Open Access
Volume 56, Number 5, September-October 2022
Page(s) 1629 - 1653
Published online 20 July 2022
  1. P.-A. Absil and J. Malick, Projection-like retractions on matrix manifolds. SIAM J. Optim. 22 (2012) 135–158. [Google Scholar]
  2. P.-A. Absil, R. Mahony and R. Sepulchre, Optimization Algorithms on Matrix Manifolds. Princeton University Press, Princeton, NJ (2008). [CrossRef] [Google Scholar]
  3. F. Alouges and C. Audouze, Preconditioned gradient flows for nonlinear eigenvalue problems and application to the Hartree-Fock functional. Numer. Meth. Part. D. E. 25 (2009) 380–400. [CrossRef] [Google Scholar]
  4. R. Altmann and D. Peterseim, Localized computation of eigenstates of random Schrödinger operators. SIAM J. Sci. Comput. 41 (2019) B1211–B1227. [CrossRef] [Google Scholar]
  5. R. Altmann, P. Henning and D. Peterseim, The J-method for the Gross-Pitaevskii eigenvalue problem. Numer. Math. 148 (2021) 575–610. [CrossRef] [MathSciNet] [Google Scholar]
  6. 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) 330–358. [Google Scholar]
  7. W. Bao and Q. Du, Computing the ground state solution of Bose–fEinstein condensates by a normalized gradient flow. SIAM J. Sci. Comput. 25 (2004) 1674–1697. [Google Scholar]
  8. D. Braess, Finite Elements – Theory, Fast Solvers, and Applications in Solid Mechanics, 3rd edition, Cambridge University Press, New York (2007). [CrossRef] [Google Scholar]
  9. E. Cancès, Self-consistent field algorithms for Kohn-Sham models with fractional occupation numbers. J. Chem. Phys. 114 (2001) 10616–10622. [CrossRef] [Google Scholar]
  10. E. Cancès and C. Le Bris, On the convergence of SCF algorithms for the Hartree-Fock equations. ESAIM: M2AN 34 (2000) 749–774. [CrossRef] [EDP Sciences] [Google Scholar]
  11. E. Cancès, R. Chakir and Y. Maday, Numerical analysis of the planewave discretization of some orbital-free and Kohn-Sham models. ESAIM: M2AN 46 (2012) 341–388. [CrossRef] [EDP Sciences] [Google Scholar]
  12. E. Cancès, G. Dusson, Y. Maday, B. Stamm and M. Vohralk, A perturbation-method-based post-processing for the planewave discretization of Kohn-Sham models. J. Comput. Phys. 307 (2016) 446–459. [CrossRef] [MathSciNet] [Google Scholar]
  13. E. Cancès, G. Kemlin and A. Levitt, Convergence analysis of direct minimization and self-consistent iterations. SIAM J. Matrix Anal. Appl. 42 (2021) 243–274. [CrossRef] [MathSciNet] [Google Scholar]
  14. A. Edelman, T.A. Arias and S.T. Smith, The geometry of algorithms with orthogonality constraints. SIAM J. Matrix Anal. Appl. 20 (1998) 303–353. [Google Scholar]
  15. G.H. Golub and C.F. Van Loan, Matrix Computations, 4th edition, The Johns Hopkins University Press, Baltimore, London (2013). [Google Scholar]
  16. P. Harms and A. Mennucci, Geodesics in infinite dimensional Stiefel and Grassmann manifolds. C. R. Math. 350 (2012) 773–776. [CrossRef] [Google Scholar]
  17. 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. [CrossRef] [Google Scholar]
  18. 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) 1744–1772. [Google Scholar]
  19. P. Hohenberg and W. Kohn, Inhomogeneous electron gas. Phys. Rev. 136 (1964) B864–B871. [CrossRef] [Google Scholar]
  20. J. Hu, X. Liu, Z.-W. Wen and Y.-X. Yuan, A brief introduction to manifold optimization. J. Oper. Res. Soc. China 8 (2020) 199–248. [CrossRef] [MathSciNet] [Google Scholar]
  21. E. Jarlebring and P. Upadhyaya, Implicit algorithms for eigenvector nonlinearities. Numer. Algorithms 90 (2022) 301–321. [CrossRef] [MathSciNet] [Google Scholar]
  22. T. Kaneko, S. Fiori and T. Tanaka, Empirical arithmetic averaging over the compact Stiefel manifold. IEEE Trans. Signal Proces. 61 (2013) 883–894. [CrossRef] [Google Scholar]
  23. W. Kohn and L.J. Sham, Self-consistent equations including exchange and correlation effects. Phys. Rev. 140 (1965) A1133–A1138. [CrossRef] [Google Scholar]
  24. P. Lancaster and M. Tismenetsky, The Theory of Matrices, 2nd edition. Academic Press, Orlando, FL (1985). [Google Scholar]
  25. C. Le Bris, Computational chemistry from the perspective of numerical analysis. Acta Numer. 14 (2005) 363–444. [CrossRef] [MathSciNet] [Google Scholar]
  26. E.H. Lieb, R. Seiringer and J. Yngvason, A rigorous derivation of the Gross-Pitaevskii energy functional for a two-dimensional Bose gas. Comm. Math. Phys. 224 (2001) 17–31. [CrossRef] [MathSciNet] [Google Scholar]
  27. J.M. MacLaren, D.P. Clougherty, M.E. McHenry and M.M. Donovan, Parameterised local spin density exchange-correlation energies and potentials for electronic structure calculations I. Zero temperature formalism. Comput. Phys. Commun. 66 (1991) 383–391. [CrossRef] [Google Scholar]
  28. J.P. Perdew and A. Zunger, Self-interaction correction to density-functional approximations for many-electron systems. Phys. Rev. B 23 (1981) 5048–5079. [Google Scholar]
  29. L.P. Pitaevskii and S. Stringari, Bose-Einstein Condensation. Oxford University Press, Oxford (2003). [Google Scholar]
  30. H. Sato and K. Aihara, Cholesky QR-based retraction on the generalized Stiefel manifold. Comput. Optim. Appl. 72 (2019) 293–308. [CrossRef] [MathSciNet] [Google Scholar]
  31. R. Schneider, T. Rohwedder, A. Neelov and J. Blauert, Direct minimization for calculating invariant subspaces in density functional computations of the electronic structure. J. Comput. Math. 27 (2009) 360–387. [Google Scholar]
  32. M.P. Teter, M.C. Payne and D.C. Allan, Solution of Schrödinger’s equation for large systems. Phys. Rev. B 40 (1989) 12255–12263. [CrossRef] [PubMed] [Google Scholar]
  33. A. Uschmajew, Well-posedness of convex maximization problems on Stiefel manifolds and orthogonal tensor product approximations. Numer. Math. 115 (2010) 309–331. [CrossRef] [MathSciNet] [Google Scholar]
  34. Z. Wen and W. Yin, A feasible method for optimization with orthogonality constraints. Math. Program. 142 (2013) 397–434. [CrossRef] [MathSciNet] [Google Scholar]
  35. C. Yang, J.C. Meza and L.-W. Wang, A constrained optimization algorithm for total energy minimization in electronic structure calculation. J. Comput. Phys. 217 (2006) 709–721. [CrossRef] [MathSciNet] [Google Scholar]
  36. C. Yang, J.C. Meza, B. Lee and L.-W. Wang, KSSOLV – a MATLAB toolbox for solving the Kohn-Sham equations. ACM Trans. Math. Softw. 36 (2009) 1–35. [CrossRef] [Google Scholar]
  37. E. Zeidler, Nonlinear Functional Analysis and its Applications IIa: Linear Monotone Operators. Springer-Verlag, New York (1990). [Google Scholar]
  38. Z. Zhang, Exponential convergence of Sobolev gradient descent for a class of nonlinear eigenproblems. Commun. Math. Sci. 20 (2022) 377–403. [CrossRef] [MathSciNet] [Google Scholar]
  39. H. Zhang and W.W. Hager, A nonmonotone line search technique and its application to unconstrained optimization. SIAM J. Optim. 14 (2004) 1043–1056. [Google Scholar]

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