Solution of Linear Systems Using Randomized Rounding

This paper gives an algorithm for solving linear systems, using a randomized version of incomplete LU factorization together with iterative improvement. The factorization uses Gaussian elimination with partial pivoting, and preserves sparsity during factorization by randomized rounding of the entries. The resulting approximate factorization is then applied to estimate the solution. This simple technique, combined with iterative improvement, is demonstrated to be effective for a range of linear systems. When applied to medium-sized sample matrices for PDEs, the algorithm is qualitatively like multigrid: the work per iteration is typically linear in the order of the matrix, and the number of iterations to achieve a small residual is typically on the order of fifteen to twenty. The technique is also tested for a sample of asymmetric matrices from the Matrix Market, and is found to have similar behavior for many of them.