Partitioning strategies for the block Cimmino algorithm

Abstract

Science and Technology Facilities Council preprints are available online at: http://epubs.stfc.ac.uk ABSTRACT In the context of the block Cimmino algorithm, we study preprocessing strategies to obtain block partitionings that can be applied to general linear systems of equations Ax = b. We study strategies that transform the matrix AA T into a matrix with a block tridiagonal structure. This provides a partitioning of the linear system for row projection methods because block Cimmino is essentially equivalent to block Jacobi on the normal equations and the resulting partition will yield a two-block partition of the original matrix. Therefore the resulting block partitioning should improve the rate of convergence of block row projection methods such as block Cimmino. We discuss a way of obtaining a partitioning using a dropping strategy that gives more blocks at the cost of relaxing the two-block partitioning. We then use a hypergraph partitioning that works directly on the matrix A to reduce directly the connections between blocks. We give numerical results showing the performance of these techniques both in their effect on the convergence of the block Cimmino algorithm and in their ability to exploit parallelism.

A Novel Partitioning Method for Accelerating the Block Cimmino Algorithm

“…One alternative variation is the block Cimmino [7] which is a block row projection method. Iterative block Cimmino has been studied extensively in [7,9,14,17,23,25,26]. A pseudo-direct version of block Cimmino based on the augmented block row projections is proposed in [24].…”

“…It is also reported that the CG accelerated Cimmino method competes favorably compared to classical preconditioned Generalized Minimum Residual (GMRES) and Conjugate Gradient on Normal Equations (CGNE) for the solution of nonsymmetric linear systems that arise in an elliptic partial differential equation. On the other hand, it should be noted that the main motivation of block Cimmino algorithm is its amenability of parallelism [23].…”

“…In summary, the convergence rate of the CG accelerated block Cimmino depends on the extreme eigenvalues and the number of clusters as well as the quality of the clustering. Therefore, the partitioning of the coefficient matrix A into block rows is crucial for improving the convergence rate of the CG accelerated block Cimmino algorithm, since it can improve the eigenvalue distribution of H. Note that the eigenvalues of H is only affected by the block-row partitioning of the coefficient matrix A and independent of any column ordering [23].…”

“…In the literature, Cuthill-Mckee (CM) [19] based partitioning strategies [23,43,47] are utilized to define block rows using CM level set information on the normal equations of the original matrix. These strategies benefit from the level sets of CM for creating the desired number of block rows.…”

“…In recent studies [23,47], a hypergraph partitioning method is used to find a "good" block-row partitioning for the CG accelerated block Cimmino algorithm. It is reported that it performs better than CM based methods (with or without filtering small elements).…”