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Sparse grids for the Schrödinger equation

Published online by Cambridge University Press:  16 June 2007

Michael Griebel  and
Jan Hamaekers
Michael Griebel
Affiliation:
Institute of Numerical Simulation, University of Bonn, Wegelerstraße 6, 53115 Bonn, Germany. [email protected]; [email protected]
Jan Hamaekers
Affiliation:
Institute of Numerical Simulation, University of Bonn, Wegelerstraße 6, 53115 Bonn, Germany. [email protected]; [email protected]
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Abstract

We present a sparse grid/hyperbolic cross discretization for many-particle problems.It involves the tensor product of a one-particle multilevel basis. Subsequent truncation of the associated series expansion then results in a sparse grid discretization.Here, depending on the norms involved, different variants of sparse grid techniques for many-particle spaces can be derivedthat, in the best case, result in complexities and error estimates which are independent of the number of particles.Furthermore we introduce an additional constraint which gives antisymmetric sparse grids which are suited to fermionic systems. We apply the antisymmetric sparse grid discretization to the electronic Schrödinger equationand compare costs, accuracy, convergence rates and scalability with respect to the number of electrons present in the system.


Keywords

Schrödinger equationnumerical approximationsparse grid methodantisymmetric sparse grids.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 41 , Issue 2: Special issue on Molecular Modelling , March 2007 , pp. 215 - 247
Copyright
© EDP Sciences, SMAI, 2007

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References

Ackad, E. and Horbatsch, M., Numerical solution of the Dirac equation by a mapped Fourier grid method. J. Phys. A: Math. General 38 (2005) 3157–3171. CrossRef
R.A. Adams, Sobolev spaces. Academic Press, New York (1975).
Arnold, A., Mathematical concepts of open quantum boundary conditions. Transport Theory Statist. Phys. 30 (2001) 561–584. CrossRef
P.W. Atkins and R.S. Friedman, Molecular quantum mechanics. Oxford University Press, Oxford (1997).
Babenko, K.I., Approximation by trigonometric polynomials in a certain class of periodic functions of several variables. Dokl. Akad. Nauk SSSR 132 (1960) 672–675.
S. Balay, K. Buschelman, V. Eijkhout, W.D. Gropp, D. Kaushik, M.G. Knepley, L.C. McInnes, B.F. Smith and H. Zhang, PETSc users manual. Tech. Report ANL-95/11 - Revision 2.1.5, Argonne National Laboratory (2004).
R. Bellmann, Adaptive control processes: A guided tour. Princeton University Press (1961).
J. Boyd, Chebyshev and Fourier spectral methods. Dover Publications, New York (2000).
H. Bungartz, Dünne Gitter und deren Anwendung bei der adaptiven Lösung der dreidimensionalen Poisson-Gleichung. Dissertation, Institut für Informatik, TU München (1992).
H. Bungartz, Finite elements of higher order on sparse grids. Habilitationsschrift, Institut für Informatik, TU München and Shaker Verlag, Aachen (1998).
Bungartz, H. and Griebel, M., A note on the complexity of solving Poisson's equation for spaces of bounded mixed derivatives. J. Complexity 15 (1999) 167–199. CrossRef
Bungartz, H. and Griebel, M., Sparse grids. Acta Numer. 13 (2004) 147–269. CrossRef
Cai, Z., Mandel, J. and McCormick, S., Multigrid methods for nearly singular linear equations and eigenvalue problems. SIAM J. Numer. Anal. 34 (1997) 178–200. CrossRef
Chan, T. and Sharapov, I., Subspace correction multi-level methods for elliptic eigenvalue problems. Numer. Linear Algebra Appl. 9 (2002) 1–20. CrossRef
Chui, C. and Wang, Y., A general framework for compactly supported splines and wavelets. J. Approx. Theory 71 (1992) 263–304. CrossRef
A. Cohen, Numerical analysis of wavelet methods, Studies in Mathematics and its Applications 32. North Holland (2003).
Condon, E., The theory of complex spectra. Phys. Rev. 36 (1930) 1121–1133. CrossRef
I. Daubechies, Ten lectures on wavelets. CBMS-NSF Regional Conf. Series in Appl. Math. 61, SIAM (1992).
Deslauriers, G. and Dubuc, S., Symmetric iterative interpolation processes. Constr. Approx. 5 (1989) 49–68. CrossRef
DeVore, R., Konyagin, S. and Temlyakov, V., Hyperbolic wavelet approximation. Constr. Approx. 14 (1998) 1–26. CrossRef
Dobrovol'skii, N. and Roshchenya, A., Number of lattice points in the hyperbolic cross. Math. Notes 11 (1998) 319–324. CrossRef
D. Donoho and P. Yu, Deslauriers-Dubuc: Ten years after, CRM Proceedings and Lecture Notes 18, G. Deslauriers and S. Dubuc Eds. (1999).
Ewald, P., Die Berechnung optischer und elektrostatischer Gitterpotentiale. Ann. Phys. 64 (1921) 253–287. CrossRef
Fattal, E., Baer, R. and Kosloff, R., Phase space approach for optimizing grid representations: the mapped Fourier method. Phys. Rev. E 53 (1996) 1217–1227. CrossRef
Fevens, T. and Jiang, H., Absorbing boundary conditions for the Schrödinger equation. SIAM J. Scientific Comput. 21 (1999) 255–282. CrossRef
R. Feynman, There's plenty of room at the bottom: An invitation to enter a new world of physics. Engineering and Science XXIII, Feb. issue (1960), http://www.zyvex.com/nanotech/feynman.html.
H.-J. Flad, W. Hackbusch, D. Kolb and R. Schneider, Wavelet approximation of correlated wavefunctions. I. Basics., J. Chem. Phys. 116 (2002) 9641–9857. CrossRef
H.-J. Flad, W. Hackbusch and R. Schneider, Best N term approximation in electronic structure calculations. I. One electron reduced density matrix. Tech. Report 05-9, Berichtsreihe des Mathematischen Seminars der Universität Kiel (2005).
Fliegl, H., Klopper, W. and Hättig, C., Coupled-cluster theory with simplified linear-r12 corrections: The CCSD(R12) model. J. Chem. Phys. 122 (2005) 084107. CrossRef
Fournais, S., Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T. and Sørensen, T., The electron density is smooth away from the nuclei. Commun. Math. Phys. 228 (2002) 401–415. CrossRef
Fournais, S., Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T. and Sørensen, T., Sharp regularity results for Coulombic many-electron wave functions. Commun. Math. Phys. 255 (2005) 183–227. CrossRef
Frank, K., Heinrich, S. and Pereverzev, S., Information complexity of multivariate Fredholm equations in Sobolev classes. J. Complexity 12 (1996) 17–34. CrossRef
G. Friesecke, The configuraton-interaction equations for atoms and molecules: Charge quantization and existence of solutions. Preprint, June 28, 1999, Mathematical Insitute, University of Oxford, UK.
Garcke, J. and Griebel, M., On the computation of the eigenproblems of hydrogen and helium in strong magnetic and electric fields with the sparse grid combination technique. J. Comput. Phys. 165 (2000) 694–716. CrossRef
Gerstner, T. and Griebel, M., Numerical integration using sparse grids. Numer. Algorithms 18 (1998) 209–232. CrossRef
Gerstner, T. and Griebel, M., Dimension-adaptive tensor-product quadrature. Computing 71 (2003) 65–87. CrossRef
Griebel, M., Multilevel algorithms considered as iterative methods on semidefinite systems. SIAM J. Sci. Stat. Comput. 15 (1994) 547–565. CrossRef
M. Griebel, Sparse grids and related approximation schemes for higher dimensional problems, in Proceedings of the conference on Foundations of Computational Mathematics (FoCM05), Santander, Spain, 2005.
Griebel, M. and Knapek, S., Optimized tensor-product approximation spaces. Constr. Approx. 16 (2000) 525–540. CrossRef
Griebel, M. and Oswald, P., On additive Schwarz preconditioners for sparse grid discretizations. Numer. Math. 66 (1994) 449–463. CrossRef
Griebel, M. and Oswald, P., On the abstract theory of additive and multiplicative Schwarz algorithms. Numer. Math. 70 (1995) 161–180. CrossRef
Griebel, M. and Oswald, P., Tensor product type subspace splitting and multilevel iterative methods for anisotropic problems. Adv. Comput. Math. 4 (1995) 171–206. CrossRef
Griebel, M., Oswald, P. and Schiekofer, T., Sparse grids for boundary integral equations. Numer. Mathematik 83 (1999) 279–312. CrossRef
Gygi, F., Adaptive Riemannian metric for plane-wave electronic-structure calculations. Europhys. Lett. 19 (1992) 617. CrossRef
Gygi, F., Electronic-structure calculations in adaptive coordinates. Phys. Rev. B 48 (1993) 11692. CrossRef
Hamann, D., Comparison of global and local adaptive coordinates for density-functional calculations. Phys. Rev. B 63 (2001) 075107. CrossRef
Hernandez, V., Roman, J. and SLEPc, V. Vidal: A scalable and flexible toolkit for the solution of eigenvalue problems. ACM Trans. Math. Software 31 (2005) 351–362. CrossRef
R. Hochmuth, Wavelet bases in numerical analysis and restricted nonlinear approximation. Habilitationsschrift, Freie Universität Berlin (1999).
R. Hochmuth, S. Knapek and G. Zumbusch, Tensor products of Sobolev spaces and applications. Tech. Report 685, SFB 256, Univ. Bonn (2000).
Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T. and Sørensen, T., Electron wavefunction and densities for atoms. Ann. Henri Poincaré 2 (2001) 77–100. CrossRef
G. Karniadakis and S. Sherwin, Spectral/hp element methods for CFD. Oxford University Press (1999).
Kato, T., On the eigenfunctions of many-particle systems in quantum mechanics. Commun. Pure Appl. Math. 10 (1957) 151–177. CrossRef
Keller, J. and Givoli, D., Exact non-reflecting boundary conditions. J. Comput. Phys. 82 (1989) 172–192. CrossRef
S. Knapek, Approximation und Kompression mit Tensorprodukt-Multiskalenräumen. Dissertation, Universität Bonn, April (2000).
S. Knapek, Hyperbolic cross approximation of integral operators with smooth kernel. Tech. Report 665, SFB 256, Univ. Bonn (2000).
Knyazev, A. and Neymeyr, K., Efficient solution of symmetric eigenvalue problem using multigrid preconditioners in the locally optimal block conjugate gradient method. Electronic Trans. Numer. Anal. 15 (2003) 38–55.
W. Kutzelnigg, Convergence of expansions in a Gaussian basis. Strategies and Applications in Quantum Chemistry, M. Defranceschi and Y. Ellinger Eds., Kluwer, Dordrecht (1996).
Kutzelnigg, W. and Mukherjee, D., Minimal parametrization of an n-electron state. Phys. Rev. A 71 (2005) 022502. CrossRef
Le Bris, C., Computational chemistry from the perspective of numerical analysis. Acta Numer. 14 (2005) 363–444. CrossRef
I. Levine, Quantum chemistry, 5th edn., Prentice-Hall (2000).
Liu, W. and Sherman, A., Comparative analysis of Cuthill-McKee and reverse Cuthill-McKee ordering algorithms for sparse matrices. SIAM J. Numer Anal. 13 (1976) 198–213. CrossRef
O. Livne and A. Brandt, O(N log N) multilevel calculation of N eigenfunctions. Multiscale Computational Methods in Chemistry and Physics, A. Brandt, J. Bernholc and K. Binder Eds., NATO Science Series III: Computer and Systems Sciences, IOS Press 177 (2001) 112–136.
Maz'ya, V. and Schmidt, G., On approximate approximations using Gaussian kernels. IMA J. Numer. Anal. 16 (1996) 13–29. CrossRef
Mazziotti, D., Variational two-electron reduced density matrix theory for many-electron atoms and molecules: Implementation of the spin- and symmetry-adapted T-2 condition through first-order semidefinite programming. Phys. Rev. A 72 (2005) 032510. CrossRef
A. Messiah, Quantum mechanics. Vol. 1 and 2, North-Holland, Amsterdam, 1961/62.
P. Nitsche, Best n term approximation spaces for sparse grids. Tech. Report 2003-11, ETH Zürich, Seminar für Angewandte Mathematik (2003).
P. Oswald, Multilevel finite element approximation. Teubner Skripten zur Numerik, Teubner, Stuttgart (1994).
R. Parr and W. Yang, Density Functional Theory of Atoms and Molecules. Oxford University Press, New York (1989).
H.-J. Schmeisser and H. Triebel, Fourier analysis and functions spaces. John Wiley, Chichester (1987).
Sims, J.S. and Hagstrom, S.A., High-precision Hy-CI variational calculations for the ground state of neutral helium and helium-like ions. Int. J. Quant. Chem. 90 (2002) 1600–1609. CrossRef
Slater, J., The theory of complex spectra. Phys. Rev. 34 (1929) 1293–1322. CrossRef
Smolyak, S., Quadrature and interpolation formulas for tensor products of certain classes of functions. Soviet Math. Dokl. 4 (1963) 240–243, Russian original in Dokl. Akad. Nauk SSSR 148 (1963) 1042–1045.
B. Szabo and I. Babuska, Finite element analysis. Wiley (1991).
Szeftel, J., Design of absorbing boundary conditions for Schrödinger equations in $\mathbb{R}^{d*}$ . SIAM J. Numer. Anal. 42 (2004) 1527–1551. CrossRef
J. Weidmann, Linear operators in Hilbert spaces. Springer, New York (1980).
H. Yserentant, On the electronic Schrödinger equation. Report 191, SFB 382, Univ. Tübingen (2003).
Yserentant, H., On the regularity of the electronic Schrödinger equation in Hilbert spaces of mixed derivatives. Numer. Math. 98 (2004) 731–759. CrossRef
Zung, D., The approximation of classes of periodic functions of many variables. Russian Math. Surveys 38 (1983) 117–118. CrossRef
Zung, D., Approximation by trigonometric polynomials of functions of several variables on the torus. Math. USSR Sbornik 59 (1988) 247–267. CrossRef