Comptes Rendus
no. 3-4
Numerical Analysis
A reduced basis method applied to the Restricted Hartree–Fock equations
[Une méthode de base réduite appliquée aux équations Restricted Hartree–Fock]

Présenté par : Olivier Pironneau

Yvon Maday1, 2 ; Ulrich Razafison1
1 UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France
2 Division of Applied Mathematics, Brown University 182 George Street, Providence, RI 02912, USA
Comptes Rendus. Mathématique, Volume 346 (2008) no. 3-4, pp. 243-248.

Dans cette Note, nous décrivons une méthode d'approximation par bases réduites pour les calculs de structures électroniques en chimie quantique basées sur le modèle Restricted Hartree–Fock. Nous présentons des résultats numériques montrant que la méthode permet des réductions de complexité et potentiellement de coûts de calculs.

In this Note, we describe a reduced basis approximation method for the computation of some electronic structure in quantum chemistry, based on the Restricted Hartree–Fock equations. Numerical results are presented to show that this approach allows for reducing the complexity and potentially the computational costs.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2007.11.015
Yvon Maday 1, 2 ; Ulrich Razafison 1

1 UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France
2 Division of Applied Mathematics, Brown University 182 George Street, Providence, RI 02912, USA 
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Yvon Maday; Ulrich Razafison. A reduced basis method applied to the Restricted Hartree–Fock equations. Comptes Rendus. Mathématique, Volume 346 (2008) no. 3-4, pp. 243-248. doi : 10.1016/j.crma.2007.11.015. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.11.015/

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[2] M. Barrault; Y. Maday; N.C. Nguyen; A.T. Patera An “empirical interpolation” method: Application to efficient reduced-basis discretization of partial differential equations, C. R. Acad. Sci. Paris, Ser. I, Volume 339 (2004), pp. 667-672

[3] E. Cancès; C. Le Bris; Y. Maday; G. Turinici Towards reduced basis approaches in ab initio electronic structure computations, J. Sci. Comput., Volume 17 (2002) no. 1–4, pp. 461-469

[4] E. Cancès; M. Defranceschi; W. Kutzelnigg; C. Le Bris; Y. Maday Computational quantum chemistry: a primer (Ph. Ciarlet; C. Le Bris, eds.), Handbook of Numerical Analysis. Volume X: Special Volume: Computational Chemistry, North-Holland, 2003

[5] E. Cancès; C. Le Bris; Y. Maday; N.C. Nguyen; A.T. Patera; G.S.H. Pau Feasibility and competitiveness of a reduced-basis approach for rapid electronic structure calculations in quantum chemistry, High-dimensional Partial Differential Equations in Science and Engineering, CRM Proceedings Series, vol. 41, American Mathematical Society, 2007, pp. 15-47

[6] L. Machiels; Y. Maday; I.B. Oliviera; A.T. Patera; D.V. Rovas Outputs bounds for the reduced-basis approximations of symmetric positive definite eigenvalue problems, C. R. Acad. Sci. Paris, Ser. I., Volume 331 (2000) no. 2, pp. 153-158

[7] A.T. Patera, G. Rozza, Reduced basis approximation and a posteriori error estimation for parametrized partial differential equations, http://augustine.mit.edu/methodology/methodology_book.htm

[8] G.S.H. Pau, Reduced-basis method for quantum models of periodic solids, Ph.D. Thesis, Massachusetts Institute of Technology, 2007

[9] A. Pinkus n-Widths in Approximation Theory, Springer-Verlag, 1985

[10] C. Prud'homme; D. Rovas; K. Veroy; Y. Maday; A.T. Patera; G. Turinici Reliable real-time solution of parametrized partial differential equations: Reduced basis output bound methods, J. Fluids Engrg., Volume 124 (2002) no. 1, pp. 70-80

[11] E.M. Rønquist; A.E. Løvgren; Y. Maday A reduced basis element method for the steady Stokes problem, M2AN Math. Model. Numer. Anal., Volume 40 (2006) no. 3, p. 529-552.h

[12] K. Veroy; A.T. Patera Certified real-time solution of the parametrized steady incompressible Navier–Stokes equations: rigourous reduced basis a posteriori error-bounds, Internat. J. Numer. Methods Fluids, Volume 47 (2005) no. 8–9, pp. 773-788

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