  We present a new method to achieve high-contrast images using segmented
and/or on-axis telescopes. Our approach relies on using two sequential
Deformable Mirrors to compensate for the large amplitude excursions in the
telescope aperture due to secondary support structures and/or segment gaps. In
this configuration the parameter landscape of Deformable Mirror Surfaces that
yield high contrast Point Spread Functions is not linear, and non-linear
methods are needed to find the true minimum in the optimization topology. We
solve the highly non-linear Monge-Ampere equation that is the fundamental
equation describing the physics of phase induced amplitude modulation. We
determine the optimum configuration for our two sequential Deformable Mirror
system and show that high-throughput and high contrast solutions can be
achieved using realistic surface deformations that are accessible using
existing technologies. We name this process Active Compensation of Aperture
Discontinuities (ACAD). We show that for geometries similar to JWST, ACAD can
attain at least 10^-7 in contrast and an order of magnitude higher for both the
future Extremely Large Telescopes and on-axis architectures reminiscent of HST.
We show that the converging non-linear mappings resulting from our Deformable
Mirror shapes actually damp near-field diffraction artifacts in the vicinity of
the discontinuities. Consequently, ACAD is a true broadband solution to the
problem of high-contrast imaging with segmented and/or on-axis apertures. We
finally show that once the non-linear solution is found, fine tuning with
linear methods used in wavefront control can be applied to further contrast by
another order of magnitude. Generally speaking, the ACAD technique can be used
to significantly improve a broad class of telescope designs for a variety of
problems.
