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Lanczos approach using out-of-core memory for eigenvalues and diagonal of inverse problems

Lanczos approach using out-of-core memory for eigenvalues and diagonal of inverse problems. Pierre Carrier , Yousef Saad, James Freericks, Ehsan Khatami, Marcos Rigol, Tarek El-Ghazawi. Description of the physical problems: 1. Numerical Linked-clusters (NLC)

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Lanczos approach using out-of-core memory for eigenvalues and diagonal of inverse problems

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  1. Lanczos approach using out-of-core memory for eigenvalues and diagonal of inverse problems Pierre Carrier, Yousef Saad, James Freericks, Ehsan Khatami, Marcos Rigol, Tarek El-Ghazawi • Description of the physical problems: • 1. Numerical Linked-clusters (NLC) • 2. (Real and imaginary time) Dynamical Mean-Field Theory (DMFT) • Description of the numerical solvers: • 3. NLC: eigenvalue problems with Lanczos • 4. DMFT: diagonal of the inverse with Lanczos (consider also direct methods, probing,...) • 5. Optimization and I/O of Lanczos basis vectors 1/16 Extreme Scale I/O and Data Analysis Workshop March 22-24 2010, Austin, Tx

  2. 1. Numerical-Linked clusters (NLC): physical problem Goal: Compute finite-temperature properties of generic quantum lattice systems at low temperatures. Principles: Based on the linked-cluster basis of high-temperature expansions [Domb & Green], where analytical expansion in 1/T are replaced by an exact numerical calculation. Main references: C.Domb and M. S. Green Phase Transitions and Critical Phenomena (Academic Press, New York, 1974) J. Oitmaa, Ch. Hamer, andW. Zheng Series Expansion Methods for Strongly Interacting Lattice Models (Cambridge Univ. Press, Melbourne, 2006) M. Rigol, T. Bryant, and R. R. P. Singh Numerical Linked-Cluster Algorithms: I. Spin systems on square, triangular, and kagomé lattices Phys. Rev. E 75, 061118 (2007); arXiv:0706.3254 Numerical Linked-Cluster Algorithms: II. t-J models on the square lattice Phys. Rev. E 75, 061119 (2007); arXiv:0706.3255 2/16

  3. 1. Numerical-Linked clusters (NLC): physical problem Goal: Compute finite-temperature properties of generic quantum lattice systems at low temperatures. 3 sites 4 sites ....N sites c = 1 2 3 4 5 6 7 8 9 Example: • number of sites: • number of clusters: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 1 1 3 4 10 19 51 112 300 746 2,042 5,450 15,197 42,192 119,561 • One example of sparse H: • Very large number of eigenvalue problems • All eigenvalues of H are required • H is sparse; pattern not fixed; dimension 2N • Very high-precision required (10-12) Phys. Rev. E 75, 061118 (2007) 3/16

  4. 2. Dynamical Mean-Field Theory (DMFT): physical problem Goal: Compute fermion-boson many-body interactions for ultracold atoms in optical lattices. Principles: Based on a mapping of the lattice problem onto an impurity problem that mimics the motion of electrons via their hopping from site to site, by solving the diagonal of the inverse of Dyson’s equations: Imaginary time: Real time: Main references: James K. Freericks Transport in multilayered nanostructures, the dynamical mean-field theory approach (Imperial College Press, London, 2006) W. Metzner and D. Volhardt, Correlated Lattice Fermions in d=infinity dimensions Phys. Rev. Lett. 62, 324 (1988) G. Kotliar and D. Vollhardt Strongly Correlated Materials: Insights from Dynamical Mean-Field Theory Physics Today 57, No. 3 (March), 53 (2004); References therein. 4/16

  5. 2. Dynamical Mean-Field Theory (DMFT): physical problem Off-diagonal elements are Imaginary time: Real time: • G is complex symmetric non-hermitian • Size of system is very large Hopping matrix tab forms the off-diagonal elements and takes values 0 or 1, only • Several values of need to be evaluated • G is sparse; pattern fixed • moderate precision required (10-8) • Imaginary time problem is diagonally dominant 5/16

  6. 2. Dynamical Mean-Field Theory (DMFT): physical problem Principles: Based on a mapping of the lattice problem onto an impurity problem that mimics the motion of electrons via their hopping from site to site, by solving the diagonal of the inverse of Dyson’s equations: DMFT loop (e.g., imaginary time): Impurity solver: Chemical potential: (for imaginary time only) Continuation: 6/16

  7. Lanczos approach using out-of-core memory for eigenvalues and diagonal of inverse problems Pierre Carrier, Yousef Saad, James Freericks, Ehsan Khatami, Marcos Rigol, Tarek El-Ghazawi • Description of the physical problems: • 1. Numerical Linked-clusters (NLC) • 2. (Real and imaginary time) Dynamical Mean-Field Theory (DMFT) • Description of the numerical solvers: • 3. NLC: eigenvalue problems with Lanczos • 4. DMFT: diagonal of the inverse with Lanczos • 5. Optimization and I/O of Lanczos basis vectors 7/16

  8. 3. NLC: eigenvalue problems with Lanczos: numerical solvers Sparse matrix: Tridiagonal matrix: Much easier to diagonalize 8/16

  9. 3. NLC: eigenvalue problems with Lanczos: numerical solvers Initialization Lanczos’ recurrence Simon’s and Kahan’s re-orthogonalization schemes Update 9/16

  10. 3. NLC: eigenvalue problems with Lanczos: numerical solvers Initialization Lanczos’ recurrence Simon’s and Kahan’s re-orthogonalization schemes Update 9/16

  11. 4. DMFT: diagonal of the inverse with Lanczos 10/16

  12. 4. DMFT: diagonal of the inverse with Lanczos Tm decomposition is After some algebra, one gets the diaginv algorithm... http://www.msi.umn.edu/~carrierp/images/DIAGINV_Lanczos.pdf 11/16

  13. 4. DMFT: diagonal of the inverse with Lanczos Lanczos routine diaginv routine 12/16

  14. 4. DMFT: diagonal of the inverse with Lanczos Example of the diagonal of inverse Green’s function small matrix of dimension 441 X 441 13/16

  15. Lanczos approach using out-of-core memory for eigenvalues and diagonal of inverse problems Pierre Carrier, Yousef Saad, James Freericks, Ehsan Khatami, Marcos Rigol, Tarek El-Ghazawi All algorithms have been tested on the 2D problems and give accurate solutions, relatively fast • Description of the physical problems: • 1. Numerical Linked-clusters (NLC) • 2. (Real and imaginary time) Dynamical Mean-Field Theory (DMFT) • Description of the numerical solvers: • 3. NLC: eigenvalue problems with Lanczos • 4. DMFT: diagonal of the inverse with Lanczos • 5. Optimization and I/O of Lanczos basis vectors (beginning of project: Sept ’09) 14/16

  16. 5. Optimization and I/O of Lanczos basis vectors Lanczos routine swap? diaginv routine Impurity solver: Chemical potential: (for imaginary time only) Continuation: 15/16

  17. 5. Optimization and I/O of Lanczos basis vectors I/O of Lanczos vectors Compression (wavelets) Load vectors by blocks from disk 16/16

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