  We consider a series of distorted black hole initial data sets, and develop
techniques to evolve them using the linearized equations of motion for the
gravitational wave perturbations on a Schwarzschild background. We apply this
to 2D and 3D distorted black hole spacetimes. In 2D, waveforms for different
modes of the radiation are presented, comparing full nonlinear evolutions for
different axisymmetric l-modes with perturbative evolutions. We show how
axisymmetric black hole codes solving the full, nonlinear Einstein equations
are capable of very accurate evolutions, and also how these techniques aid in
studying nonlinear effects. In 3D we show how the initial data for the
perturbation equations can be computed, and we compare with analytic solutions
given from a perturbative expansion of the initial value problem. In addition
to exploring the physics of these distorted black hole data sets, in particular
allowing an exploration of linear, nonlinear, and mode mixing effects, this
approach provides an important testbed for any fully nonlinear numerical code
designed to evolve black hole spacetimes in 2D or 3D.
