Fast Flexible LSQR with a Hybrid Variant for Efficient Large-Scale Regularization

Abstract. A wide range of applications necessitates solving large-scale ill-posed problems contaminated by noise, which are commonly formulated as linear approximation problems (possibly overdetermined or underdetermined). Krylov subspace regularization methods are particularly advantageous in this context, as they rely solely on matrix-vector multiplication. Among the most widely used techniques are least squares QR (LSQR) and conjugate gradient least squares (CGLS), both of which can be extended with flexible preconditioning to enforce solution properties such as nonnegativity or sparsity. Flexible LSQR (FLSQR) can also be combined with direct methods to create efficient hybrid approaches. The flexible Golub–Kahan process underlying FLSQR requires two long-term recurrences. Here, we introduce a novel fast flexible Golub–Kahan method that employs one long-term and one short-term recurrence. Using this, we develop the fast flexible LSQR (FaFLSQR) algorithm, which offers comparable computational cost to flexible CGLS (FCGLS) while also supporting hybrid regularization like FLSQR. We analyze the properties of FaFLSQR and prove its mathematical equivalence to FCGLS. Numerical experiments demonstrate that in floating-point arithmetic, FaFLSQR outperforms both FCGLS and FLSQR in terms of computational efficiency.