  Initial-boundary value problems for integrable nonlinear partial differential
equations have become tractable in recent years due to the development of
so-called unified transform techniques. The main obstruction to applying these
methods in practice is that calculation of the spectral transforms of the
initial and boundary data requires knowledge of too many boundary conditions,
more than are required make the problem well-posed. The elimination of the
unknown boundary values is frequently addressed in the spectral domain via the
so-called global relation, and types of boundary conditions for which the
global relation can be solved are called \emph{linearizable}. For the
defocusing nonlinear Schr\"odinger equation, the global relation is only known
to be explicitly solvable in rather restrictive situations, namely homogeneous
boundary conditions of Dirichlet, Neumann, and Robin (mixed) type. General
nonhomogeneous boundary conditions are not known to be linearizable. In this
paper, we propose an explicit approximation for the nonlinear
Dirichlet-to-Neumann map supplied by the defocusing nonlinear Schr\"odinger
equation and use it to provide approximate solutions of general nonhomogeneous
boundary value problems for this equation posed as an initial-boundary value
problem on the half-line. Our method sidesteps entirely the solution of the
global relation. The accuracy of our method is proven in the semiclassical
limit, and we provide explicit asymptotics for the solution in the interior of
the quarter-plane space-time domain.
