Regularization of linear inverse problems by rational Krylov methods

For approximately solving linear ill-posed problems in Hilbert spaces, we investigate the regularization properties of the aggregation method and the RatCG method. These recent algorithms use previously calculated solutions of Tikhonov regularization (respectively, Landweber iterations) to set up a new search space on which the least-squares functional is minimized. We outline how these methods can be understood as rational Krylov space methods, i.e., based on the space of rational functions of the forward operator. The main result is that these methods form an optimal-order regularization schemes when combined with the discrepancy principle as stopping rule and when the underlying regularization parameters are sufficiently large.

💡 Research Summary

The paper addresses the regularization of linear ill‑posed problems in Hilbert spaces, focusing on two recent algorithms: the aggregation method and the RatCG method. Both algorithms exploit a collection of Tikhonov‑regularized solutions (x^{\delta}{\alpha_i}) (and, for RatCG, also Landweber or CGNE iterates) that are typically already computed when solving a problem with multiple regularization parameters. The aggregation method builds a low‑dimensional subspace (\mathcal{R}n = \operatorname{span}{x^{\delta}{\alpha_1},\dots,x^{\delta}{\alpha_n}}) and solves a small least‑squares problem (\min_{x\in\mathcal{R}n}|Ax-y^{\delta}|^2). RatCG constructs a mixed space (\mathcal{KR}n = \mathcal{R}{\lfloor n/2\rfloor}\cup\mathcal{K}{\lfloor n/2\rfloor}) where (\mathcal{K}_k) is the classical Krylov space generated by (A^*y^{\delta}). RatCG alternates between solving a Tikhonov‑type subproblem (even steps) and performing a CGNE update (odd steps), yielding a recursion that reuses previous information.

The central theoretical contribution is the reinterpretation of these methods as rational Krylov subspace methods. While a standard Krylov space consists of polynomial functions of (A^*A) applied to (A^*y^{\delta}), the spaces (\mathcal{R}_n) and (\mathcal{KR}_n) can be written as \