  We discuss the initial value problem of general relativity in its recently
unified Lagrangian and Hamiltonian pictures and present a multi-domain
pseudo-spectral collocation method to solve the resulting coupled nonlinear
partial differential equations. Using this code, we explore several approaches
to construct initial data sets containing one or two black holes: We compute
quasi-circular orbits for spinning equal mass black holes and unequal mass
(nonspinning) black holes using the effective potential method with Bowen-York
extrinsic curvature. We compare initial data sets resulting from different
decompositions, and from different choices of the conformal metric with each
other. Furthermore, we use the quasi-equilibrium method to construct initial
data for single black holes and for binary black holes in quasi-circular
orbits. We investigate these binary black hole data sets and examine the limits
of large mass-ratio and wide separation. Finally, we propose a new method for
constructing spacetimes with superposed gravitational waves of possibly very
large amplitude.
