Model Reduction in Symbolically Semi-separable Systems with Application to Pre-conditioners for 3D Sparse Systems of Equations

Preconditioned iterative solvers are considered to be one of the most promising methods for solving large and sparse linear systems. It has been shown in the literature that their impact can be fairly easily extended to semi-separable systems or even larger classes build on semi-separable ideas. In this paper, we propose and evaluate a new type of preconditioners for the class of matrices that have a two level deep ‘symbolically hierarchical semiseparable form’ meaning that the matrices have a semi-separable like block structure with blocks that are (sequentially) semi-separable themselves. The new preconditioners are based on approximations of Schur complements in a sequential or hierarchical decomposition of the original block matrix. The type of matrices considered commonly occur in 3D modeling problems. Mathematics Subject Classification (2000). 65F10, 65F15; 65F50; 93A13.