메뉴 건너뛰기




Volumn 41, Issue 2, 2007, Pages 215-247

Sparse grids for the Schrödinger equation

Author keywords

Antisymmetric sparse grids; Numerical approximation; Schr dinger equation; Sparse grid method

Indexed keywords


EID: 34547545863     PISSN: 0764583X     EISSN: 12903841     Source Type: Journal    
DOI: 10.1051/m2an:2007015     Document Type: Article
Times cited : (84)

References (80)
  • 1
    • 17044389388 scopus 로고    scopus 로고
    • Numerical solution of the Dirac equation by a mapped Fourier grid method
    • E. Ackad and M. Horbatsch, Numerical solution of the Dirac equation by a mapped Fourier grid method. J. Phys. A: Math. General 38 (2005) 3157-3171.
    • (2005) J. Phys. A: Math. General , vol.38 , pp. 3157-3171
    • Ackad, E.1    Horbatsch, M.2
  • 2
  • 3
    • 0035594730 scopus 로고    scopus 로고
    • Mathematical concepts of open quantum boundary conditions
    • A. Arnold, Mathematical concepts of open quantum boundary conditions. Transport Theory Statist. Phys. 30 (2001) 561-584.
    • (2001) Transport Theory Statist. Phys , vol.30 , pp. 561-584
    • Arnold, A.1
  • 5
    • 0040214718 scopus 로고
    • Approximation by trigonometric polynomials in a certain class of periodic functions of several variables
    • K.I. Babenko, Approximation by trigonometric polynomials in a certain class of periodic functions of several variables. Dokl. Akad. Nauk SSSR 132 (1960) 672-675.
    • (1960) Dokl. Akad. Nauk SSSR , vol.132 , pp. 672-675
    • Babenko, K.I.1
  • 10
    • 0004258344 scopus 로고    scopus 로고
    • Habilitationsschrift, Institut für Informatik, TU München and Shaker Verlag, Aachen
    • H. Bungartz, Finite elements of higher order on sparse grids. Habilitationsschrift, Institut für Informatik, TU München and Shaker Verlag, Aachen (1998).
    • (1998) Finite elements of higher order on sparse grids
    • Bungartz, H.1
  • 11
    • 0001335132 scopus 로고    scopus 로고
    • A note on the complexity of solving Poisson's equation for spaces of bounded mixed derivatives
    • H. Bungartz and M. Griebel, A note on the complexity of solving Poisson's equation for spaces of bounded mixed derivatives. J. Complexity 15 (1999) 167-199.
    • (1999) J. Complexity , vol.15 , pp. 167-199
    • Bungartz, H.1    Griebel, M.2
  • 13
    • 0000323375 scopus 로고    scopus 로고
    • Multigrid methods for nearly singular linear equations and eigenvalue problems
    • Z. Cai, J. Mandel and S. McCormick, Multigrid methods for nearly singular linear equations and eigenvalue problems. SIAM J. Numer. Anal. 34 (1997) 178-200.
    • (1997) SIAM J. Numer. Anal , vol.34 , pp. 178-200
    • Cai, Z.1    Mandel, J.2    McCormick, S.3
  • 14
    • 0036103889 scopus 로고    scopus 로고
    • Subspace correction multi-level methods for elliptic eigenvalue problems
    • T. Chan and I. Sharapov, Subspace correction multi-level methods for elliptic eigenvalue problems. Numer. Linear Algebra Appl. 9 (2002) 1-20.
    • (2002) Numer. Linear Algebra Appl , vol.9 , pp. 1-20
    • Chan, T.1    Sharapov, I.2
  • 15
    • 36148972266 scopus 로고
    • A general framework for compactly supported splines and wavelets
    • C. Chui and Y. Wang, A general framework for compactly supported splines and wavelets. J. Approx. Theory 71 (1992) 263-304.
    • (1992) J. Approx. Theory , vol.71 , pp. 263-304
    • Chui, C.1    Wang, Y.2
  • 17
    • 0001695987 scopus 로고
    • The theory of complex spectra
    • E. Condon, The theory of complex spectra. Phys. Rev. 36 (1930) 1121-1133.
    • (1930) Phys. Rev , vol.36 , pp. 1121-1133
    • Condon, E.1
  • 19
    • 34249975165 scopus 로고
    • Symmetric iterative interpolation processes
    • G. Deslauriers and S. Dubuc, Symmetric iterative interpolation processes. Constr. Approx. 5 (1989) 49-68.
    • (1989) Constr. Approx , vol.5 , pp. 49-68
    • Deslauriers, G.1    Dubuc, S.2
  • 21
    • 29344456764 scopus 로고    scopus 로고
    • Number of lattice points in the hyperbolic cross
    • N. Dobrovol'skii and A. Roshchenya, Number of lattice points in the hyperbolic cross. Math. Notes 11 (1998) 319-324.
    • (1998) Math. Notes , vol.11 , pp. 319-324
    • Dobrovol'skii, N.1    Roshchenya, A.2
  • 23
    • 84977266737 scopus 로고
    • Die Berechnung optischer und elektrostatischer Gitterpotentiale
    • P. Ewald, Die Berechnung optischer und elektrostatischer Gitterpotentiale. Ann. Phys. 64 (1921) 253-287.
    • (1921) Ann. Phys , vol.64 , pp. 253-287
    • Ewald, P.1
  • 24
    • 0001164755 scopus 로고    scopus 로고
    • Phase space approach for optimizing grid representations: The mapped Fourier method
    • E. Fattal, R. Baer and R. Kosloff, Phase space approach for optimizing grid representations: the mapped Fourier method. Phys. Rev. E 53 (1996) 1217-1227.
    • (1996) Phys. Rev. E , vol.53 , pp. 1217-1227
    • Fattal, E.1    Baer, R.2    Kosloff, R.3
  • 25
    • 0033295857 scopus 로고    scopus 로고
    • Absorbing boundary conditions for the Schrödinger equation
    • T. Fevens and H. Jiang, Absorbing boundary conditions for the Schrödinger equation. SIAM J. Scientific Comput. 21 (1999) 255-282.
    • (1999) SIAM J. Scientific Comput , vol.21 , pp. 255-282
    • Fevens, T.1    Jiang, H.2
  • 26
    • 34547507756 scopus 로고    scopus 로고
    • 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.
    • 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.
  • 27
    • 0037042658 scopus 로고    scopus 로고
    • Wavelet approximation of correlated wavefunctions. I. Basics
    • H.-J. Fiad, W. Hackbusch, D. Kolb and R. Schneider, Wavelet approximation of correlated wavefunctions. I. Basics., J. Chem. Phys. 116 (2002) 9641-9857.
    • (2002) J. Chem. Phys , vol.116 , pp. 9641-9857
    • Fiad, H.-J.1    Hackbusch, W.2    Kolb, D.3    Schneider, R.4
  • 28
    • 34547544100 scopus 로고    scopus 로고
    • 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
    • H.-J. Fiad, 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).
    • (2005)
    • Fiad, H.-J.1    Hackbusch, W.2    Schneider, R.3
  • 29
    • 22944452424 scopus 로고    scopus 로고
    • Coupled-cluster theory with simplified linear-rl2 corrections: The CCSD(R12) model
    • H. Fliegl, W. Klopper and C. Hättig, Coupled-cluster theory with simplified linear-rl2 corrections: The CCSD(R12) model. J. Chem. Phys. 122 (2005) 084107.
    • (2005) J. Chem. Phys , vol.122 , pp. 084107
    • Fliegl, H.1    Klopper, W.2    Hättig, C.3
  • 32
    • 0030099613 scopus 로고    scopus 로고
    • Information complexity of multivariate Fredholm equations in Sobolev classes
    • K. Frank, S. Heinrich and S. Pereverzev, Information complexity of multivariate Fredholm equations in Sobolev classes. J. Complexity 12 (1996) 17-34.
    • (1996) J. Complexity , vol.12 , pp. 17-34
    • Frank, K.1    Heinrich, S.2    Pereverzev, S.3
  • 33
    • 34547507126 scopus 로고    scopus 로고
    • 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.
    • 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.
  • 34
    • 0001032475 scopus 로고    scopus 로고
    • On the computation of the eigenproblems of hydrogen and helium in strong magnetic and electric fields with the sparse grid combination technique
    • J. Garcke and M. Griebel, 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.
    • (2000) J. Comput. Phys , vol.165 , pp. 694-716
    • Garcke, J.1    Griebel, M.2
  • 35
    • 0242440688 scopus 로고    scopus 로고
    • Numerical integration using sparse grids
    • T. Gerstner and M. Griebel, Numerical integration using sparse grids. Numer. Algorithms 18 (1998) 209-232.
    • (1998) Numer. Algorithms , vol.18 , pp. 209-232
    • Gerstner, T.1    Griebel, M.2
  • 36
    • 0141991028 scopus 로고    scopus 로고
    • Dimension-adaptive tensor-product quadrature
    • T. Gerstner and M. Griebel, Dimension-adaptive tensor-product quadrature. Computing 71 (2003) 65-87.
    • (2003) Computing , vol.71 , pp. 65-87
    • Gerstner, T.1    Griebel, M.2
  • 37
    • 0001610807 scopus 로고
    • Multilevel algorithms considered as iterative methods on semidefinite systems
    • M. Griebel, Multilevel algorithms considered as iterative methods on semidefinite systems. SIAM J. Sci. Stat. Comput. 15 (1994) 547-565.
    • (1994) SIAM J. Sci. Stat. Comput , vol.15 , pp. 547-565
    • Griebel, M.1
  • 39
    • 0034337983 scopus 로고    scopus 로고
    • Optimized tensor-product approximation spaces
    • M. Griebel and S. Knapek, Optimized tensor-product approximation spaces. Constr. Approx. 16 (2000) 525-540.
    • (2000) Constr. Approx , vol.16 , pp. 525-540
    • Griebel, M.1    Knapek, S.2
  • 40
    • 0242503926 scopus 로고
    • On additive Schwarz preconditioners for sparse grid discretizations
    • M. Griebel and P. Oswald, On additive Schwarz preconditioners for sparse grid discretizations. Numer. Math. 66 (1994) 449-463.
    • (1994) Numer. Math , vol.66 , pp. 449-463
    • Griebel, M.1    Oswald, P.2
  • 41
    • 21844511143 scopus 로고
    • On the abstract theory of additive and multiplicative Schwarz algorithms
    • M. Griebel and P. Oswald, On the abstract theory of additive and multiplicative Schwarz algorithms. Numer. Math. 70 (1995) 161-180.
    • (1995) Numer. Math , vol.70 , pp. 161-180
    • Griebel, M.1    Oswald, P.2
  • 42
    • 0000277689 scopus 로고
    • Tensor product type subspace splitting and multilevel iterative methods for anisotropic problems
    • M. Griebel and P. Oswald, Tensor product type subspace splitting and multilevel iterative methods for anisotropic problems. Adv. Comput. Math. 4 (1995) 171-206.
    • (1995) Adv. Comput. Math , vol.4 , pp. 171-206
    • Griebel, M.1    Oswald, P.2
  • 43
    • 0033443380 scopus 로고    scopus 로고
    • Sparse grids for boundary integral equations
    • M. Griebel, P. Oswald and T. Schiekofer, Sparse grids for boundary integral equations. Numer. Mathematik 83 (1999) 279-312.
    • (1999) Numer. Mathematik , vol.83 , pp. 279-312
    • Griebel, M.1    Oswald, P.2    Schiekofer, T.3
  • 44
    • 84956246254 scopus 로고
    • Adaptive Riemannian metric for plane-wave electronic-structure calculations
    • F. Gygi, Adaptive Riemannian metric for plane-wave electronic-structure calculations. Europhys. Lett. 19 (1992) 617.
    • (1992) Europhys. Lett , vol.19 , pp. 617
    • Gygi, F.1
  • 45
    • 4244217973 scopus 로고
    • Electronic-structure calculations in adaptive coordinates
    • F. Gygi, Electronic-structure calculations in adaptive coordinates. Phys. Rev. B 48 (1993) 11692.
    • (1993) Phys. Rev. B , vol.48 , pp. 11692
    • Gygi, F.1
  • 46
    • 0035099429 scopus 로고    scopus 로고
    • Comparison of global and local adaptive coordinates for density-functional calculations
    • D. Hamann, Comparison of global and local adaptive coordinates for density-functional calculations. Phys. Rev. B 63 (2001) 075107.
    • (2001) Phys. Rev. B , vol.63 , pp. 075107
    • Hamann, D.1
  • 47
    • 33745196196 scopus 로고    scopus 로고
    • SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems
    • V. Hernandez, J. Roman and V. Vidal, SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems. ACM Trans. Math. Software 31 (2005) 351-362.
    • (2005) ACM Trans. Math. Software , vol.31 , pp. 351-362
    • Hernandez, V.1    Roman, J.2    Vidal, V.3
  • 49
    • 34547517374 scopus 로고    scopus 로고
    • R. Hochmuth, S. Knapek and G. Zumbusch, Tensor products of Sobolev spaces and applications. Tech. Report 685, SFB 256, Univ. Bonn (2000).
    • R. Hochmuth, S. Knapek and G. Zumbusch, Tensor products of Sobolev spaces and applications. Tech. Report 685, SFB 256, Univ. Bonn (2000).
  • 52
    • 84980086883 scopus 로고
    • On the eigenfunctions of many-particle systems in quantum mechanics
    • T. Kato, On the eigenfunctions of many-particle systems in quantum mechanics. Commun. Pure Appl. Math. 10 (1957) 151-177.
    • (1957) Commun. Pure Appl. Math , vol.10 , pp. 151-177
    • Kato, T.1
  • 53
    • 0000271460 scopus 로고
    • Exact non-reflecting boundary conditions
    • J. Keller and D. Givoli, Exact non-reflecting boundary conditions. J. Comput. Phys. 82 (1989) 172-192.
    • (1989) J. Comput. Phys , vol.82 , pp. 172-192
    • Keller, J.1    Givoli, D.2
  • 55
    • 34547511039 scopus 로고    scopus 로고
    • S. Knapek, Hyperbolic cross approximation of integral operators with smooth kernel. Tech. Report 665, SFB 256, Univ. Bonn (2000).
    • S. Knapek, Hyperbolic cross approximation of integral operators with smooth kernel. Tech. Report 665, SFB 256, Univ. Bonn (2000).
  • 56
    • 2942668468 scopus 로고    scopus 로고
    • Efficient solution of symmetric eigenvalue problem using multigrid preconditioners in the locally optimal block conjugate gradient method
    • A. Knyazev and K. Neymeyr, 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.
    • (2003) Electronic Trans. Numer. Anal , vol.15 , pp. 38-55
    • Knyazev, A.1    Neymeyr, K.2
  • 57
    • 34547528034 scopus 로고    scopus 로고
    • Convergence of expansions in a Gaussian basis
    • M. Defranceschi and Y. Ellinger Eds, Kluwer, Dordrecht
    • W. Kutzelnigg, Convergence of expansions in a Gaussian basis. Strategies and Applications in Quantum Chemistry, M. Defranceschi and Y. Ellinger Eds., Kluwer, Dordrecht (1996).
    • (1996) Strategies and Applications in Quantum Chemistry
    • Kutzelnigg, W.1
  • 58
    • 33344477526 scopus 로고    scopus 로고
    • Minimal parametrization of an n-electron state
    • W. Kutzelnigg and D. Mukherjee, Minimal parametrization of an n-electron state. Phys. Rev. A 71 (2005) 022502.
    • (2005) Phys. Rev. A , vol.71 , pp. 022502
    • Kutzelnigg, W.1    Mukherjee, D.2
  • 59
    • 19844366465 scopus 로고    scopus 로고
    • Computational chemistry from the perspective of numerical analysis
    • C. Le Bris, Computational chemistry from the perspective of numerical analysis. Acta Numer. 14 (2005) 363-444.
    • (2005) Acta Numer , vol.14 , pp. 363-444
    • Le Bris, C.1
  • 61
    • 0016940739 scopus 로고
    • Comparative analysis of Cuthill-McKee and reverse Cuthill-McKee ordering algorithms for sparse matrices
    • W. Liu and A. Sherman, Comparative analysis of Cuthill-McKee and reverse Cuthill-McKee ordering algorithms for sparse matrices. SIAM J. Numer Anal. 13 (1976) 198-213.
    • (1976) SIAM J. Numer Anal , vol.13 , pp. 198-213
    • Liu, W.1    Sherman, A.2
  • 62
    • 34547525226 scopus 로고    scopus 로고
    • 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.
    • 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.
  • 63
    • 0039863907 scopus 로고    scopus 로고
    • On approximate approximations using Gaussian kernels
    • V. Maz'ya and G. Schmidt, On approximate approximations using Gaussian kernels. IMA J. Numer. Anal. 16 (1996) 13-29.
    • (1996) IMA J. Numer. Anal , vol.16 , pp. 13-29
    • Maz'ya, V.1    Schmidt, G.2
  • 64
    • 28844460348 scopus 로고    scopus 로고
    • 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
    • D. Mazziotti, 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.
    • (2005) Phys. Rev. A , vol.72 , pp. 032510
    • Mazziotti, D.1
  • 65
    • 34547545065 scopus 로고
    • and 2, North-Holland, Amsterdam
    • A. Messiah, Quantum mechanics. Vol. 1 and 2, North-Holland, Amsterdam, 1961/62.
    • (1961) Quantum mechanics , vol.1
    • Messiah, A.1
  • 67
    • 0004139365 scopus 로고
    • Multilevel finite element approximation
    • Teubner, Stuttgart
    • P. Oswald, Multilevel finite element approximation. Teubner Skripten zur Numerik, Teubner, Stuttgart (1994).
    • (1994) Teubner Skripten zur Numerik
    • Oswald, P.1
  • 70
    • 0037147161 scopus 로고    scopus 로고
    • High-precision Hy-CI variational calculations for the ground state of neutral helium and heliumlike ions
    • J.S. Sims and S.A. Hagstrom, High-precision Hy-CI variational calculations for the ground state of neutral helium and heliumlike ions. Int. J. Quant. Chem. 90 (2002) 1600-1609.
    • (2002) Int. J. Quant. Chem , vol.90 , pp. 1600-1609
    • Sims, J.S.1    Hagstrom, S.A.2
  • 71
    • 36149026675 scopus 로고
    • The theory of complex spectra
    • J. Slater, The theory of complex spectra. Phys. Rev. 34 (1929) 1293-1322.
    • (1929) Phys. Rev , vol.34 , pp. 1293-1322
    • Slater, J.1
  • 72
    • 0001048297 scopus 로고
    • Quadrature and interpolation formulas for tensor products of certain classes of functions
    • S. Smolyak, Quadrature and interpolation formulas for tensor products of certain classes of functions. Soviet Math. Dokl. 4 (1963) 240-243.
    • (1963) Soviet Math. Dokl , vol.4 , pp. 240-243
    • Smolyak, S.1
  • 73
    • 0001048298 scopus 로고
    • Russian original in Dokl
    • Russian original in Dokl. Akad. Nauk SSSR 148 (1963) 1042-1045.
    • (1963) Akad. Nauk SSSR , vol.148 , pp. 1042-1045
  • 77
    • 34547522251 scopus 로고    scopus 로고
    • H. Yserentant, On the electronic Schrödinger equation. Report 191, SFB 382, Univ. Tübingen (2003).
    • H. Yserentant, On the electronic Schrödinger equation. Report 191, SFB 382, Univ. Tübingen (2003).
  • 78
    • 2942635358 scopus 로고    scopus 로고
    • On the regularity of the electronic Schrödinger equation in Hilbert spaces of mixed derivatives
    • H. Yserentant, On the regularity of the electronic Schrödinger equation in Hilbert spaces of mixed derivatives. Numer. Math. 98 (2004) 731-759.
    • (2004) Numer. Math , vol.98 , pp. 731-759
    • Yserentant, H.1
  • 79
    • 0001918867 scopus 로고
    • The approximation of classes of periodic functions of many variables
    • D. Zung, The approximation of classes of periodic functions of many variables. Russian Math. Surveys 38 (1983) 117-118.
    • (1983) Russian Math. Surveys , vol.38 , pp. 117-118
    • Zung, D.1
  • 80
    • 33751113798 scopus 로고
    • Approximation by trigonometric polynomials of functions of several variables on the torus
    • D. Zung, Approximation by trigonometric polynomials of functions of several variables on the torus. Math. USSR Sbornik 59 (1988) 247-267.
    • (1988) Math. USSR Sbornik , vol.59 , pp. 247-267
    • Zung, D.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.