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Theses

Techniques d'intervalles pour la résolution de systèmes d'équations

Abstract : This dissertation is devoted to solving systems of nonlinear equations. It presents a survey of various theories based on interval computations (interval analysis, modal intervals and constraint programming over the reals) as well as some new results in each of these areas. A special care is brought to the linear case, one of our main field of study. This case is of considerable significance since all nonlinear methods are based upon it. In interval analysis, we give an extension of the Hansen-Bliek method which computes an optimal outer approximation of the solution set of interval linear systems. This extension allows more freedom in the choice of the quantifiers (existential or universal) associated to the coeficients, thus handling a wider variety of problems. A generalization of the LU decomposition based on Kaucher's interval arithmetic is also given. We also propose a new formulation of the modal intervals theory, with the underlying concept of quantified range - a natural generalization of the range of a function. This new approach allows us to introduce Kaucher's arithmetic with a vivid meaning, and not only as an abstract algebraic extension of the classical interval arithmetic. In constraint programming, we study local consistencies with domains represented by unions of intervals. This structure is more adapted to store refinements of domains throughout propagation. We show to which extent arc-consistency can be achieved with this structure.
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Gilles Chabert. Techniques d'intervalles pour la résolution de systèmes d'équations. Autre [cs.OH]. Université Nice Sophia Antipolis, 2007. Français. ⟨tel-00260907⟩

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