TY - GEN

T1 - Efficiently generalizing ultra-cold atomic simulations via inhomogeneous dynamical mean-field theory from two-to three-dimensions

AU - Freericks, J. K.

AU - Krishnamurthy, H. R.

AU - Carrier, P.

AU - Saad, Yousef

PY - 2011

Y1 - 2011

N2 - We describe techniques that we are implementing to move inhomogeneous dynamical mean-field theory simulations from two-to three-dimensions. Two-dimensional simulations typically run on 2,000-10,000 lattice sites, while three-dimensional simulations typically need to run on 1,000,000 or more lattice sites. The inhomogeneous dynamical mean-field theory requires the diagonal of the inverse of many sparse matrices with the same sparsity pattern, and a dimension equal to the number of lattice-sites. For two-dimensional systems, we have employed general dense LAPACK routines since the matrices are small enough. For three-dimensional systems, we need to employ sparse matrix techniques. Here, we present one possible strategy for the sparse matrix routine, based on the well-known Lanczos technique, with a long run of the algorithm and (partial) reorthogonalization. This approach is about two-times faster than the LAPACK routines with identical accuracy, and hence will become the standard we use on the two-dimensional problems. We illustrate this approach on the problem of increasing the efficiency for pre-forming dipolar molecules in K-Rb mixtures on a lattice. We compare the local density approximation to inhomogeneous dynamical mean-field theory to illustrate how the local density approximation fails at low-temperature, and to illustrate the benefits of the new algorithms. For a three-dimensional problem, a speed-up of 1,000 or more times is needed. We end by discussing some options that are promising toward reaching this goal.

AB - We describe techniques that we are implementing to move inhomogeneous dynamical mean-field theory simulations from two-to three-dimensions. Two-dimensional simulations typically run on 2,000-10,000 lattice sites, while three-dimensional simulations typically need to run on 1,000,000 or more lattice sites. The inhomogeneous dynamical mean-field theory requires the diagonal of the inverse of many sparse matrices with the same sparsity pattern, and a dimension equal to the number of lattice-sites. For two-dimensional systems, we have employed general dense LAPACK routines since the matrices are small enough. For three-dimensional systems, we need to employ sparse matrix techniques. Here, we present one possible strategy for the sparse matrix routine, based on the well-known Lanczos technique, with a long run of the algorithm and (partial) reorthogonalization. This approach is about two-times faster than the LAPACK routines with identical accuracy, and hence will become the standard we use on the two-dimensional problems. We illustrate this approach on the problem of increasing the efficiency for pre-forming dipolar molecules in K-Rb mixtures on a lattice. We compare the local density approximation to inhomogeneous dynamical mean-field theory to illustrate how the local density approximation fails at low-temperature, and to illustrate the benefits of the new algorithms. For a three-dimensional problem, a speed-up of 1,000 or more times is needed. We end by discussing some options that are promising toward reaching this goal.

KW - Bose atoms

KW - Inhomogeneous dynamical mean-field theory

KW - mxitures of Fermi

KW - preforming dipolar matter

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U2 - 10.1109/HPCMP-UGC.2010.17

DO - 10.1109/HPCMP-UGC.2010.17

M3 - Conference contribution

AN - SCOPUS:80053374360

SN - 9780769543925

T3 - Proceedings - 2010 DoD High Performance Computing Modernization Program Users Group Conference, HPCMP UGC 2010

SP - 221

EP - 227

BT - Proceedings - 2010 DoD High Performance Computing Modernization Program Users Group Conference, HPCMP UGC 2010

T2 - 2010 DoD High Performance Computing Modernization Program Users Group Conference, HPCMP UGC 2010

Y2 - 14 June 2010 through 17 June 2010

ER -