Efficiently generalizing ultracold atomic simulations via inhomogeneous dynamical mean-field theory from two to three dimensions
|Title||Efficiently generalizing ultracold atomic simulations via inhomogeneous dynamical mean-field theory from two to three dimensions|
|Publication Type||Book Chapter|
|Year of Publication||2011|
|Authors||Freericks, J. K., Krishnamurthy H. R., Carrier P., and Saad Y.|
|Book Title||Proceedings of the HPCMP Users Group Conference 2010, Chicago, IL, June 14--17, 2010, edited by D. E. Post|
|Publisher||IEEE Computer Society|
|City||Los Alamitos, CA|
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.
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