A Novel Partitioning Method for Accelerating the Block Cimmino Algorithm
This work addresses the problem of slow convergence in the block Cimmino algorithm for solving sparse linear systems, offering a practical improvement for numerical linear algebra applications.
The paper proposes a novel block-row partitioning method that uses a row inner-product graph model to minimize inter-block inner products, improving the eigenvalue spectrum and reducing the number of iterations for the block Cimmino algorithm. Experiments on a large set of matrices show significant reduction in iterations compared to a state-of-the-art method.
We propose a novel block-row partitioning method in order to improve the convergence rate of the block Cimmino algorithm for solving general sparse linear systems of equations. The convergence rate of the block Cimmino algorithm depends on the orthogonality among the block rows obtained by the partitioning method. The proposed method takes numerical orthogonality among block rows into account by proposing a row inner-product graph model of the coefficient matrix. In the graph partitioning formulation defined on this graph model, the partitioning objective of minimizing the cutsize directly corresponds to minimizing the sum of inter-block inner products between block rows thus leading to an improvement in the eigenvalue spectrum of the iteration matrix. This in turn leads to a significant reduction in the number of iterations required for convergence. Extensive experiments conducted on a large set of matrices confirm the validity of the proposed method against a state-of-the-art method.