  Unmatched pairs of forward and back projectors are common in X-ray CT
computations for large-scale problems; they are caused by the need for fast
algorithms that best utilize the computer hardware, and it is an interesting
and challenging task to develop fast and easy-to-use algorithms for these
cases. Our approach is to use preconditioned GMRES, in the form of the AB- and
BA-GMRES algorithms, to handle the unmatched normal equations associated with
an unmatched pair. These algorithms are simple to implement, they rely only on
computations with the available forward and back projectors, and they do not
require the tuning of any algorithm parameters. We show that these algorithms
are equivalent to well-known LSQR and LSMR algorithms in the case of a matched
projector. Our numerical experiments demonstrate that AB- and BA-GMRES exhibit
a desired semi-convergence behavior that is comparable with LSQR/LSMR and that
standard stopping rules work well. Hence, AB- and BA-GMRES are suited for
large-scale CT reconstruction problems with noisy data and unmatched projector
pairs.
