  Gravitational waves, like light, can be gravitationally lensed by massive
astrophysical objects such as galaxies and galaxy clusters. Strong
gravitational-wave lensing, forecasted at a reasonable rate in ground-based
gravitational-wave detectors such as Advanced LIGO, Advanced Virgo, and KAGRA,
produces multiple images separated in time by minutes to months. These images
appear as repeated events in the detectors: gravitational-wave pairs, triplets,
or quadruplets with identical frequency evolution originating from the same sky
location. To search for these images, we need to, in principle, analyze all
viable combinations of individual events present in the gravitational-wave
catalogs. An increasingly pressing problem is that the number of candidate
pairs that we need to analyse grows rapidly with the increasing number of
single-event detections. At design sensitivity, one may have as many as
$\mathcal O(10^5)$ event pairs to consider. To meet the ever-increasing
computational requirements, we develop a fast and precise Bayesian methodology
to analyse strongly lensed event pairs, enabling future searches. The
methodology works by replacing the prior used in the analysis of one strongly
lensed gravitational-wave image by the posterior of another image; the
computation is then further sped up by a pre-computed lookup table. We
demonstrate how the methodology can be applied to any number of lensed images,
enabling fast studies of strongly lensed quadruplets.
