Extension and optimization of the FIND algorithm: Computing Green’s and less-than Green’s functions

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 Nx × Ny mesh, the best known algorithms have a running time of O(Nx3Ny), whereas FIND only requires O(Nx2Ny). 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 130 × 130 2D mesh down to a 40 × 40 mesh. These improvements make the optimized FIND algorithm even more competitive for real-life applications.

► FIND is an algorithm for calculating entries of the inverse of a sparse matrix.
► We extend the algorithm to other matrix inverse related calculations.
► We exploit sparsity and symmetry to improve performance.