Preconditioning linear least-squares problems by identifying a basis matrix

We study the solution of the linear least-squares problem minx ∥b − Ax∥2 where the matrix A ∈ Rm×n (m ≥ n) has rank n and is large and sparse. We assume that A is available as a matrix, not an operator. The preconditioning of this problem is difficult because the matrix A does not have the properties of differential problems that make standard preconditioners effective. Incomplete Cholesky techniques applied to the normal equations do not produce a well-conditioned problem. We attempt to bypass the ill-conditioning by finding an n × n nonsingular submatrix B of A that reduces the Euclidean norm of AB−1. We use B to precondition a symmetric quasi-definite linear system whose condition number is then independent of the condition number of A and has the same solution as the original least-squares problem. We illustrate the performance of our approach on some standard test problems and show it is competitive with other approaches.