximation errors and a similar growth of the Lebesgue constant.

Fast Calculation of Scattering by 3-D Inhomogeneities in Uniaxial Anisotropic Multilayers

A fast algorithm to calculate the response of uniaxial layered media on a rectilinear mesh is proposed in order to handle scattering phenomena due to inhomogeneities within them. The uniaxial layered media are with their optical axes parallel to the interfaces of the layered structure. The generalized Poisson summation formula is used to efficiently calculate the discrete Fourier transform of spatial responses of the layeredmedia from its continuous Fourier spectrum. Using the windowing technique, the integral involved in constructing the discrete Fourier spectrum can be performed in a small domain without compromising the accuracy of the result. Furthermore, a fast numerical integration method based on the recently proposed Padua points is used, which, together with the windowing technique, yields a fast solution to such scattering problems. Numerical simulations illustrate the accuracy and efficiency of the proposed algorithm.

Lagrange polynomials, reproducing kernels and cubature in two dimensions

We obtain by elementary methods necessary and sufficient conditions for a k-dimensional cubature formula to hold for all polynomials of degree up to 2m−1 when the nodes of the formula have Lagrange polynomials of degree at most m. The main condition is that the Lagrange polynomial at each node is a scalar multiple of the reproducing kernel of degree m−1 evaluated at the node plus an orthogonal polynomial of degree m. Stronger conditions are given for the case where the cubature formula holds for all polynomials of degree up to 2m.This result is applied in one dimension to obtain a quadrature formula where the nodes are the roots of a quasi-orthogonal polynomial of order 2. In two dimensions the result is applied to obtain constructive proofs of cubature formulas of degree 2m−1 for the Geronimus and the Morrow–Patterson classes of nodes. A cubature formula of degree 2m is obtained for a subclass of Morrow–Patterson nodes. Our discussion gives new proofs of previous theorems for the Chebyshev points and the Padua points, which are special cases.

Padua point interpolation and L p -norm minimisation in colour-based image indexing and retrieval

Colour has proven to be a very powerful feature for image indexing. Many examples of image retrieval systems based on colour or chromaticity histograms have been proposed, following on from the histogram intersection method of Swain and Ballard. Here the authors introduce a compact representation of the chromaticity histogram which is based on the Padua point interpolation technique. Specifically, the histogram is represented as a linear combination of Chebyshev polynomials. This bounds a certain maximum deviation, as opposed to a least-squares criterion used in previous work. With this in mind, the minimisation of different Lp norms and the L∞ norm of the error is compared.After presenting the Padua point image indexing and retrieval method, the authors compare its performance to the histogram intersection, the discrete cosine transform, and a dataset oriented method based on principal component analysis. The experiments show that the Padua points match and, in some cases, improve the performance of these methods. This is significant as the proposed method is not tuned (unlike the PCA approach to any dataset). Finally, the behaviour of the Padua point method is analysed in relation to the minimisation of different norms.

Padua2DM: fast interpolation and cubature at the Padua points in Matlab/Octave

We have implemented in Matlab/Octave two fast algorithms for bivariate Lagrange interpolation at the so-called Padua points on rectangles, and the corresponding versions for algebraic cubature.

Algorithm 886

We present a stable and efficient Fortran implementation of polynomial interpolation at the Padua points on the square [ − 1,1] 2 . These points are unisolvent and their Lebesgue constant has minimal order of growth (log square of the degree). The algorithm is based on the representation of the Lagrange interpolation formula in a suitable orthogonal basis, and takes advantage of a new matrix formulation together with the machine-specific optimized BLAS subroutine for the matrix-matrix product. Extension to interpolation on rectangles, triangles and ellipses is also described.

Nontensorial Clenshaw–Curtis cubature

We extend Clenshaw–Curtis quadrature to the square in a nontensorial way, by using Sloan’s hyperinterpolation theory and two families of points recently studied in the framework of bivariate (hyper)interpolation, namely the Morrow–Patterson–Xu points and the Padua points. The construction is an application of a general approach to product-type cubature, where we prove also a relevant stability theorem. The resulting cubature formulas turn out to be competitive on nonentire integrands with tensor-product Clenshaw–Curtis and Gauss–Legendre formulas, and even with the few known minimal formulas.

Histogram compression and image retrieval through Padua points interpolation

Histogram compression and image retrieval through Padua points interpolation

Bivariate Lagrange interpolation at the Padua points: Computational aspects

The so-called “Padua points” give a simple, geometric and explicit construction of bivariate polynomial interpolation in the square. Moreover, the associated Lebesgue constant has minimal order of growth O ( log 2 ( n ) ) . Here we show four families of Padua points for interpolation at any even or odd degree n , and we present a stable and efficient implementation of the corresponding Lagrange interpolation formula, based on the representation in a suitable orthogonal basis. We also discuss extension of (non-polynomial) Padua-like interpolation to other domains, such as triangles and ellipses; we give complexity and error estimates, and several numerical tests.

Bivariate Lagrange interpolation at the Padua points: the ideal theory approach

The Padua points are a family of points on the square [−1, 1]2 given by explicit formulas that admits unique Lagrange interpolation by bivariate polynomials. Interpolation polynomials and cubature formulas based on the Padua points are studied from an ideal theoretic point of view, which leads to the discovery of a compact formula for the interpolation polynomials. The Lp convergence of the interpolation polynomials is also studied.

Bivariate Lagrange interpolation at the Padua points: The generating curve approach

We give a simple, geometric and explicit construction of bivariate interpolation at certain points in a square (called Padua points), giving compact formulas for their fundamental Lagrange polynomials. We show that the associated norms of the interpolation operator, i.e., the Lebesgue constants, have minimal order of growth of O ( ( log n ) 2 ) . To the best of our knowledge this is the first complete, explicit example of near optimal bivariate interpolation points.

Bivariate polynomial interpolation on the square at new nodal sets

As known, the problem of choosing “good” nodes is a central one in polynomial interpolation. While the problem is essentially solved in one dimension (all good nodal sequences are asymptotically equidistributed with respect to the arc-cosine metric), in several variables it still represents a substantially open question. In this work we consider new nodal sets for bivariate polynomial interpolation on the square. First, we consider fast Leja points for tensor-product interpolation. On the other hand, for interpolation in P n 2 on the square we experiment four families of points which are (asymptotically) equidistributed with respect to the Dubiner metric, which extends to higher dimension the arc-cosine metric. One of them, nicknamed Padua points, gives numerically a Lebesgue constant growing like log square of the degree.

On the experiment of Lissajous

In this article, we study bivariate polynomial interpolation on the node points of degenerate Lissajous figures. These node points form Chebyshev lattices of rank 1 and are generalizations of the well-known Padua points. We show that these node points allow unique interpolation in appropriately defined spaces of polynomials and give explicit formulas for the Lagrange basis polynomials. Further, we prove mean and uniform convergence of the interpolating schemes. For the uniform convergence the growth of the Lebesgue constant has to be taken into consideration. It turns out that this growth is of logarithmic nature.