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

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  1. Breden, Maxime; Kuehn, Christian: Rigorous validation of stochastic transition paths (2019)
  2. Colombo, Alessandro; Del Buono, Nicoletta; Lopez, Luciano; Pugliese, Alessandro: Computational techniques to locate crossing/sliding regions and their sets of attraction in non-smooth dynamical systems (2018)
  3. Cyranka, Jacek; Wanner, Thomas: Computer-assisted proof of heteroclinic connections in the one-dimensional Ohta-Kawasaki model (2018)
  4. Lessard, Jean-Philippe: Continuation of solutions and studying delay differential equations via rigorous numerics (2018)
  5. Capiński, Maciej J.; Mireles James, J. D.: Validated computation of heteroclinic sets (2017)
  6. van den Berg, Jan Bouwe; Williams, J. F.: Validation of the bifurcation diagram in the 2D Ohta-Kawasaki problem (2017)
  7. de la Llave, R.; Mireles James, J. D.: Connecting orbits for compact infinite dimensional maps: computer assisted proofs of existence (2016)
  8. Gameiro, Marcio; Lessard, Jean-Philippe; Pugliese, Alessandro: Computation of smooth manifolds via rigorous multi-parameter continuation in infinite dimensions (2016)
  9. Hungria, Allan; Lessard, Jean-Philippe; Mireles James, J. D.: Rigorous numerics for analytic solutions of differential equations: the radii polynomial approach (2016)
  10. Lessard, Jean-Philippe; Mireles James, J. D.; Ransford, Julian: Automatic differentiation for Fourier series and the radii polynomial approach (2016)