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Convergence of a normalized gradient algorithm for computing ground states

Erwan Faou 1, * Tiphaine Jézéquel 1
* Corresponding author
1 MINGUS - Multi-scale numerical geometric schemes
IRMAR - Institut de Recherche Mathématique de Rennes, ENS Rennes - École normale supérieure - Rennes, Inria Rennes – Bretagne Atlantique
Abstract : We consider the approximation of the ground state of the one-dimensional cubic nonlinear Schrödinger equation by a normalized gradient algorithm combined with linearly implicit time integrator, and finite difference space approximation. We show that this method, also called imaginary time evolution method in the physics literature, is con-vergent, and we provide error estimates: the algorithm converges exponentially towards a modified solitons that is a space discretization of the exact soliton, with error estimates depending on the discretization parameters.
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Erwan Faou, Tiphaine Jézéquel. Convergence of a normalized gradient algorithm for computing ground states. IMA Journal of Numerical Analysis, Oxford University Press (OUP), 2018, 38 (1), pp.360-376. ⟨10.1093/imanum/drx009⟩. ⟨hal-01284679⟩

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