  Integration against, and hence sampling from, high-dimensional probability
distributions is of essential importance in many application areas and has been
an active research area for decades. One approach that has drawn increasing
attention in recent years has been the generation of samples from a target
distribution $\mathbb{P}_{\mathrm{tar}}$ using transport maps: if
$\mathbb{P}_{\mathrm{tar}} = T_\# \mathbb{P}_{\mathrm{ref}}$ is the pushforward
of an easily-sampled probability distribution $\mathbb{P}_{\mathrm{ref}}$ under
the transport map $T$, then the application of $T$ to
$\mathbb{P}_{\mathrm{ref}}$-distributed samples yields
$\mathbb{P}_{\mathrm{tar}}$-distributed samples. This paper proposes the
application of transport maps not just to random samples, but also to
quasi-Monte Carlo points, higher-order nets, and sparse grids in order for the
transformed samples to inherit the original convergence rates that are often
better than $N^{-1/2}$, $N$ being the number of samples/quadrature nodes. Our
main result is the derivation of an explicit transport map for the case that
$\mathbb{P}_{\mathrm{tar}}$ is a mixture of simple distributions, e.g.\ a
Gaussian mixture, in which case application of the transport map $T$ requires
the solution of an \emph{explicit} ODE with \emph{closed-form} right-hand side.
Mixture distributions are of particular applicability and interest since many
methods proceed by first approximating $\mathbb{P}_{\mathrm{tar}}$ by a mixture
and then sampling from that mixture (often using importance reweighting).
Hence, this paper allows for the sampling step to provide a better convergence
rate than $N^{-1/2}$ for all such methods.
