Localized computation of eigenstates of random Schrödinger operators. (English) Zbl 1486.65233
The paper concerns the numerical approximation of low-energy eigenstates of the linear random Schrödinger operator. The authors are interested in both the identification of the regions of localization and also actual approximations of the lowermost eigenstates. A finite element method is proposed, which integrates several techniques including the inverse iteration, a multigrid-based pre-conditioner, an appropriate starting subspace, and parallelizations. Numerical examples in two and three dimensions, including a nonlinear problem, are shown and the results are compared with those of the Matlab solver eigs.
Reviewer: Jiguang Sun (Houghton)
MSC:
| 65N25 | Numerical methods for eigenvalue problems for boundary value problems involving PDEs |
| 47B80 | Random linear operators |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N55 | Multigrid methods; domain decomposition for boundary value problems involving PDEs |
| 35R60 | PDEs with randomness, stochastic partial differential equations |
Keywords:
Schrödinger operator; Anderson localization; random potential; iterative eigensolver; multigrid solverReferences:
| [1] | D. N. Arnold, G. David, D. Jerison, S. Mayboroda, and M. Filoche, Effective confining potential of quantum states in disordered media, Phys. Rev. Lett., 116 (2016), 056602. |
| [2] | R. Altmann, P. Henning, and D. Peterseim, Quantitative Anderson Localization of Schrödinger Eigenstates under Disorder Potentials, preprint, https://arxiv.org/abs/1803.09950, 2018. · Zbl 1444.65064 |
| [3] | R. Altmann, P. Henning, and D. Peterseim, The \(J\)-Method for the Gross-Pitaevskii Eigenvalue Problem, preprint, https://arxiv.org/abs/1908.00333, 2019. · Zbl 1510.65289 |
| [4] | P. W. Anderson, Absence of diffusion in certain random lattices, Phys. Rev., 109 (1958), pp. 1492-1505. |
| [5] | R. Altmann, D. Peterseim, and D. Varga, Localization studies for ground states of the Gross-Pitaevskii equation, Proc. Appl. Math. Mech., 18 (2018), e201800343. |
| [6] | W. Bao and Q. Du, Computing the ground state solution of Bose-Einstein condensates by a normalized gradient flow, SIAM J. Sci. Comput., 25 (2004), pp. 1674-1697, https://doi.org/10.1137/S1064827503422956. · Zbl 1061.82025 |
| [7] | D. Boiron, C. Mennerat-Robilliard, J.-M. Fournier, L. Guidoni, C. Salomon, and G. Grynberg, Trapping and cooling cesium atoms in a speckle field, Eur. Phys. J. D, 7 (1999), pp. 373-377. |
| [8] | S. N. Bose, Plancks Gesetz und Lichtquantenhypothese, Z. Phys., 26 (1924), pp. 178-181. · JFM 51.0732.04 |
| [9] | J. H. Bramble, J. E. Pasciak, and J. Xu, Parallel multilevel preconditioners, Math. Comp., 55 (1990), pp. 1-22. · Zbl 0703.65076 |
| [10] | S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, 3rd ed., Springer, New York, 2008. · Zbl 1135.65042 |
| [11] | F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Theory of Bose-Einstein condensation in trapped gases, Rev. Modern Phys., 71 (1999), pp. 463-512. |
| [12] | D. D. Duncan, S. Kirkpatrick, and R. Wang, Statistics of local speckle contrast, J. Opt. Soc. Amer. A, 25 (2008), pp. 9-15. |
| [13] | A. Einstein, Quantentheorie des einatomigen idealen Gases, Sitzungber. Kgl. Preuss. Akad. Wiss., 1924, pp. 261-267. |
| [14] | L. Fallani, C. Fort, and M. Inguscio, Bose-Einstein condensates in disordered potentials, in Advances in Atomic, Molecular, and Optical Physics, Vol. 56, Academic Press, 2008, pp. 119-160. |
| [15] | M. Filoche and S. Mayboroda, Universal mechanism for Anderson and weak localization, Proc. Natl. Acad. Sci. USA, 109 (2012), pp. 14761-14766. |
| [16] | A. Gloria, Reduction of the resonance error – Part 1: Approximation of homogenized coefficients, Math. Models Methods Appl. Sci., 21 (2011), pp. 1601-1630. · Zbl 1233.35016 |
| [17] | J. W. Goodman, Statistical Properties of Laser Speckle Patterns, Springer, Berlin, Heidelberg, 1975, pp. 9-75. |
| [18] | W. Hackbusch, Multigrid Methods and Applications, Springer-Verlag, Berlin, 1985. · Zbl 0595.65106 |
| [19] | P. Henning and D. Peterseim, Sobolev Gradient Flow for the Gross-Pitaevskii Eigenvalue Problem: Global Convergence and Computational Efficiency, preprint, https://arxiv.org/abs/1812.00835, 2018. · Zbl 1512.35538 |
| [20] | A. V. Knyazev and K. Neymeyr, Efficient solution of symmetric eigenvalue problems using multigrid preconditioners in the locally optimal block conjugate gradient method, Electron. Trans. Numer. Anal., 15 (2003), pp. 38-55. · Zbl 1031.65126 |
| [21] | A. V. Knyazev, Preconditioned eigensolvers, in Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide, Software Environ. Tools 11, Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, and H. van der Vorst, eds., SIAM, Philadelphia, 2000, pp. 337-368, https://doi.org/10.1137/1.9780898719581.ch11. |
| [22] | A. V. Knyazev, Toward the optimal preconditioned eigensolver: Locally optimal block preconditioned conjugate gradient method, SIAM J. Sci. Comput., 23 (2001), pp. 517-541, https://doi.org/10.1137/S1064827500366124. · Zbl 0992.65028 |
| [23] | R. Kornhuber and H. Yserentant, Numerical homogenization of elliptic multiscale problems by subspace decomposition, Multiscale Model. Simul., 14 (2016), pp. 1017-1036, https://doi.org/10.1137/15M1028510. · Zbl 1352.65521 |
| [24] | J. E. Lye, L. Fallani, M. Modugno, D. S. Wiersma, C. Fort, and M. Inguscio, Bose-Einstein condensate in a random potential, Phys. Rev. Lett., 95 (2005), 070401. |
| [25] | J. Lu and S. Steinerberger, Detecting localized eigenstates of linear operators, Res. Math. Sci., 5 (2018), 33. · Zbl 1425.35128 |
| [26] | S. F. McCormick, Multigrid Methods, SIAM, Philadelphia, 1987, https://doi.org/10.1137/1.9781611971057. |
| [27] | L. P. Pitaevskii and S. Stringari, Bose-Einstein Condensation, Oxford University Press, New York, 2003. · Zbl 1110.82002 |
| [28] | J. Rogel-Salazar, The Gross-Pitaevskii equation and Bose-Einstein condensates, European J. Phys., 34 (2013), pp. 247-257. · Zbl 1267.82037 |
| [29] | G. Strang and G. J. Fix, An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, NJ, 1973. · Zbl 0356.65096 |
| [30] | S. Steinerberger, Localization of quantum states and landscape functions, Proc. Amer. Math. Soc., 145 (2017), pp. 2895-2907. · Zbl 1365.35098 |
| [31] | D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, Localization of light in a disordered medium, Nature, 390 (1997), pp. 671-673. |
| [32] | H. Xie, L. Zhang, and H. Owhadi, Fast Eigenpairs Computation with Operator Adapted Wavelets and Hierarchical Subspace Correction, preprint, https://arxiv.org/abs/1806.00565, 2018. · Zbl 1447.65165 |
| [33] | H. Yserentant, On the multilevel splitting of finite element spaces, Numer. Math., 49 (1986), pp. 379-412. · Zbl 0608.65065 |
| [34] | H. Yserentant, Old and new convergence proofs for multigrid methods, Acta Numer., 2 (1993), pp. 285-326. · Zbl 0788.65108 |
| [35] | H. Yserentant, A short theory of the Rayleigh-Ritz method, Comput. Methods Appl. Math., 13 (2013), pp. 495-502. · Zbl 1447.65168 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
