Preconditioning via Randomized Range Deflation (RandRAND)

Abstract

We introduce RandRAND, a new class of randomized preconditioning methods for large-scale linear systems. RandRAND deflates the spectrum via efficient orthogonal projections onto random subspaces, without computing eigenpairs or low-rank approximations. This leads to advantages in computational cost and numerical stability. We establish rigorous condition number bounds that depend only weakly on the problem size and that reduce to a small constant when the dimension of the deflated subspace is comparable to the effective spectral dimension. RandRAND can be employed without explicit operations with the deflation basis, enabling the effective use of fast randomized transforms. In this setting, the costly explicit basis orthogonalization is bypassed by using fast randomized Q-less QR factorizations or iterative methods for computing orthogonal projections. These strategies balance the cost of constructing RandRAND preconditioners and applying them within linear solvers, and can ensure robustness to rounding errors.