  A least-squares method for solving the hyperbolic Monge-Amp\`ere equation
with transport boundary condition is introduced. The method relies on an
iterative procedure for the gradient of the solution, the so-called mapping. By
formulating error functionals for the interior domain, the boundary, both
separately and as linear combination, three minimization problems are solved
iteratively to compute the mapping. After convergence, a fourth minimization
problem, to compute the solution of the Monge-Amp\`ere equation, is solved. The
approach is based on a least-squares method for the elliptic Monge-Amp\`ere
equation by Prins et al., and is improved upon by the addition of analytical
solutions for the minimization on the interior domain and by the introduction
of two new boundary methods. Lastly, the iterative method is tested on a
variety of examples. It is shown that, when the iterative method converges,
second-order global convergence as function of the spatial discretization is
obtained.
