Patchwork algorithm for the parallel computation of the Green's function in open systems

The efficient calculation of the Green's function is a central issue for assessing electronic transport at the nanoscale. In a near-to-equilibrium description, it can be obtained from a matrix inversion, combined with iterative algorithms developed in the 80s. However, this procedure becomes computationally challenging when dealing with very large systems. A set of algorithms (known as knitting and sewing) based on the recursive application of Dyson's equation were recently proposed, where the Green's function elements are obtained in a selective way and without the need of explicit matrix inversion, by including one matrix element at a time. Here we propose a variation of these algorithms adapted to parallel computing. The approach is based on the division of the system in a set of domains whose individual Green's functions are computed independently. The domains are then merged to yield the necessary elements of the Green's function for subsequent evaluation of the electronic transport properties. Promising scaling behavior is found, depending on the details of the domain decomposition.