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)2. (Real and imaginary time) Dynamical Mean-Field Theory (DMFT)

Description of the numerical solvers: 3. NLC: eigenvalue problems with Lanczos4. DMFT: diagonal of the inverse with Lanczos (consider also direct methods, probing,...)

5. Optimization and I/O of Lanczos basis vectors

1/16Extreme Scale I/O and Data Analysis Workshop March 22-24 2010, Austin, Tx

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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.

C.Domb and M. S. GreenPhase Transitions and Critical Phenomena (Academic Press, New York, 1974)

J. Oitmaa, Ch. Hamer, andW. ZhengSeries Expansion Methods for Strongly Interacting Lattice Models (Cambridge Univ. Press, Melbourne, 2006)

M. Rigol, T. Bryant, and R. R. P. SinghNumerical Linked-Cluster Algorithms: I. Spin systems on square, triangular, and kagomé latticesPhys. Rev. E 75, 061118 (2007); arXiv:0706.3254Numerical Linked-Cluster Algorithms: II. t-J models on the square latticePhys. Rev. E 75, 061119 (2007); arXiv:0706.3255

Main references:

Goal: Compute finite-temperature properties of generic quantum lattice systems at low temperatures.

1. Numerical-Linked clusters (NLC): physical problem

http://link.aps.org/abstract/PRE/v75/e061118





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Goal: Compute finite-temperature properties of generic quantum lattice systems at low temperatures.

Example:

Phys. Rev. E 75, 061118 (2007)

3 sites 4 sites ....N sites

1. Numerical-Linked clusters (NLC): physical problem

One example of sparse H:

1 2 3 4 5 6 7 8 9

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 161 1 1 3 4 10 19 51 112 300 746 2,042 5,450 15,197 42,192 119,561

number of sites:number of clusters:

•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)

c =




2. Dynamical Mean-Field Theory (DMFT): physical problem

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James K. FreericksTransport 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 dimensionsPhys. 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.

Main references:

Goal: Compute fermion-boson many-body interactions for ultracold atoms in optical lattices.

Imaginary time:

Real time:

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:



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Imaginary time:

Real time:

Off-diagonal elements are

Hopping matrix tab forms the off-diagonal elements

and takes values 0 or 1, only

•G is complex symmetric non-hermitian•Size of system is very large •Several values of need to be evaluated•G is sparse; pattern fixed•moderate precision required (10-8)•Imaginary time problem is diagonally dominant


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DMFT loop (e.g., imaginary time):

Impurity solver:

Continuation:

Chemical potential: (for imaginary time only)

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:




Description of the numerical solvers: 3. NLC: eigenvalue problems with Lanczos4. DMFT: diagonal of the inverse with Lanczos


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3. NLC: eigenvalue problems with Lanczos: numerical solvers

Sparse matrix:Tridiagonal matrix:

Much easier to diagonalize8/16

3. NLC: eigenvalue problems with Lanczos: numerical solvers

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Initialization

Update

Lanczos’ recurrence

Simon’s and Kahan’s re-orthogonalization schemes

4. DMFT: diagonal of the inverse with Lanczos

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After some algebra,

one gets the diaginv algorithm...

Tm decomposition is

http://www.msi.umn.edu/~carrierp/images/DIAGINV_Lanczos.pdf


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Lanczosroutine

diaginvroutine

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Example of the diagonal of inverse Green’s functionsmall matrix of dimension 441 X 441





Description of the numerical solvers: 3. NLC: eigenvalue problems with Lanczos4. DMFT: diagonal of the inverse with Lanczos

5. Optimization and I/O of Lanczos basis vectors (beginning of project: Sept ’09)

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All algorithms have been tested on the 2D problems and give accurate solutions, relatively fast


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Impurity solver:

Continuation:

Chemical potential: (for imaginary time only)

Lanczos routine

diaginv routine

swap?

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I/O of Lanczos vectorsCompression (wavelets)


Load vectors by blocksfrom disk

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

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