Encyclopedia of Applied and Computational Mathematics

2015 Edition
| Editors: Björn Engquist

A Priori and A Posteriori Error Analysis in Chemistry

  • Yvon Maday
Reference work entry
DOI: https://doi.org/10.1007/978-3-540-70529-1_255

Synonyms

Convergence analysis; Error estimates; Guaranteed accuracy; Refinement

Definition

For a numerical discretization chosen to approximate the solution of a given problem or for an algorithm used to solve the discrete problem resulting from the previous discretization, a priori analysis explains how the method behaves and to which extent the numerical solution that is produced from the discretization/algorithm is close to the exact one. It also allows to compare the numerical method of interest with another one. With a priori analysis though, there is no definite certainty that a given computation provides a good enough approximation. It is only when the number of degrees of freedom and the complexity of the computation is large enough that the convergence of the numerical method can be guaranteedly achieved. On the contrary, a posteriori analysis provides bounds on the error on the solution or the output coming out of the simulation. The concept of a posteriori analysis can even...

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References

  1. 1.
    Boulton, L., Boussa, N., Lewin, M.: Generalized Weyl theorem and spectral pollution in the Galerkin method. http://arxiv.org/pdf/1011.3634v2
  2. 2.
    Boys, S.F.: Electronic wavefunction I. A general method of calculation for the stationary states of any molecular system. Proc. R. Soc. A 200, 542–554 (1950)CrossRefzbMATHGoogle Scholar
  3. 3.
    Braess, D.: Asymptotics for the approximation of wave functions by sums of exponential sums. J. Approx. Theory 83, 93–103 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brezzi, F., Canuto, C., Russo, A.: A self-adaptive formulation for the Euler/NavierStokes coupling. CMAME Arch. 73(3), 317–330 (1989)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Cancès, E., Ehrlacher, V., Maday, Y.: Periodic Schrdinger operators with local defects and spectral pollution, arXiv:1111.3892Google Scholar
  6. 6.
    Cancès, E., LeBris, C.: On the convergence of SCF algorithms for the HartreeFock equations. Math. Model. Numer. Anal. 34, 749–774 (2000)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Cancès, E., LeBris, C., Maday, Y., Turinici, G.: Towards reduced basis approaches in ab initio electronic structure computations. J. Sci. Comput. 17(1), 461–469 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cancès, E., LeBris, C., Maday, Y.: Méthodes Mathématiques en chimie quantique: une Introduction (in French). Mathématiques and Applications (Berlin), vol. 53. Springer, Berlin (2006)Google Scholar
  9. 9.
    Conte, D., Lubich, C.: An error analysis of the multi-configuration time-dependent Hartree method of quantum dynamics. ESAIM M2AN 44, 759–780 (2010)Google Scholar
  10. 10.
    El Alaoui, L., Ern, A., Vohralk, M.: Guaranteed and robust a posteriori error estimates and balancing discretization and linearization errors for monotone nonlinear problems. Comput. Method Appl. Mech. Eng. 200, 2782–2795 (2011)CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Friesecke, G.: The multiconfiguration equations for atoms and molecules: charge quantization and existence of solutions. Arch. Ration. Mech. Anal. 169, 35–71 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hohenberg, P., Kohn, W.: Inhomogeneous electron gas. Phys. Rev. 136(3B), B864–B871 (1964)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Kobus, J., Quiney, H.M., Wilson, S.: A comparison of finite difference and finite basis set Hartree-Fock calculations for the N2 molecule with finite nuclei. J. Phys. B Atomic Mol. Opt. Phys. 34, 10 (2001)Google Scholar
  14. 14.
    Kohn, W., Sham, L.J.: Self-consistent equations including exchange and correlation effects. Phys. Rev. 140(4A), A1133–A1138 (1965)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Kudin, K., Scuseria, G.E., Cancès, E.: A black-box self-consistent field convergence algorithm: one step closer. J. Chem. Phys. 116, 8255–8261 (2002)CrossRefGoogle Scholar
  16. 16.
    Kutzelnigg, W.: Theory of the expansion of wave functions in a Gaussian basis. Int. J. Quantum Chem. 51, 447–463 (1994)CrossRefGoogle Scholar
  17. 17.
    Lewin, M.: Solution of multiconfiguration equations in quantum chemistry. Arch. Ration. Mech. Anal. 171, 83–114 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Lewin, M., Séré, É.: Spectral pollution and how to avoid it (with applications to Dirac and periodic Schrödinger operators). Proc. Lond. Math. Soc. 100(3), 864–900 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Maday, Y., Razafison, U.: A reduced basis method applied to the restricted HartreeFock equations. Comptes Rendus Math. 346(3–4), 243–248 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Maday, Y., Turinici, G.: Error bars and quadratically convergent methods for the numerical simulation of the Hartree-Fock equations. Numer. Math. 94, 739–770 (2003)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Yvon Maday
    • 1
  1. 1.Sorbonne Universités, UPMC Univ Paris 06UMR 7598, Laboratoire Jacques-Louis LionsParisFrance