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Journal of Scientific Computing

, Volume 42, Issue 3, pp 447–455 | Cite as

Hierarchical Matrices in Computations of Electron Dynamics

  • Othmar KochEmail author
  • Christopher Ede
  • Gerald Jordan
  • Armin Scrinzi
Article
  • 97 Downloads

Abstract

We discuss the approximation of the meanfield terms appearing in computations of the multi-configuration time-dependent Hartree–Fock method for the solution of the time-dependent multi-particle (electronic) Schrödinger equation by hierarchical matrices. We give theoretical error bounds for the cross approximation defined by low rank approximations of admissible matrix sub-blocks, and illustrate the gain in performance by numerical experiments.

Keywords

Multi-configuration time-dependent Hartree–Fock method Time-dependent multi-particle Schrödinger equation Hierarchical matrices 

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References

  1. 1.
    Axelsson, O., Barker, V.: Finite Element Solution of Boundary Value Problems: Theory and Computation. Academic Press, Orlando (1984) zbMATHGoogle Scholar
  2. 2.
    Bebendorf, M., Hackbusch, W.: Stabilized rounded addition of hierarchical matrices. Numer. Linear Algebra Appl. 14, 407–423 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Beck, M.H., Jäckle, A., Worth, G.A., Meyer, H.-D.: The multiconfiguration time-dependent Hartree (MCTDH) method: A highly efficient algorithm for propagating wavepackets. Phys. Rep. 324, 1–105 (2000) CrossRefGoogle Scholar
  4. 4.
    Braess, D.: Finite Elements, 2nd edn. Cambridge University Press, Cambridge (2001) zbMATHGoogle Scholar
  5. 5.
    Brenner, S., Scott, L.: The Mathematical Theory of Finite Element Methods, 2nd edn. Springer, New York (2002) zbMATHGoogle Scholar
  6. 6.
    Caillat, J., Zanghellini, J., Kitzler, M., Kreuzer, W., Koch, O., Scrinzi, A.: Correlated multielectron systems in strong laser pulses—an MCTDHF approach. Phys. Rev. A 71, 012712 (2005) CrossRefGoogle Scholar
  7. 7.
    Dirac, P.: Note on exchange phenomena in the Thomas atom. Proc. Camb. Philos. Soc. 26, 376–385 (1930) zbMATHCrossRefGoogle Scholar
  8. 8.
    Frenkel, J.: Wave Mechanics, Advanced General Theory. Clarendon Press, Oxford (1934) zbMATHGoogle Scholar
  9. 9.
    Grasedyck, L., Hackbusch, W.: Construction and arithmetic of ℋ-matrices. Computing 70, 295–334 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Hackbusch, W.: A sparse matrix arithmetic based on ℋ-matrices. Part I: Introduction to ℋ-matrices. Computing 62, 89–108 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Hackbusch, W., Khoromskij, B.: A sparse ℋ-matrix arithmetic. Part II: Application to multi-dimensional problems. Computing 64, 21–47 (1999) MathSciNetGoogle Scholar
  12. 12.
    Koch, O.: Efficient computation of the MCTDHF approximation to the time-dependent Schrödinger equation. Opuscula Math. 26, 473–487 (2006) Google Scholar
  13. 13.
    Koch, O.: Approximation of the meanfield terms in MCTDHF computations by ℋ-matrices. ASC Report 5/2008, Inst. for Anal. and Sci. Comput., Vienna Univ. of Technology (2008) Google Scholar
  14. 14.
    Meyer, H.-D., Manthe, U., Cederbaum, L.S.: The multi-configurational time-dependent Hartree approach. Chem. Phys. Lett. 165, 73–78 (1990) CrossRefGoogle Scholar
  15. 15.
    Zanghellini, J., Kitzler, M., Brabec, T., Scrinzi, A.: Testing the multi-configuration time-dependent Hartree–Fock method. J. Phys. B, At. Mol. Phys. 37, 763–773 (2004) CrossRefGoogle Scholar
  16. 16.
    Zanghellini, J., Kitzler, M., Fabian, C., Brabec, T., Scrinzi, A.: An MCTDHF approach to multi-electron dynamics in laser fields. Laser Phys. 13(8), 1064–1068 (2003) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Othmar Koch
    • 1
    Email author
  • Christopher Ede
    • 2
  • Gerald Jordan
    • 2
  • Armin Scrinzi
    • 2
  1. 1.Institute for Analysis and Scientific Computing (E101)Vienna University of TechnologyViennaAustria
  2. 2.Photonics Institute (E387)Vienna University of TechnologyViennaAustria

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