A Schur complement approach to preconditioning sparse linear least-squares problems with some dense rows

The effectiveness of sparse matrix techniques for directly solving large-scale linear least-squares problems is severely limited if the system matrix A has one or more nearly dense rows. In this paper, we partition the rows of A into sparse rows and dense rows (As and Ad) and apply the Schur complement approach. A potential difficulty is that the reduced normal matrix AsTAs is often rank-deficient, even if A is of full rank. To overcome this, we propose explicitly removing null columns of As and then employing a regularization parameter and using the resulting Cholesky factors as a preconditioner for an iterative solver applied to the symmetric indefinite reduced augmented system. We consider complete factorizations as well as incomplete Cholesky factorizations of the shifted reduced normal matrix. Numerical experiments are performed on a range of large least-squares problems arising from practical applications. These demonstrate the effectiveness of the proposed approach when combined with either a sparse parallel direct solver or a robust incomplete Cholesky factorization algorithm.