Enhancing Block Cimmino for Sparse Linear Systems with Dense Columns via Schur Complement

Abstract

. The block Cimmino is a parallel hybrid row-block projection iterative method suc-4 cessfully used for solving general sparse linear systems. However, the convergence of the method 5 degrades when angles between subspaces spanned by the row-blocks are far from being orthogonal. 6 The density of columns as well as the numerical values of their nonzeros are more likely to contribute 7 to the non-orthogonality between row blocks. We propose a novel scheme to handle such “dense” 8 columns. The proposed scheme forms a reduced system by separating these columns and the re-9 spective rows from the original coefficient matrix and handling them via Schur complement. Then, 10 the angles between subspaces spanned by the row-blocks of the reduced system are expected to be 11 closer to orthogonal and the reduced system is solved efficiently by the block Conjugate Gradient 12 accelerated block Cimmino in fewer iterations. We also propose a novel metric for selecting “dense” 13 columns considering the numerical values. The proposed metric establishes an upper bound on the 14 sum of inner–products between row-blocks. Then, we propose an efficient algorithm for computing 15 the proposed metric for the columns. Extensive numerical experiments for a wide range of linear 16 systems confirm the effectiveness of the proposed scheme by achieving fewer iterations and faster 17 parallel solution time compared to the classical CG accelerated block Cimmino algorithm. 18