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            <funder>This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 810367).</funder>
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              <p>This article is concerned with the numerical solution of subspace optimization problems, consisting of minimizing a smooth functional over the set of orthogonal projectors of fixed rank. Such problems are encountered in particular in electronic structure calculation (Hartree-Fock and Kohn-Sham Density Functional Theory -DFT- models). We compare from a numerical analysis perspective two simple representatives, the damped self-consistent field (SCF) iterations and the gradient descent algorithm, of the two classes of methods competing in the field: SCF and direct minimization methods. We derive asymptotic rates of convergence for these algorithms and analyze their dependence on the spectral gap and other properties of the problem. Our theoretical results are complemented by numerical simulations on a variety of examples, from toy models with tunable parameters to realistic Kohn-Sham computations. We also provide an example of chaotic behavior of the simple SCF iterations for a nonquadratic functional.</p>
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              <p>Cet article aborde la résolution numérique de problèmes d'optimisation qui consistent à minimiser une fonctionnelle régulière sur un ensemble de projecteurs orthogonaux de rang fixe. De tels problèmes sont rencontrés en particulier en calcul de structure électronique (modèles de Hartree-Fock et de Kohn-Sham en Théorie de la Fonctionnelle de la Densité - DFT-). Nous comparons d'un point de vue de l'analyse numérique deux représentants simples, un algorithme de champ auto-cohérent amorti et une descente de gradient, issus des deux classes de méthodes rencontrées dans le domaine: les champs auto-cohérents et les méthodes de minimisation directe. Nous dérivons des taux de convergence asymptotique pour ces algorithmes et analysons leur dépendance en fonction du trou spectral et d’autres propriétés du problème. Nos résultats théoriques sont complétés par des simulations numériques sur différents exemples, d'un modèle simple où l'on contrôle explicitement les paramètres au modèle plus réaliste de Kohn-Sham. Nous fournissons également un exemple de comportement chaotique des itérations de champ auto-cohérent non-amorti pour une fonctionnelle non-quadratique.</p>
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