Volume 60, pp. 327-350, 2024.

A short-term rational Krylov method for linear inverse problems

Stefan Kindermann and Werner Zellinger

Abstract

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.

Full Text (PDF) [363 KB], BibTeX

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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