Fast algorithm for border bases of Artinian Gorenstein algebras

Given a multi-index sequence $$\sigma$$, we present a new efficient algorithm to compute generators of the linear recurrence relations between the terms of $$\sigma$$. We transform this problem into an algebraic one, by identifying multi-index sequences, multivariate formal power series and linear functionals on the ring of multivariate polynomials. In this setting, the recurrence relations are the elements of the kerne l$I$\sigma of the Hankel operator $H$\sigma associated to $$\sigma$$. We describe the correspondence between multi-index sequences with a Hankel operator of finite rank and Artinian Gorenstein Algebras. We show how the algebraic structure of the Artinian Gorenstein algebra $A$\sigma$$associated to the sequence$$\sigma yields the structure of the terms $\sigma\alpha$for all$$\alpha$ $\in$ N n$. This structure is explicitly given by a border basis of$A$\sigma$$, which is presented as a quotient of the polynomial ring$K[x 1 ,. .. , xn$] by the kernel$I$\sigma$$of the Hankel operator$H$\sigma$$. The algorithm provides generators of$I$\sigma$$constituting a border basis, pairwise orthogonal bases of$A$\sigma$$ and the tables of multiplication by the variables in these bases. It is an extension of Berlekamp-Massey-Sakata (BMS) algorithm, with improved complexity bounds. We present applications of the method to different problems such as the decomposition of functions into weighted sums of exponential functions, sparse interpolation, fast decoding of algebraic codes, computing the vanishing ideal of points, and tensor decomposition. Some benchmarks illustrate the practical behavior of the algorithm

An efficient technique based on polynomial chaos to model the uncertainty in the resonance frequency of textile antennas due to bending

The generalized polynomial chaos theory is combined with a dedicated cavity model for curved textile antennas to statistically quantify variations in the antenna's resonance frequency under randomly varying bending conditions. The nonintrusive stochastic method solves the dispersion relation for the resonance frequencies of a set of radius of curvature realizations corresponding to the Gauss quadrature points belonging to the orthogonal polynomials having the probability density function of the random variable as a weighting function. The formalism is applied to different distributions for the radius of curvature, either using a priori known or on-the-fly constructed sets of orthogonal polynomials. Numerical and experimental validation shows that the new approach is at least as accurate as Monte Carlo simulations while being at least 100 times faster. This makes the method especially suited as a design tool to account for performance variability when textile antennas are deployed on persons with varying body morphology