Row-Splitting ILU Preconditioners for Sparse Least-Squares Problems
This addresses memory-efficient preconditioning for sparse least-squares problems, which is important for computational applications but has received little attention, representing an incremental improvement.
The authors tackled the challenge of preconditioning overdetermined least-squares problems by proposing a row-splitting ILU-based preconditioner that identifies a well-conditioned square submatrix and combines it with algebraic corrections. Numerical experiments showed this approach outperforms incomplete Cholesky factorization methods on normal equations, including for sparse-dense problems.
Preconditioning for overdetermined least-squares problems has received comparatively little attention, and designing methods that are both effective and memory-efficient remains challenging. We propose a class of ILU-based preconditioners built around a row-splitting strategy that identifies a well-conditioned square submatrix via an incomplete LU factorization and combines its incomplete factors with algebraic corrections from the remaining rows. This construction avoids forming the normal equations and is well suited to problems for which the normal matrix is ill-conditioned or relatively dense. Numerical experiments on test problems arising from practical applications illustrate the effectiveness of the proposed approach when used with a Krylov subspace solver and demonstrate it can outperform preconditioners based on incomplete Cholesky factorization of the normal equations, including for sparse-dense problems, where the splitting naturally isolates dense rows.