## Abstract

Inhomogeneous dynamical mean-field theory has been employed to solve many interesting strongly interacting problems from transport in multilayered devices to the properties of ultracold atoms in a trap. The main computational step, especially for large systems, is the problem of calculating the inverse of a large sparse matrix to solve Dyson's equation and determine the local Green's function at each lattice site from the corresponding local self-energy. We present a new efficient algorithm, the Lanczos-based low-rank algorithm, for the calculation of the inverse of a large sparse matrix which yields this local (imaginary time) Green's function. The Lanczos-based low-rank algorithm is based on a domain decomposition viewpoint, but avoids explicit calculation of Schur complements and relies instead on low-rank matrix approximations derived from the Lanczos algorithm, for solving the Dyson equation. We report at least a 25-fold improvement of performance compared to explicit decomposition (such as sparse LU) of the matrix inverse. We also report that scaling relative to matrix sizes, of the low-rank correction method on the one hand and domain decomposition methods on the other, are comparable.

Original language | English (US) |
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Pages (from-to) | 22-28 |

Number of pages | 7 |

Journal | Physics Procedia |

Volume | 15 |

DOIs | |

State | Published - 2011 |

Event | 24th Annual Workshop on Recent Developments in Computer Simulation Studies in Condensed Matter Physics, CSP 2011 - Athens, GA, United States Duration: Feb 21 2011 → Feb 25 2011 |

### Bibliographical note

Funding Information:We thank Daniel Osei-Kuffuor for his help with the preconditioners with the software ARMS. We acknowledge Shuxia Zhang, David Porter, and the University of Minnesota Supercomputing Institute for support and resources. This work is sponsored by NSF under grants OCI-0904587 and OCI-0904597, through TeraGrid resources, under the American Recovery and Reinvestment Act (ARRA) of 2009. J.K.F. also acknowledges the McDevitt endowment trust from Georgetown University.

## Keywords

- Green's functions
- Inhomogeneous dynamical mean-field theory
- Lanczos
- Matrix inversion