Solving mixed sparse-dense linear least-squares problems by preconditioned iterative methods

The efficient solution of large linear least-squares problems in which the system matrix A contains rows with very different densities is challenging. Previous work has focused on direct methods for problems in which A has a few relatively dense rows. These rows are initially ignored, a factorization of the sparse part is computed using a sparse direct solver, and then the solution is updated to take account of the omitted dense rows. In some practical applications the number of dense rows can be significant and for very large problems, using a direct solver may not be feasible. We propose processing rows that are identified as dense separately within a conjugate gradient method using an incomplete factorization preconditioner combined with the factorization of a dense matrix of size equal to the number of dense rows. Numerical experiments on large-scale problems from real applications are used to illustrate the effectiveness of our approach. The results demonstrate that we can efficiently solve problems that could not be solved by a preconditioned conjugate gradient method without exploiting the dense rows