Sensitivity Analysis Using a Fixed Point Interval Iteration

Proving the existence of a solution to a system of real equations is a central issue in numerical analysis. In many situations, the system of equations depend on parameters which are not exactly known. It is then natural to aim proving the existence of a solution for all values of these parameters in some given domains. This is the aim of the parametrization of existence tests. A new parametric existence test based on the Hansen-Sengupta operator is presented and compared to a similar one based on the Krawczyk operator. It is used as a basis of a fixed point iteration dedicated to rigorous sensibility analysis of parametric systems of equations

An equioscillation theorem for multivariate Chebyshev approximation

The equioscillation theorem interleaves the Haar condition, the existence and uniqueness and strong uniqueness of the optimal Chebyshev approximation and its characterization by the equioscillation condition in a way that cannot extend to multivariate approximation: Rice~[\emph{Transaction of the AMS}, 1963] says ''A form of alternation is still present for functions of several variables. However, there is apparently no simple method of distinguishing between the alternation of a best approximation and the alternation of other approximating functions. This is due to the fact that there is no natural ordering of the critical points.'' In addition, in the context of multivariate approximation the Haar condition is typically not satisfied and strong uniqueness may hold or not. The present paper proposes an multivariate equioscillation theorem, which includes such a simple alternation condition on error extrema, existence and a sufficient condition for strong uniqueness. To this end, the relationship between the properties interleaved in the univariate equioscillation theorem is clarified: first, a weak Haar condition is proposed, which simply implies existence. Second, the equioscillation condition is shown to be equivalent to the optimality condition of convex optimization, hence characterizing optimality independently from uniqueness. It is reformulated as the synchronized oscillations between the error extrema and the components of a related Haar matrix kernel vector, in a way that applies to multivariate approximation. Third, an additional requirement on the involved Haar matrix and its kernel vector, called strong equioscillation, is proved to be sufficient for the strong uniqueness of the solution. These three disconnected conditions give rise to a multivariate equioscillation theorem, where existence, characterization by equioscillation and strong uniqueness are separated, without involving the too restrictive Haar condition. Remarkably, relying on optimality condition of convex optimization gives rise to a quite simple proof. Instances of multivariate problems with strongly unique, non-strong but unique and non-unique solutions are presented to illustrate the scope of the theorem

Modal Intervals Revisited Part 1: A Generalized Interval Natural Extension

The modal intervals theory is an extension of the classical intervals theory which provides richer interpretations (including in particular inner and outer approximations of the ranges of real functions). In spite of its promising potential, the modal intervals theory is not widely used today because of its original and complicated construction. The present paper proposes a new formulation of the modal intervals theory. New extensions of continuous real functions to generalized intervals (intervals whose bounds are not constrained to be ordered) are defined. They are called AE-extensions. These AE-extensions provide the same interpretations as the ones provided by the modal intervals theory, thus enhancing the interpretation of the classical interval extensions. The construction of AE-extensions follows the model of the classical intervals theory: starting from a generalization of the definition of the extensions to classical intervals, the minimal AE-extensions of the elementary operations are first built leading to a generalized interval arithmetic. This arithmetic is proved to coincide with the well known Kaucher arithmetic. Then the natural AE-extensions are constructed similarly to the classical natural extensions. The natural AE-extensions represent an important simplification of the formulation of the four "theorems of $\ast$ and $\ast\ast$ interpretation of a modal rational extension" and "theorems of coercion to $\ast$ and $\ast\ast$ interpretability" of the modal intervals theory. With a construction similar to the classical intervals theory, the new formulation of the modal intervals theory proposed in this paper should facilitate the understanding of the underlying mechanisms, the addition of new items to the theory (e.g. new extensions) and its utilization. In particular, a new mean-value extension to generalized intervals will be introduced in the second part of this paper

On the Exponentiation of Interval Matrices

The numerical computation of the exponentiation of a real matrix has been intensively studied. The main objective of a good numerical method is to deal with round-off errors and computational cost. The situation is more complicated when dealing with interval matrices exponentiation: Indeed, the main problem will now be the dependency loss of the different occurrences of the variables due to interval evaluation, which may lead to so wide enclosures that they are useless. In this paper, the problem of computing a sharp enclosure of the interval matrix exponential is proved to be NP-hard. Then the scaling and squaring method is adapted to interval matrices and shown to drastically reduce the dependency loss w.r.t. the interval evaluation of the Taylor series

Search Strategies for an Anytime Usage of the Branch and Prune Algorithm

International audienceWhen applied to numerical CSPs, the branch and prune algorithm (BPA) computes a sharp covering of the solution set. The BPA is therefore impractical when the solution set is large, typically when it has a dimension larger than four or five which is often met in underconstrained problems. The purpose of this paper is to present a new search tree exploration strategy for BPA that hybridizes depth-first and breadth-first searches. This search strategy allows the BPA discovering potential solutions in different areas of the search space in early stages of the exploration, hence allowing an anytime usage of the BPA. The merits of the proposed search strategy are experimentally evaluated

Contrainte de non-chevauchement entre objets décrits par des inégalités non-linéaires

National audiencePacking 2D objects in a limited space is an ubiquitous problem with many academic and industrial variants. In any case, solving this problem requires the ability to determine where a first object can be placed so that it does not intersect a second, previously placed, object. This subproblem is called the non-overlapping constraint. The complexity of this non-overlapping constraint depends on the type of objects considered. It is simple in the case of rectangles. It has also been studied in the case of polygons. This paper proposes a numerical approach for the wide class of objects described bynon-linear inequalities. Our goal here is to calculate the non-overlapping constraint, that is, to describe the set of all positions and orientations that can be assigned to the first object so that intersection with the second one is empty. This is done using a dedicated branch & bound approach. We first show that the non-overlapping constraint can be cast into a Minkowski sum, even if we take into account orientation. We derive from this an innercontractor, that is, an operator that removes from the current domain a subset of positions and orientations that necessarily violate the non-overlapping constraint. This inner contractor is then embedded in a sweeping loop, a pruning technique that was only used with discrete domains so far. We finally come up with a branch & bound algorithm that outperforms the generic state-of-the-art solver Rsolver.Le placement d'objets 2D dans un espace limité est un problème omniprésent aussi bien sur le plan académique qu'industriel. Quel que soit le contexte, la résolution de ce problème exige la capacité de pouvoir déterminer où un premier objet peu être placé de telle façon qu'il ne chevauche pas un second objet, précédemment placé. Ce sous-problème s'appelle la contrainte de non-chevauchement. La complexité de cette contrainte de non-chevauchement dépend du type d'objets considérés. Elle est simple dans le cas de rectangles. Elle a également été étudiée dans le cas de polygones. Cet article propose une approche numérique pour la classe générale des objets décrits par des inégalités non-linéaires. Notre objectif ici est de calculer la contrainte de non-chevauchement, c'est à dire, de décrire l'ensemble de toutes les positions et orientations qui peuvent être attribuées au premier objet de telle sorte que l'intersection avec le second soit vide. Nous nous basons sur un algorithme de branch & prune dédié. Nous montrons d'abord que la contrainte de non-chevauchement, équivaut à une somme de Minkowski, même lorsque l'orientation est prise en compte. Nous en déduisons un contracteur intérieur, c'est à dire, un opérateur qui élimine du domaine courant un sous-ensemble de positions et orientations qui violent nécessairement la contrainte de non-chevauchement. Ce contracteur intérieur est intégré dans une boucle de sweep, une technique utilisée jusqu'ici uniquement pour les domaines discrets. Nous aboutissons ainsi à un algorithme de branch & prune présentant de bien meilleures performances que Rsolver, outil de référence pour la résolution de contraintes quantifiées en domaines continus

Constraint Based Computation of Periodic Orbits of Chaotic Dynamical Systems

International audienceThe chaos theory emerged at the end of the 19th century, and it has given birth to a deep mathematical theory in the 20th century, with a strong practical impact (e.g., weather forecast, turbulence analysis). Periodic orbits play a key role in understanding chaotic systems. Their rigorous computation provides some insights on the chaotic behavior of the system and it enables computer assisted proofs of chaos related properties (e.g., topological entropy). In this paper, we show that the (numerical) constraint programming framework provides a very convenient and efficient method for computing periodic orbits of chaotic dynamical systems: Indeed, the flexibility of CP modeling allows considering various models as well as including additional constraints (e.g., symmetry breaking constraints). Furthermore, the richness of the different solving techniques (tunable local propagators, search strategies, etc.) leads to highly efficient computations. These strengths of the CP framework are illustrated by experimental results on classical chaotic systems from the literature

A New Methodology for Tolerance Synthesis of Parallel Manipulators

International audienceComputing the maximal pose error given an upper bound on perturbations is challenging for parallel robots, mainly because the direct kinematic problem has several solutions, which become unstable near or at parallel singularities. In this paper, we propose a local uniqueness hypothesis that will allow safely computing pose error upper bounds using nonlinear optimization. This hypothesis , together with a corresponding maximal allowed perturbation domain and a certified pose error upper bound valid over the complete workspace, will be proved numerically using a parametric version of Kantorovich theorem and certified nonlinear global optimization. We will then show how to synthesize tolerances reaching a prescribed maximal pose error over a workspace using approximate linearizations. This approximate tolerance synthesis will finally be checked using the certified pose error upper bound we propose. Preliminary experiments on a RPRPR and a 3RPR with fixed orientation parallel manipulators are presented