Computation of smooth manifolds via rigorous multi-parameter continuation in infinite dimensions

Authors: Gameiro, Marcio; Lessard, Jean-Philippe; Pugliese, Alessandro
Abstract: In this paper, we introduce a constructive rigorous numerical method to compute smooth manifolds implicitly defined by infinite-dimensional nonlinear operators. We compute a simplicial triangulation of the manifold using a multi-parameter continuation method on a finite-dimensional projection. The triangulation is then used to construct local charts and an atlas of the manifold in the infinite-dimensional domain of the operator. The idea behind the construction of the smooth charts is to use the radii polynomial approach to verify the hypotheses of the uniform contraction principle over a simplex. The construction of the manifold is globalized by proving smoothness along the edge of adjacent simplices. We apply the method to compute portions of a two-dimensional manifold of equilibria of the Cahn–Hilliard equation.
Document Type: Article de recherche
Issue Date: 4 June 2015
Open Access Date: 4 June 2016
Document version: VoR
This document was published in: Foundations of Computational Mathematics, Vol. 16 (2), 531–575 (2016)
Alternative version: 10.1007/s10208-015-9259-7
Collection:Articles publiés dans des revues avec comité de lecture

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