Volume 60, pp. 327-350, 2024.

A short-term rational Krylov method for linear inverse problems

Stefan Kindermann and Werner Zellinger


Motivated by the aggregation method, we present an iterative method for finding approximate solutions of least-squares problems for linear ill-posed problems over (mixed) rational Krylov spaces. The mixed rational Krylov spaces where the solution is sought consist of Tikhonov-regularized solutions mixed with usual Krylov space elements from the normal equations. We present an algorithm based on the Arnoldi–Lanczos iteration, and, as main result, derive the rational CG method, a short-term iteration that, similar as the usual conjugate gradient method, does not requires orthogonalization or saving of the Krylov basis vectors. Some numerical experiments illustrate the performance of the method.

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Key words

rational Krylov space, rational conjugate gradient method, aggregation method, short-term recurrence

AMS subject classifications

65F10, 65F22

Links to the cited ETNA articles

[34]Vol. 38 (2011), pp. 233-257 Stefan Kindermann: Convergence analysis of minimization-based noise level-free parameter choice rules for linear ill-posed problems
[54]Vol. 51 (2019), pp. 451-468 Niel Van Buggenhout, Marc Van Barel, and Raf Vandebril: Biorthogonal rational Krylov subspace methods

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