On Complex Preconditioning for the solution of the Kohn-Sham Equation for Molecular Transport

This paper analyzes the performance of a few preconditioners for solving the complex linear systems which originate from the application of real space pseudopotential methods in the study of electron transport properties of nanoscale junctions. These linear systems of equations are part of a self-consistent loop to compute the charge density and corresponding potential at the nanoscale junctions. The coefficient matrices for these systems have a regular structure with two dense blocks associated with boundary conditions. These dense blocks cause difficulties to general preconditioners, due to the amount of fill-in which they tend to generate. The preconditioners studied are of the general-purpose kind, such as ILU with threshold (ILUT), ILU with level of fill (ILUK), and the Algebraic Recursive Multilevel Solvers (ARMS). The study shows that ARMS with diagonal dominance – based nonsymmetric ordering (ddPQ) is generally more robust than the other preconditioners which were tested.