Extension and optimization of the FIND algorithm: computing Green's and less-than Green's functions (with technical appendix)

TL;DR

This paper extends and optimizes the FIND algorithm for calculating Green's functions in sparse matrices, significantly reducing computation time and making it more practical for real-world quantum transport simulations.

Contribution

The authors extended the FIND algorithm to compute less-than Green's functions and optimized it to reduce constant factors, enhancing its efficiency for larger problems.

Findings

Reduced constant factors in FIND algorithm computations.

Decreased problem size threshold where FIND outperforms other methods.

Enhanced applicability of FIND in real-life quantum transport simulations.

Abstract

The FIND algorithm is a fast algorithm designed to calculate certain entries of the inverse of a sparse matrix. Such calculation is critical in many applications, e.g., quantum transport in nano-devices. We extended the algorithm to other matrix inverse related calculations. Those are required for example to calculate the less-than Green's function and the current density through the device. For a 2D device discretized as an N_x x N_y mesh, the best known algorithms have a running time of O(N_x^3 N_y), whereas FIND only requires O(N_x^2 N_y). Even though this complexity has been reduced by an order of magnitude, the matrix inverse calculation is still the most time consuming part in the simulation of transport problems. We could not reduce the order of complexity, but we were able to significantly reduce the constant factor involved in the computation cost. By exploiting the sparsity and symmetry, the size of the problem beyond which FIND is faster than other methods typically decreases from a 130x130 2D mesh down to a 40x40 mesh. These improvements make the optimized FIND algorithm even more competitive for real-life applications.