Row-Splitting ILU Preconditioners for Sparse Least-Squares Problems

TL;DR

This paper introduces a novel ILU-based row-splitting preconditioner for sparse least-squares problems, improving efficiency and effectiveness without forming normal equations.

Contribution

It proposes a new preconditioning strategy that isolates dense rows and avoids normal equations, enhancing performance for ill-conditioned or dense problems.

Findings

Outperforms incomplete Cholesky-based preconditioners in tests

Effective for problems with dense rows or ill-conditioned matrices

Works well with Krylov subspace solvers

Abstract

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.