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Fourier transform on sparse grids: Code design and the time dependent Schrödinger equation

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Abstract

The pseudo-spectral method together with a Strang-splitting are well suited for the discretization of the time-dependent Schrödinger equation with smooth potential. The curse of dimensionality limits this approach to low dimensions, if we stick to full grids. Theoretically, sparse grids allow accurate computations in (moderately) higher dimensions, provided that we supply an efficient Fourier transform. Motivated by this application, the design of the Fourier transform on sparse grids in multiple dimensions is described in detail. The focus of this presentation is on issues of flexible implementation and numerical studies of the convergence.

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References

  • Achatz, S.: Adaptive finite Dünngitter-Elemente höherer Ordnung für elliptische partielle Differentialgleichungen mit variablen Koeffizienten, PhD thesis, TU München 2003.

  • G. Baszenski F. Delvos (1989) A discrete Fourier transform scheme for Boolean sums of trigonometric operators K. Z. C. K. Chui W. Schempp (Eds) Multivariate approximation theory IV Birkhäuser Basel 15–24

    Google Scholar 

  • H. Bungartz (1998) Finite elements of higher order on sparse grids Shaker Verlag Aachen

    Google Scholar 

  • H. Bungartz M. Griebel (2004) ArticleTitleSparse grids Acta Numer. 13 1–123 Occurrence Handle2249147 Occurrence Handle10.1017/S0962492904000182

    Article  MathSciNet  Google Scholar 

  • Brandt, S., Dahmen, H.: The picture book of quantum mechanics. Springer 1995.

  • Fenn, M., Kunis, S., Potts, D.: Fast evaluation of trigonometric polynomials from hyperbolic crosses. Numer. Algorithms (to appear).

  • Feuersänger, C.: Dünngitterverfahren für hochdimensionale elliptische partielle Differentialgleichungen. Master's thesis, Institut für Numerische Simulation, Universität Bonn 2005.

  • Gradinaru, V.: Strang splitting for the time dependent Schrödinger equation on sparse grids. SIAM J. Numer. Anal. (submitted).

  • Gradinaru, V.: Whitney elements on sparse grids. PhD thesis, Universität Tübingen, 2002. Electronically published http://w210.ub.uni-tuebingen.de/dbt/volltexte/2002/543/.

  • M. Griebel (1998) ArticleTitleAdaptive sparse grid multilevel methods for elliptic PDEs based on finite differences Computing 61 151–179 Occurrence Handle0918.65078 Occurrence Handle1650985

    MATH  MathSciNet  Google Scholar 

  • K. Hallatschek (1992) ArticleTitleFouriertransformation auf dünnen Gittern mit hierarchischen Basen Numer. Math. 63 83–97 Occurrence Handle0762.65098 Occurrence Handle10.1007/BF01385849 Occurrence Handle1182513

    Article  MATH  MathSciNet  Google Scholar 

  • T. Jahnke C. Lubich (2000) ArticleTitleError bounds for exponential operator splittings BIT 40 735–744 Occurrence Handle0972.65061 Occurrence Handle10.1023/A:1022396519656 Occurrence Handle1799313

    Article  MATH  MathSciNet  Google Scholar 

  • Knapek, S.: Approximation und Kompression mit Tensorprodukt-Multiskalen-Approximationsräumen. PhD thesis, Universität Bonn 2000.

  • Kupka, F.: Sparse grid spectral methods for the numerical solution of partial differential equations with periodic boundary conditions. PhD thesis, Univerität Wien 1997.

  • C. Lasser S. Teufel (2005) ArticleTitlePropagation through conical crossings: an asymptotic semigroup Commun. Pure Appl. Math. 58 1188–1230 Occurrence Handle02207891 Occurrence Handle10.1002/cpa.20087 Occurrence Handle2163659

    Article  MATH  MathSciNet  Google Scholar 

  • Lasser, C., Swart, T., Teufel, S.: Construction and validation of a rigorous surface hopping algorithm for conical crossings preprint (2006).

  • Sedgewick, R.: Algorithmen in C++. Addison-Wesley 1992.

  • S. Smolyak (1963) ArticleTitleQuadrature and interpolation formulas for tensor products of certain classes of functions Dokl. Akad. Nauk 148 1042–1045 Occurrence Handle0202.39901

    MATH  Google Scholar 

  • V. Temlyakov (1985) ArticleTitleApproximate recovery of periodic fucntions of several variables Mat. Sbornik 128 256–268 Occurrence Handle809488

    MathSciNet  Google Scholar 

  • Trefethen, L.: Spectral methods in Matlab. SIAM 2000.

  • G. Worth M. Robb (2002) ArticleTitleApplying direct molecular dynamics to non-adiabatic systems Adv. Chem. Phys. 124 355–431 Occurrence Handle10.1002/0471433462.ch7

    Article  Google Scholar 

  • Zenger, C.: Sparse grids. Tech. Rep. 18-90, SFB 342, TU München, 1990.

  • Zumbusch, G.: A sparse grid pde solver. In: Advances in Software Tools for Scientific Computing (Proc. SciTools '98) (Langtangen, H. P., Bruaset, A. M., Quak, E., eds.). Lecture Notes in Computational Science and Engineering, vol. 10. Springer 2000, chap. 4, pp. 133–178.

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  1. Universität Tübingen, Auf der Morgenstelle 10, 72076, Tübingen, Germany

    V. Gradinaru

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  1. V. Gradinaru
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Gradinaru, V. Fourier transform on sparse grids: Code design and the time dependent Schrödinger equation. Computing 80, 1–22 (2007). https://doi.org/10.1007/s00607-007-0225-3

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