Computing and Visualization in Science

, Volume 11, Issue 2, pp 115–122 | Cite as

A parallel multigrid accelerated Poisson solver for ab initio molecular dynamics applications

  • H. Köstler
  • R. Schmid
  • U. Rüde
  • Ch. Scheit
Regular article


In this paper we present an application for a parallel multigrid solver in 3D to solve the Coulomb problem for the charge self interaction in a quantum-chemical program used to perform ab initio molecular dynamics. Techniques such as Mehrstellendiscretization and τ-extrapolation are used to improve the discretization error. The results show that the expected convergence rates and parallel performance of the multigrid solver are achieved. Within the applied Carr–Parrinello Molecular Dynamics scheme the quality of the solution also determines the accuracy in energy conservation. All forms of discretization employed lead to energy conserving dynamics. In order to test the applicability of our code to larger systems in a massively parallel environment, we investigated a 256 atom periodic supercell of bulk gallium nitride.


Real Space Multigrid Method Discretization Error Multigrid Solver Coulomb Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Dacapo: Scholar
  2. 2.
    Hpc-cluster: Scholar
  3. 3.
    The MPI forum: The MPI message-passing interface standard. Scholar
  4. 4.
    Numerical python: Scholar
  5. 5.
    Python: http://www.python.orgGoogle Scholar
  6. 6.
    Rsdft: http://www.rsdft.orgGoogle Scholar
  7. 7.
    Ancilotto, F., Blandin, P., Toigo, F.: Real-space full multigrid study of the fragmentation of \(_{11}^+\) clusters. Phys. Rev. B 59, 7868 (1999)Google Scholar
  8. 8.
    Beck T.L. (2000). Real-space mesh techniques in density-functional theory. Rev. Mod. Phys. 72: 1041–1080 CrossRefGoogle Scholar
  9. 9.
    Bernert K. (1997). τ-extrapolation—theoretical foundation, numerical experiment and application to Navier–Stokes equations. SIAM J. Sci. Comp. 18: 460–478 CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Bernholc J. (1999). Computational materials science: the era of apllied quantum mechanics. Phys. Today 52(9): 30–35 CrossRefGoogle Scholar
  11. 11.
    Brandt, A.: Multigrid methods: 1984 guide with applications to fluid dynamics. The Weizmann Institute of Science, Rehovot, Israel (1984)Google Scholar
  12. 12.
    Castro A., Marques M.A.L., Alonso J.A., Bertsch G.F., Yabana K. and Rubio A. (2002). Can optical spectroscopy directly elucidate the ground state of c20?. J. Chem. Phys. 116: 1930–1933 CrossRefGoogle Scholar
  13. 13.
    Ceperley D.M. and Alder B.J. (1980). Ground state of the electron gas by a stochastic method. Phys. Rev. Lett. 45: 566–569 CrossRefGoogle Scholar
  14. 14.
    Chelikowsky J.R., Saad Y., Ögüt S., Vasiliev I. and Stathopoulos A. (2000). Electronic structure methods for predicting the properties of materials: grids in space. Phys. Stat. Sol. B 217: 173–195 CrossRefGoogle Scholar
  15. 15.
    Douglas, C., Haase, G., Langer, U.: A Tutorial on Elliptic PDE Solvers and their Parallelization, SIAM (2003)Google Scholar
  16. 16.
    Gropp W., Lusk E. and Skjellum A. (1999). Using MPI, Portable Parallel Programming with the Mesage-Passing Interface , 2nd edn. MIT Press, Cambridge Google Scholar
  17. 17.
    Hackbusch W. (1985). Multi-Grid Methods and Applications. Springer, Heidelberg zbMATHGoogle Scholar
  18. 18.
    Hinsen K. (2000). The molecular modeling toolkit: a new approach to molecular simulations. J. Comput. Chem. 21: 79–85 CrossRefGoogle Scholar
  19. 19.
    Hohenberg P. and Kohn W. (1964). Inhomogeneous electron gas. Phys. Rev. 136: B864–B871 CrossRefMathSciNetGoogle Scholar
  20. 20.
    Hülsemann, F., Kowarschik, M., Mohr, M., Rüde, U.: Parallel geometric multigrid. In: Bruaset, A., Tveito, A. (eds.) Numerical Solution of Partial Differential Equations on Parallel Computers, chap. 5, vol. 51 of LNCSE. Springer, Heidelberg, pp. 165–208 (2005). ISBN 3-540-29076-1Google Scholar
  21. 21.
    Jin Y.G. and Chang K.J. (2002). Efficient real-space multigrid method and applications to clusters and defects in SiO2. J. Korean Phys. Soc. 40: 406–415 Google Scholar
  22. 22.
    Kendall R.A., Apra E., Bernholdt D.E., Bylaska E.J., Dupuis M., Fann G.I., Harrison R.J., Ju J., Nichols J.A., Nieplocha J., Straatsma T.P., Windus T.L. and Wong, A.T. (2000). High performance computational chemistry: an overview of nwchem a distributed parallel application. Comput. Phys. Commun. 128: 260–283 CrossRefzbMATHGoogle Scholar
  23. 23.
    Kohn W. and Sham L.J. (1965). Self-consistent equations including exchange and correlation effects. Phys. Rev. 140: A1133–A1138 CrossRefMathSciNetGoogle Scholar
  24. 24.
    Martín I. and Tirado F. (1997). Relationships between efficiency and execution time of full multigrid methods on parallel computers. IEEE Trans. Parallel Distrib. Syst. 8: 562–573 CrossRefGoogle Scholar
  25. 25.
    Marx, D., Hutter, J.: Ab Inition Molecular Dynamics: Theory and Implementation, vol. 1 of NIC Series. John von Neumann Institute for Computing, Julich, pp. 301–449 (2000)Google Scholar
  26. 26.
    Mortensen J.J., Hansen L.B. and Jacobsen K.W. (2005). Real-space grid implementation of the projector augmented wave method. Phys. Rev. B 71: 035109–103510911 CrossRefGoogle Scholar
  27. 27.
    Ono T. and Hirose K. (1999). Timesaving double-grid method for real-space electronic-structure calculations. Phys. Rev. Lett. 81: 5016–5019 CrossRefGoogle Scholar
  28. 28.
    Parrinello M. and Car R. (1985). Unified approach for molecular dynamics and density-functional theory. Phys. Rev. Lett. 55: 2471–2474 CrossRefGoogle Scholar
  29. 29.
    Perdew J.P. and Zunger A. (1981). Self-interaction correction to density-functional approximations for many-electron systems. Phys. Rev. B 23: 5048–5079 CrossRefGoogle Scholar
  30. 30.
    Richardson L. (1927). The deferred approach to the limit. I. Single lattice. Philos. Trans. R. Soc. Lond. A 226: 229–349 CrossRefGoogle Scholar
  31. 31.
    Rüde, U.: Multiple τ-extrapolation for multigrid methods. Tech. Rep. I-8701, Technische Universität München (1987)Google Scholar
  32. 32.
    Schmid R. (2004). Car–Parrinello molecular-dynamics simulations with real space methods. J. Comput. Chem. 25: 799–812 CrossRefGoogle Scholar
  33. 33.
    Schmid R., Tafipolsky M., König P.H. and Köstler H. (2006). Car–Parrinello molecular dynamics using real space wavefunctions. Phys. Status solidi b 243: 1001–1015 CrossRefGoogle Scholar
  34. 34.
    Shimojo F., Kalia R.K., Nakano A. and Vashishta P. (2001). Linear-scaling density-functional-theory calculations of electronic structure based on real-space grids: design, analysis, and scalability test of parallel algorithms. Comput. Phys. Commun. 140: 303–314 CrossRefzbMATHGoogle Scholar
  35. 35.
    Sterk M. and Trobec R. (2003). Parallel performances of a multigrid poisson solver. ISPDC 00: 238 Google Scholar
  36. 36.
    Tafipolsky, M., Schmid, R.: A general and efficient pseudopotential fourier filtering scheme for real space methods using mask functions. J. Chem. Phys. 243(5) (2005)Google Scholar
  37. 37.
    Torsti T., Heiskanen M., Puska M.J. and Nieminen R.M. (2003). Mika: multigrid-based program package for electronic structure calculations. Int. J. Quantum Chem. 91: 171–176 CrossRefGoogle Scholar
  38. 38.
    Trottenberg U., Oosterlee C. and Schüller A. (2001). Multigrid. Academic, New York zbMATHGoogle Scholar
  39. 39.
    Waghmare U.V., Kim H., Park I.J., Modine N., Maragakis P. and Kaxiras E. (2001). Hares: an efficient method for first-principles electronic structure calculations of complex systems. Comput. Phys. Commun. 137: 341–360 CrossRefzbMATHGoogle Scholar
  40. 40.
    Wang J. and Beck T.L. (2000). Efficient real-space solution of the Kohn-Sham equations with multiscale techniques. J. Chem. Phys. 112: 9223–9228 CrossRefGoogle Scholar
  41. 41.
    Wang J., Wang Y., Yu S. and Kolb D. (2005). Nonlinear algorithm for the solution of the Kohn-Sham equations in solids. J. Phys. Cond. Mat. 17: 3701–3715 CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Ruhr-University BochumBochumGermany
  2. 2.University of Erlangen-NurembergErlangenGermany

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