Let X be a set of points whose coordinates are known with limited accuracy; our aim is to give a characterization of the vanishing ideal I(X) independent of the data uncertainty. We present a method to compute a polynomial basis B of I(X) which exhibits structural stability, that is, if e X is any set of points differing only slightly from X, there exists a polynomial set e B structurally similar to B, which is a basis of the perturbed ideal I( e X).

From the numerical point of view, given a set X, subset of R^n of s points whose coordinates are known with only limited precision, each set XP of s points whose elements differ from those of X of a quantity less than the data uncertainty can be considered equivalent to X. We present an algorithm that, given X and a tolerance on the data error, computes a set G of polynomials such that each element of G ``almost vanishing'' at X and at all its equivalent sets XP. Even if G is not, in the general case, a basis of the vanishing ideal I(X), we show that, differently from the basis of I(X) that can be greatly influenced by the data uncertainty, G can determine a geometrical configuration simultaneously characterizing the set X and all its equivalent sets XP

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We present a symbolic-numeric approach for the analysis of a given set of noisy data, represented as a finite set X of limited precision points. Starting from X and a permitted tolerance ε on its coordinates, our method automatically determines a low degree monic polynomial whose associated variety passes close to each point of X by less than the given tolerance ε.Proof. Letᾱ j = α j (ē) and f = t − t j ∈Oᾱ j t j . We observe that f i (e i ) = t(p i (e i )) + t j ∈Oᾱ j t j (p i (e i )) is a polynomial function of e i and its Jacobian J f i (e i ) ∈ Mat 1×n (R) is a Lipschitz function inB(ē i , r i ). We prove that

SUMMARYGroundwater¯ow in partially saturated porous media is modelled by using the non-linear Richards equation, which is discretized in the present work by using linear mixed-hybrid ®nite elements.The discretization produces an algebraic non-linear system, which can be solved by an iterative ®xed-point algorithm, the Picard method. The convergence rate is linear, and may be too poor for practical applications. A superlinear convergence rate is obtained by considering a Broyden-type approach, based on the Shermann±Morrison formula.The local character of the Broyden method can be overcome by an accurate estimate of the initial solution, that is by appropriately initializing the computation via some (relaxed) Picard iterations. This strategy needs a convergence criterion to decide when switching from the Picard to the quasi-Newton method, which is crucial for the eectiveness of the scheme, as illustrated by some numerical experiments.We also consider the non-linear algebraic problem from a dierent viewpoint. Instead of applying the quasi-Newton method directly to such a non-linear system, we applied it to the non-linear function tied to the Picard scheme. Each function evaluation requested by such an algorithm corresponds to a local step of the Picard method, which is then used to compute a Broyden displacement. The present technique can be seen as an accelerated Picard algorithm.We compare the performances of these algorithms when applied to a stationary and a time-dependent benchmark problem. #

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

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