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ETNA - Electronic Transactions on Numerical Analysis
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Verlag der Österreichischen Akademie der Wissenschaften Austrian Academy of Sciences Press
A-1011 Wien, Dr. Ignaz Seipel-Platz 2
Tel. +43-1-515 81/DW 3420, Fax +43-1-515 81/DW 3400 https://verlag.oeaw.ac.at, e-mail: verlag@oeaw.ac.at |
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DATUM, UNTERSCHRIFT / DATE, SIGNATURE
BANK AUSTRIA CREDITANSTALT, WIEN (IBAN AT04 1100 0006 2280 0100, BIC BKAUATWW), DEUTSCHE BANK MÜNCHEN (IBAN DE16 7007 0024 0238 8270 00, BIC DEUTDEDBMUC)
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ETNA - Electronic Transactions on Numerical Analysis, pp. 327-350, 2024/06/06
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.
Keywords: rational Krylov space, rational conjugate gradient method, aggregation method, short-term recurrence