Location of Repository

On large-scale diagonalization techniques for the Anderson model of localization

By Olaf Schenk, Matthias Bollhofer and Rudolf A. Roemer

Abstract

We propose efficient preconditioning algorithms for an eigenvalue problem arising in quantum physics, namely the computation of a few interior eigenvalues and their associated eigenvectors for large-scale sparse real and symmetric indefinite matrices of the Anderson model\ud of localization. We compare the Lanczos algorithm in the 1987 implementation by Cullum and Willoughby with the shift-and-invert techniques in the implicitly restarted Lanczos method and in the Jacobi–Davidson method. Our preconditioning approaches for the shift-and-invert symmetric indefinite linear system are based on maximum weighted matchings and algebraic multilevel incomplete\ud LDLT factorizations. These techniques can be seen as a complement to the alternative idea of using more complete pivoting techniques for the highly ill-conditioned symmetric indefinite Anderson matrices. We demonstrate the effectiveness and the numerical accuracy of these algorithms. Our numerical examples reveal that recent algebraic multilevel preconditioning solvers can accelerate the computation of a large-scale eigenvalue problem corresponding to the Anderson model of localization\ud by several orders of magnitude

Topics: QC
Publisher: Society for Industrial and Applied Mathematics
Year: 2006
OAI identifier: oai:wrap.warwick.ac.uk:346

Suggested articles

Preview

Citations

  1. (1998). A fast and high quality multilevel scheme for partitioning irregular graphs, doi
  2. (1999). A Fast Maximum-Weight-Bipartite-Matching Algorithm for Reducing Pivoting in Sparse Gaussian Elimination,
  3. (1996). A Jacobi–Davidson iteration for linear eigenvalue problems, doi
  4. (1996). A new pivoting strategy for Gaussian elimination, doi
  5. (1985). A proposal for an extended set of Fortran basic linear algebra subprograms, doi
  6. (1997). A QMR-based interior-point algorithm for solving linear programs, doi
  7. (1999). A supernodal approach to sparse partial pivoting, doi
  8. (1958). Absence of diffusion in certain random lattices, doi
  9. (1996). An approximate minimum degree ordering algorithm, doi
  10. (1979). An estimate for the condition number of a matrix, doi
  11. (1985). An implementation of Gaussian elimination with partial pivoting for sparse systems, doi
  12. (1996). Critical dynamics and multifractal exponents at the Anderson transition in 3d disordered systems, doi
  13. (2003). Crout versions of ILU for general sparse matrices, doi
  14. (1998). Electronic states in topologically disordered systems, doi
  15. (2000). Energy-level statistics at the metal-insulator transition in anisotropic systems, doi
  16. (2004). Felice, Charge transport in DNA-based devices,i n Long-Range Charge Transfer doi
  17. (1984). Fractal character of eigenstates in disordered systems, doi
  18. (1986). Fractal dimensionality of wave functions at the mobility edge: Quantum fractal in the Landau levels, doi
  19. (1986). GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, doi
  20. (1992). Implicit application of polynomial filters in a k-step Arnoldi method, doi
  21. (2005). Inner Iterations in Eigenvalue Solvers,
  22. (1985). Issues relating to extension of the basic linear algebra subprograms, doi
  23. (2003). Iterative Methods for Sparse Linear Systems, doi
  24. (1998). Jacobi–Davidson style QR and QZ algorithms for the reduction of matrix pencils, doi
  25. (1985). Lanczos Algorithms for Large Symmetric Eigenvalue Computations, Volume 1: Theory, Birkh¨ auser doi
  26. (1985). Lanczos Algorithms for Large Symmetric Eigenvalue Computations, Volume 2: Programs, Birkh¨ auser doi
  27. (1992). Localization of electronic states in 2D disordered systems, doi
  28. (1993). Localization: Theory and experiment, doi
  29. (1997). Multifractal analysis of the metal-insulator transition in anisotropic systems, doi
  30. (2005). Multilevel preconditioned iterative eigensolvers for Maxwell eigenvalue problems, doi
  31. (2006). Multilevel preconditioners constructed from inverse-based ILUs, doi
  32. (2005). Nearly Optimal Preconditioned Methods for Hermitian Eigenproblems under Limited Memory. Part I: Seeking One Eigenvalue, doi
  33. (2003). Numerical investigations of scaling at the Anderson transition, in The Anderson Transition and Its Ramifications—Localisation, Quantum Interference, doi
  34. (2005). Numerical solution of saddle point problems, doi
  35. (1981). Off-diagonal disorder in one-dimensional systems, doi
  36. (2004). On Fast Factorization Pivoting Methods for Symmetric Indefi-nite Systems,
  37. (1974). Partial pivoting strategies for symmetric matrices, doi
  38. (1999). Phase diagram of the three-dimensional Anderson model of localization with random hopping,
  39. (2000). Preconditioning highly indefinite and nonsymmetric matrices, doi
  40. (2002). Preconditioning techniques for large linear systems: A survey, doi
  41. (1993). Predicting structure in nonsymmetric sparse matrix factorizations, doi
  42. (1997). Scaling properties in highly anisotropic systems, doi
  43. (1995). Software for simplified Lanczos and QMR algorithms, doi
  44. (1975). Solution of sparse indefinite systems of linear equations, doi
  45. (2004). Solving unsymmetric sparse systems of linear equations with PARDISO, doi
  46. (1977). Some stable methods for calculating inertia and solving symmetric linear systems, doi
  47. (2004). Strategies for Scaling and Pivoting for Sparse Symmetric Indefinite Problems, doi
  48. (2003). SuperLU DIST: A scalable distributed-memory sparse direct solver for unsymmetric linear systems, doi
  49. (2002). Symmetric weighted matching for indefinite systems, talk presented at the Householder Symposium XV,
  50. (1999). The Anderson model of localization: A challenge for modern eigenvalue methods, doi
  51. (1990). The Anderson transition: New numerical results for the critical exponents, doi
  52. (1999). The design and use of algorithms for permuting large entries to the diagonal of sparse matrices, doi
  53. (2004). The effects of unsymmetric matrix permutations and scalings in semiconductor device and circuit simulation, doi
  54. (2004). The Iterative Eigensolver Template Library,
  55. (2002). The Jacobi–Davidson Algorithm for Solving Large Sparse Symmetric Eigenvalue Problems with Application to the Design of Accelerator Cavities,
  56. (1983). The multifrontal solution of indefinite sparse symmetric linear equations, doi
  57. (1980). The Symmetric Eigenvalue Problem, Prentice-Hall, doi
  58. (1998). The two-dimensional Anderson model of localization with random hopping, doi
  59. (2003). Three-dimensional Anderson model of localization with binary random potential, doi
  60. (2002). Use of cluster computing for the Anderson model of localization, doi
  61. (2006). Weighted matchings for preconditioning symmetric indefinite linear systems, doi

To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.