  Ulam's method is a rigorous numerical scheme for approximating invariant
densities of dynamical systems. The phase space is partitioned into connected
sets and an inter-set transition matrix is computed from the dynamics; an
approximate invariant density is read off as the leading left eigenvector of
this matrix. When a hole in phase space is introduced, one instead searches for
\emph{conditional} invariant densities and their associated escape rates. For
Lasota-Yorke maps with holes we prove that a simple adaptation of the standard
Ulam scheme provides convergent sequences of escape rates (from the leading
eigenvalue), conditional invariant densities (from the corresponding left
eigenvector), and quasi-conformal measures (from the corresponding right
eigenvector). We also immediately obtain a convergent sequence for the
invariant measure supported on the survivor set. Our approach allows us to
consider relatively large holes. We illustrate the approach with several
families of examples, including a class of Lorenz maps.
