Multivariate polynomial interpolation with perturbed data

Abstract

Given a finite set of points X in R^n, one may ask for polynomials p which belong to a subspace V and which attain given values at the points of X. We focus\ud on subspaces V of R[x_1,...,x_n], generated by low order monomials. Such V werecomputed by the BM-algorithm, which is essentially based on an LU-decomposition. In this paper we present a new algorithm based on the numerical more stable QR-decomposition. If X contains only points perturbed by measurement or rounding errors, the homogeneous interpolation problem is replaced by the problem of finding (normalized) polynomials minimizing Sum(p(u)^2, u in X). We show that such polynomials can be found easily as byproduct in the QR-decomposition and present an errorbound showing the quality of the approximation

Noise-tolerant algebraic method for reconstruction of nonlinear dynamical systems

“…(13) were perturbed by 5 % Gaussian noise and those in Eq. (14) were perturbed by 10 % Gaussian noise, respectively. The dynamics of these PDSs and of the target PDS (FHN) are shown in Fig.…”

“…As can be seen, though Eqs. (13) and (14) consist of different monomials from the target PDS, their dynamics are similar to the target ones, and the perturbed data points are approximately on the graphs of their dynamics. If the number of monomials is empirically reduced, and if there are more monomials being considered than those in the order ideal, then it might be necessary to consider Eq.…”

“…This is partly because computational algebraic approaches require strict symbolic computations, and thus, they are very sensitive to perturbations of the data. To deal with noisy data, some noisetolerant algebraic methods have recently been developed [10,11,14,20,33,38]. Instead of strict symbolic computations, they perform symbolic-numeric computations, and thus, they may be used for more general applications, including reconstruction of continuoustime dynamical systems from noisy data.…”

“…and(14) (solid lines; blue) have PDSs that are different from those in the FHN, but their behavior is similar to that of the target (dashed lines; red). a y as a function of t for the PDS of Eq.…”

“…(13) and for the FHN, with the data points contaminated with 5 % Gaussian noise. b y as a function of t for the PDS of Eq (14). and for the FHN, with the data points contaminated with 10 % Gaussian noise ber of monomials is empirically reduced, and if there are fewer monomials being considered than those in the order ideal, then Eq.…”