  We develop a perturbative expansion scheme for solving general boundary value
problems in a broad class of type IIB flux compactifications. The background
solution is any conformally Calabi-Yau compactification with imaginary
self-dual (ISD) fluxes. Upon expanding in small deviations from the ISD
solution, the equations of motion simplify dramatically: we find a simple basis
in which the n-th order equations take a triangular form. This structure
implies that the system can be solved iteratively whenever the individual,
uncoupled equations can be solved. We go on to demonstrate the solution of the
system for a general warped Calabi-Yau cone: we present an algorithm that
yields an explicit Green's function solution for all the supergravity fields,
to any desired order, in terms of the harmonic functions on the base of the
cone. Our results provide a systematic procedure for obtaining the corrections
to a warped throat geometry induced by attachment to a compact bulk. We also
present a simple method for determining the sizes of physical effects mediated
through warped geometries.
