Abstract

This paper provides a provably quasi-optimal preconditioning strategy of the linear Schrödinger eigenvalue problem with periodic potentials for a possibly nonuniform spatial expansion of the domain. The quasi-optimality is achieved by having the iterative eigenvalue algorithms converge in a constant number of iterations for different domain sizes. In the analysis, we derive an analytic factorization of the spectrum and asymptotically describe it using concepts from the homogenization theory. This decomposition allows us to express the eigenpair as an easy-to-calculate cell problem solution combined with an asymptotically vanishing remainder. We then prove that the easy-to-calculate limit eigenvalue can be used in a shift-and-invert preconditioning strategy to bound the number of eigensolver iterations uniformly. Several numerical examples illustrate the effectiveness of this quasi-optimal preconditioning strategy.

Keywords

  1. periodic Schrödinger equation
  2. iterative eigenvalue solvers
  3. preconditioner
  4. asymptotic eigenvalue analysis
  5. factorization principle
  6. directional homogenization

MSC codes

  1. 65N25
  2. 65F15
  3. 65N30
  4. 35B27
  5. 35B40

Get full access to this article

View all available purchase options and get full access to this article.

References

1.
P. M. Ajayan and O. Z. Zhou, Applications of carbon nanotubes, in Carbon Nanotubes: Synthesis, Structure, Properties, and Applications, M. S. Dresselhaus, G. Dresselhaus, and P. Avouris, eds., Topics in Applied Physics, Springer, Berlin, 2001, pp. 391--425, https://doi.org/10.1007/3-540-39947-X_14.
2.
G. Allaire, Numerical Analysis and Optimization: An Introduction to Mathematical Modelling and Numerical Simulation, Numerical Mathematics and Scientific Computation, Oxford University Press, Oxford, UK, 2007.
3.
G. Allaire, A brief introduction to homogenization and miscellaneous applications, ESAIM Proc., 37 (2012), pp. 1--49, https://doi.org/10.1051/proc/201237001.
4.
G. Allaire and G. Bal, Homogenization of the criticality spectral equation in neutron transport, ESAIM Math. Model. Numer. Anal., 33 (1999), pp. 721--746, https://doi.org/10.1051/m2an:1999160.
5.
G. Allaire and Y. Capdeboscq, Homogenization of a spectral problem in neutronic multigroup diffusion, Comput. Methods Appl. Mech. Eng., 187 (2000), pp. 91--117, https://doi.org/10.1016/S0045-7825(99)00112-7.
6.
G. Allaire and Y. Capdeboscq, Homogenization and localization for a 1-D eigenvalue problem in a periodic medium with an interface, Ann. Mat. Pura Appl., 181 (2002), pp. 247--282, https://doi.org/10.1007/s102310100040.
7.
G. Allaire and C. Conca, Bloch wave homogenization and spectral asymptotic analysis, J. Math. Pures Appl., 77 (1998), pp. 153--208, https://doi.org/10.1016/S0021-7824(98)80068-8.
8.
G. Allaire and F. Malige, Analyse asymptotique spectrale d'un problème de diffusion neutronique, C. R. Acad. Sci. Ser. I Math., 324 (1997), pp. 939--944, https://doi.org/10.1016/S0764-4442(97)86972-8.
9.
G. Allaire and A. Piatnitski, Homogenization of the Schrödinger equation and effective mass theorems, Comm. Math. Phys., 258 (2005), pp. 1--22, https://doi.org/10.1007/s00220-005-1329-2.
10.
M. Aln\aes, J. Blechta, J. Hake, A. Johansson, B. Kehlet, A. Logg, C. Richardson, J. Ring, M. E. Rognes, and G. N. Wells, The FEniCS Project Version 1.5, Arch. Numer. Softw., 3 (2015), https://doi.org/10.11588/ans.2015.100.20553.
11.
R. Altmann, P. Henning, and D. Peterseim, Quantitative Anderson localization of Schrödinger eigenstates under disorder potentials, Math. Models Methods Appl. Sci., 30 (2020), pp. 917--955, https://doi.org/10.1142/S0218202520500190.
12.
R. Altmann, P. Henning, and D. Peterseim, The J-method for the Gross--Pitaevskii eigenvalue problem, Numer. Math., 148 (2021), pp. 575--610, https://doi.org/10.1007/s00211-021-01216-5.
13.
R. Altmann, P. Henning, and D. Peterseim, Localization and delocalization of ground states of Bose--Einstein condensates under disorder, SIAM J. Appl. Math., 82 (2022), pp. 330--358, https://doi.org/10.1137/20M1342434.
14.
R. Altmann and D. Peterseim, Localized computation of eigenstates of random Schrödinger operators, SIAM J. Sci. Comput., 41 (2019), pp. B1211--B1227, https://doi.org/10.1137/19M1252594.
15.
X. Antoine and R. Duboscq, GPELab, a Matlab toolbox to solve Gross--Pitaevskii equations I: Computation of stationary solutions, Comput. Phys. Commun., 185 (2014), pp. 2969--2991, https://doi.org/10.1016/j.cpc.2014.06.026.
16.
X. Antoine and R. Duboscq, GPELab, a Matlab toolbox to solve Gross--Pitaevskii equations II: Dynamics and stochastic simulations, Comput. Phys. Commun., 193 (2015), pp. 95--117, https://doi.org/10.1016/j.cpc.2015.03.012.
17.
X. Antoine, A. Levitt, and Q. Tang, Efficient spectral computation of the stationary states of rotating Bose-Einstein condensates by the preconditioned nonlinear conjugate gradient method, J. Comput. Phys., 343 (2017), pp. 92--109, https://doi.org/10.1016/j.jcp.2017.04.040.
18.
I. Babuška and J. E. Osborn, Finite element--Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems, Math. Comput., 52 (1989), p. 24.
19.
S. Badia and F. Verdugo, Gridap: An extensible finite element toolbox in Julia, J. Open Source Softw., 5 (2020), 2520, https://doi.org/10.21105/joss.02520.
20.
Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, and H. van der Vorst, eds., Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide, Society for Industrial and Applied Mathematics, Philadelphia, 2000, https://doi.org/10.1137/1.9780898719581.
21.
J. Bezanson, A. Edelman, S. Karpinski, and V. B. Shah, Julia: A fresh approach to numerical computing, SIAM Rev., 59 (2017), pp. 65--98, https://doi.org/10.1137/141000671.
22.
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer New York, 2010, https://doi.org/10.1007/978-0-387-70914-7.
23.
E. Cancès, L. Garrigue, and D. Gontier, Second-Order Homogenization of Periodic Schrödinger Operators with Highly Oscillating Potentials, 2021, https://arxiv.org/abs/2112.12008.
24.
E. Cancès, G. Kemlin, and A. Levitt, Convergence analysis of direct minimization and self-consistent iterations, SIAM J. Matrix Anal. Appl., 42 (2021), pp. 243--274, https://doi.org/10.1137/20M1332864.
25.
S. Carr, D. Massatt, S. B. Torrisi, P. Cazeaux, M. Luskin, and E. Kaxiras, Relaxation and domain formation in incommensurate two-dimensional heterostructures, Phys. Rev. B, 98 (2018), 224102, https://doi.org/10.1103/PhysRevB.98.224102.
26.
P. Cazeaux, M. Luskin, and D. Massatt, Energy minimization of two dimensional incommensurate heterostructures, Arch. Rat. Mech. Anal., 235 (2020), pp. 1289--1325, https://doi.org/10.1007/s00205-019-01444-y.
27.
M. Chipot, On some anisotropic singular perturbation problems, Asymptot. Anal., (2007), p. 21, https://doi.org/10.5167/uzh-21524.
28.
M. Chipot, L goes to plus infinity: An update, J. Korean Soc. Ind. Appl. Math., 18 (2014), pp. 107--127, https://doi.org/10.12941/JKSIAM.2014.18.107.
29.
M. Chipot, A. Elfanni, and A. Rougirel, Eigenvalues, eigenfunctions in domains becoming unbounded, in Hyperbolic Problems and Regularity Questions, M. Padula and L. Zanghirati, eds., Birkhäuser, Basel, 2007, pp. 69--78, https://doi.org/10.1007/978-3-7643-7451-8_8.
30.
M. Chipot, W. Hackbusch, S. Sauter, and A. Veit, Numerical approximation of poisson problems in long domains, Vietnam J. Math., (2021), https://doi.org/10.1007/s10013-021-00512-9.
31.
M. Chipot and A. Rougirel, On the asymptotic behaviour of the solution of elliptic problems in cylindrical domains becoming unbounded, Commun. Contemp. Math., 04 (2002), pp. 15--44, https://doi.org/10.1142/S0219199702000555.
32.
M. Chipot and A. Rougirel, On the asymptotic behaviour of the eigenmodes for elliptic problems in domains becoming unbounded, Trans. Amer. Math. Soc., 360 (2008), pp. 3579--3603, https://doi.org/10.1090/S0002-9947-08-04361-4.
33.
M. Chipot, P. Roy, and I. Shafrir, Asymptotics of eigenstates of elliptic problems with mixed boundary data on domains tending to infinity, Asymptot. Anal., 85 (2013), pp. 199--227, https://doi.org/10.3233/ASY-131182.
34.
M. Chipot and Y. Xie, On the asymptotic behaviour of elliptic problems with periodic data, C. R. Math., 339 (2004), pp. 477--482, https://doi.org/10.1016/j.crma.2004.09.007.
35.
R. Courant and D. Hilbert, Methods of Mathematical Physics. Volume I, Wiley, New York, 1989, https://doi.org/10.1002/9783527617210.
36.
R. Dong, D. Li, and L. Wang, Regularity of elliptic systems in divergence form with directional homogenization, Discrete Contin. Dyn. Syst., 38 (2018), pp. 75--90, https://doi.org/10.3934/dcds.2018004.
37.
R. Dong, D. Li, and L. Wang, Directional homogenization of elliptic equations in non-divergence form, J. Differential Equations, 268 (2020), pp. 6611--6645, https://doi.org/10.1016/j.jde.2019.11.041.
38.
M. A. Freitag and A. Spence, Convergence of inexact inverse iteration with application to preconditioned iterative solves, BIT Numer. Math., 47 (2007), pp. 27--44, https://doi.org/10.1007/s10543-006-0100-1.
39.
M. Galewski, B. Galewska, and E. Schmeidel, Conditions for having a diffeomorphism between two Banach spaces, Electron. J. Differential Equations, 99 (2014), pp. 1--6.
40.
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Classics in Mathematics, Springer-Verlag, Berlin, 2001, https://doi.org/10.1007/978-3-642-61798-0.
41.
P. Heid, B. Stamm, and T. P. Wihler, Gradient flow finite element discretizations with energy-based adaptivity for the Gross-Pitaevskii equation, J. Comput. Phys., 436 (2021), 110165, https://doi.org/10.1016/j.jcp.2021.110165.
42.
P. Henning and D. Peterseim, Sobolev gradient flow for the Gross--Pitaevskii eigenvalue problem: Global convergence and computational efficiency, SIAM J. Numer. Anal., 58 (2020), pp. 1744--1772, https://doi.org/10.1137/18M1230463.
43.
A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Frontiers in Mathematics, Birkhäuser, Basel, 2006, https://doi.org/10.1007/3-7643-7706-2.
44.
M. F. Herbst and A. Levitt, Black-box inhomogeneous preconditioning for self-consistent field iterations in density functional theory, J. Phys. Condensed Matter, 33 (2021), 085503, https://doi.org/10.1088/1361-648X/abcbdb.
45.
V. V. Jikov, S. M. Kozlov, and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994, https://doi.org/10.1007/978-3-642-84659-5.
46.
S. Kesavan, Homogenization of elliptic eigenvalue problems: Part 1, Appl. Math. Optim., 5 (1979), pp. 153--167, https://doi.org/10.1007/BF01442551.
47.
S. Kesavan, Homogenization of elliptic eigenvalue problems: Part 2, Appl. Math. Optim., 5 (1979), pp. 197--216, https://doi.org/10.1007/BF01442554.
48.
A. Kutner and A.-M. Sändig, Some Applications of Weighted Sobolev Spaces, Teubner-Texte Zur Mathematik 100, Vieweg+Teubner Verlag, Wiesbaden, 1987, https://doi.org/10.1007/978-3-663-11385-0.
49.
S. Leonardi, The best constant in weighted Poincaré and Friedrichs inequalities, Rend. Semin. Mat. Univ. Padova, 92 (1994), pp. 195--208.
50.
G. Papanicolau, A. Bensoussan, and J.-L. Lions, Asymptotic Analysis for Periodic Structures, Vol. 5, 1st ed., North Holland, Amsterdam, 1978.
51.
Y. Saad, Numerical Methods for Large Eigenvalue Problems, Classics Appl. Math. 66, Society for Industrial and Applied Mathematics, Philadelphia, 2011, https://doi.org/10.1137/1.9781611970739.
52.
F. Santosa and M. Vogelius, First-order corrections to the homogenized eigenvalues of a periodic composite medium, SIAM J. Appl. Math., 53 (1993), pp. 1636--1668, https://doi.org/10.1137/0153076.
53.
J. Sun and A. Zhou, Finite Element Methods for Eigenvalue Problems, Chapman and Hall/CRC, London, 2016, https://doi.org/10.1201/9781315372419.
54.
T. A. Suslina, On homogenization for a periodic elliptic operator in a strip, St. Petersburg Math. J., 16 (2004), pp. 237--258, https://doi.org/10.1090/S1061-0022-04-00849-0.
55.
L. Theisen and B. Stamm, ddEigenLab.jl: Domain-Decomposition Eigenvalue Problem Lab. Zenodo, May 2022, https://doi.org/10.5281/zenodo.6576197.
56.
L. Theisen and M. Torrilhon, fenicsR13: A Tensorial Mixed Finite Element Solver for the Linear R13 Equations Using the FEniCS Computing Platform, ACM Trans. Math. Softw., 47 (2021), pp. 17:1--17:29, https://doi.org/10.1145/3442378.
57.
M. Vanninathan, Homogenization of eigenvalue problems in perforated domains, Proc. Indian Acad. Sci. Sec. A, 90 (1981), pp. 239--271, https://doi.org/10.1007/BF02838079.
58.
K. Varga and J. A. Driscoll, Computational Nanoscience: Applications for Molecules, Clusters, and Solids, Cambridge University Press, Cambridge, UK, 2011, https://doi.org/10.1017/CBO9780511736230.
59.
K. B. Vu, V. V. Vu, H. P. Thi Thu, H. N. Giang, N. M. Tam, and S. T. Ngo, Conjugated polymers: A systematic investigation of their electronic and geometric properties using density functional theory and semi-empirical methods, Synth. Met., 246 (2018), pp. 128--136, https://doi.org/10.1016/j.synthmet.2018.10.007.
60.
Y. Zhang, Estimates of eigenvalues and eigenfunctions in elliptic homogenization with rapidly oscillating potentials, J. Differential Equations, 292 (2021), pp. 388--415, https://doi.org/10.1016/j.jde.2021.05.006.
61.
V. V. Zhikov, Weighted Sobolev spaces, Sb. Math., 189 (1998), pp. 1139--1170, https://doi.org/10.1070/SM1998v189n08ABEH000344.

Information & Authors

Information

Published In

cover image SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Pages: 2508 - 2537
ISSN (online): 1095-7170

History

Submitted: 3 November 2021
Accepted: 1 June 2022
Published online: 15 September 2022

Keywords

  1. periodic Schrödinger equation
  2. iterative eigenvalue solvers
  3. preconditioner
  4. asymptotic eigenvalue analysis
  5. factorization principle
  6. directional homogenization

MSC codes

  1. 65N25
  2. 65F15
  3. 65N30
  4. 35B27
  5. 35B40

Authors

Affiliations

Funding Information

Deutsche Forschungsgemeinschaft https://doi.org/10.13039/501100001659 : 411724963

Metrics & Citations

Metrics

Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By

There are no citations for this item

View Options

View options

PDF

View PDF

Media

Figures

Other

Tables

Share

Share

Copy the content Link

Share with email

Email a colleague

Share on social media