Séminaire MACS (Modélisation, Analyse et Calcul Scientifique). Commun ICJ & UMPA.

An introduction to computer-assisted proofs for nonlinear equations: how to turn a numerical simulation into a theorem.

par Maxime Breden (Ecole Polytechnique)

Europe/Paris
Fokko du Cloux (Bâtiment Braconnier, La Doua)

Fokko du Cloux

Bâtiment Braconnier, La Doua

Description

The goal of a posteriori validation methods is to get a quantitative and
rigorous description of some specific solutions of nonlinear ODEs or PDEs,
based on numerical simulations. The general strategy consists in combining
a priori and a posteriori error estimates, interval arithmetic, and a fixed
point theorem applied to a quasi-Newton operator. Starting from a numerically computed approximate solution, one can then prove the existence
of a true solution in a small and explicit neighborhood of the numerical
approximation.
In this talk I will present the main ideas behind these techniques, describe
a rather general framework in which they can be applied, and showcase their
interest by presenting examples of application in population dynamics, fluid
dynamics and corrosion problems.