  The discovery and subsequent study of optical counterparts to transient
sources is crucial for their complete astrophysical understanding. Various
gamma ray burst (GRB) detectors, and more notably the ground--based
gravitational wave detectors, typically have large uncertainties in the sky
positions of detected sources. Searching these large sky regions spanning
hundreds of square degrees is a formidable challenge for most ground--based
optical telescopes, which can usually image less than tens of square degrees of
the sky in a single night. We present algorithms for optimal scheduling of such
follow--up observations in order to maximize the probability of imaging the
optical counterpart, based on the all--sky probability distribution of the
source position. We incorporate realistic observing constraints like the
diurnal cycle, telescope pointing limitations, available observing time, and
the rising/setting of the target at the observatory location. We use
simulations to demonstrate that our proposed algorithms outperform the default
greedy observing schedule used by many observatories. Our algorithms are
applicable for follow--up of other transient sources with large positional
uncertainties, like Fermi--detected GRBs, and can easily be adapted for
scheduling radio or space--based X--ray followup.
