  We discuss an algorithm to compute transport maps that couple the uniform
measure on $[0,1]^d$ with a specified target distribution $\pi$ on $[0,1]^d$.
The primary objectives are either to sample from or to compute expectations
w.r.t. $\pi$. The method is based on leveraging a polynomial surrogate of the
target density, which is obtained by a least-squares or interpolation
approximation. We discuss the design and construction of suitable sparse
approximation spaces, and provide a complete error and cost analysis for target
densities belonging to certain smoothness classes. Further, we explore the
relation between our proposed algorithm and related approaches that aim to find
suitable transports via optimization over a class of parametrized transports.
Finally, we discuss the efficient implementation of our algorithm and report on
numerical experiments which confirm our theory.
