Partitioning strategies for the block Cimmino algorithm

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 \(\mathbf{A}\mathbf{x}= \mathbf{b}\). We study strategies that transform the matrix \(\mathbf{A}\mathbf{A}^\mathrm{{T}}\) 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 method for 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 \(\mathbf{A}\) 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.