Bencteux, Guy (2008) Amélioration d'une méthode de décomposition de domaine pour le calcul de structures électroniques. PhD thesis Mathématique, Informatique, Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), ENPC p.141.

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Abstract

Abstract :

This work is about a domain decomposition method for electronic structure computations, with Hartree-Fock or DFT (Density Functional Theory) models. Usually, the numerical simulation of these models involve the solution of a generalized eigenvalue problem, which is a bottleneck due to the cubic scaling of the number of operations.

The MDD (Multilevel Domain Decomposition) method, that have been introduced in a previous PhD (Maxime Barrault, 2005), replace the generalized eigen-value problem with a constrained minimization problem, for which it is easier to take benefit of the localization properties of the solution.

Results produced by the present work are :

-the numerical analysis of the algorithm : a local convergence result has been proved, on a simplified instance of the problem that exhibits the same mathematical difficulties ;

-improvement of speed and accuracy, with one-dimensional sub-domain arrangements, as well as demonstration of scalability up to one thousand processors ;

-extension of the algorithm and its numerical implementation to cases with 2D/3D subdomains arrangement.

References

Bibliographie

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Liens vers des codes d'utilisation courante

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Références en Informatique : Calcul Haute Performance (HPC)

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Table of content

Sommaire

1 Introduction générale - 1

1.1 Résolution numérique des modèles Hartree-Fock et Kohn-Sham - 2

1.1.1 Bases de discrétisation - 6

1.1.2 Le calcul de la densité - 7

1.2 L'algorithme MDD - 10

1.2.1 Principe - 10

1.2.2 Présentation du travail réalisé - 12

2 Analyse numérique dans un cas simplifié - 15

2.1 Introduction - 16

2.2 Optimality Conditions - 20

2.3 The local step and continuity - 24

2.4 Convergence of the decomposition algorithm - 31

2.5 Numerical experiments - 35

2.6 Conclusions - 39

3 Domaines alignés : implémentation parallèle - 41

3.1 Introduction and motivation - 43

3.2 A new domain decomposition approach - 46

3.3 The Multilevel Domain Decomposition (MDD) algorithm - 49

3.4 Parallel implementation - 52

3.5 Numerical tests - 53

3.5.1 General presentation - 53

3.5.2 Sequential computations - 55

3.5.3 Parallel computations - 55

3.6 Conclusions and perspectives - 58

4 Domaines alignés : comportement numérique - 59

4.1 Calcul distribué sur un grand nombre de processeurs . 60

4.2 Relaxation de la contrainte d'orthogonalité - 63

4.2.1 Localisation des orbitales moléculaires et orthogonalité - 63

4.2.2 Orthogonalité et convergence de l'algorithme - 67

4.2.3 Importance de l'algorithme d'orthogonalisation . . . 70

4.3 Conclusion de la partie 1D - 72

4.3.1 Synthèse des améliorations par rapport à [11] - 72

4.3.2 Améliorations envisageables pour l'étape locale . . 72

4.3.3 Comparaison de deux algorithmes de calcul de la densité : MDD version 1D et diagonalisation de la matrice de Fock - 74

5 Répartition quelconque des domaines - 79

5.1 Adaptation de l'étape globale - 81

5.1.1 Retour sur la construction de l'étape globale dans le cas "1D" - 81

5.1.2 Adaptation au cas "2D/3D" - 84

5.2 Implémentation de la version multidimensionnelle de MDD - 86

5.2.1 Structure des données - 87

5.2.2 Implémentation - 90

5.2.3 Estimation de la mémoire totale - 94

5.2.4 Complexité des étapes - 96

6 Conclusion générale - 97

A Présentation succincte des méthodes d'ordre N - 99

A.1 Les méthodes variationnelles - 100

A.1.1 Minimisation sur les orbitales - 101

A.1.2 Minimisation sur la densité - 102

A.2 Les méthodes de projection - 102

A.2.1 Approximation de l'opérateur de Fermi - 103

A.2.2 Méthode de purification de la densité - 103

A.3 Les méthodes de décomposition de domaine - 105

A.4 Le passage au cas S ≠ INb - 106

B Calcul de dérivées pour l'étape globale - 107

B.1 Notations et rappel des formules - 107

B.2 Formules communes aux deux calculs - 108

B.2.1 Dérivées partielles de Ji-1 - 108

B.2.2 Expression des dérivées pour une étape globale à 2 blocs - 109

B.2.3 Simplification des traces de produit de matrices . . 109

B.3 Calcul du gradient - 111

B.3.1 Le gradient en U ≠ 0 - 112

B.3.2 Le gradient en U = 0 - 112

B.4 Calcul du hessien pour 2 blocs - 113

B.4.1 Expression du hessien complet pour 2 blocs - 117

B.5 Implémentation du calcul du gradient et du hessien pour 2 blocs - 119

C Mémoire et nombre d'opérations pour chaque étape . 121

C.1 Mémoire - 122

C.1.1 Données du problème et solution - 123

C.1.2 Etape locale - 123

C.1.3 Etape globale - 125

C.1.4 Mémoire totale en fonction des options - 127

C.2 Nombre d'opérations - 127

C.2.1 Initialisation - 128

C.2.2 Etape locale - 129

C.2.3 Etape globale - 129

C.2.4 Nombre d'opérations pour une itération de MDD . . 131

Bibliographie générale - 133

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