Transporting Higher-Order Quadrature Rules: Quasi-Monte Carlo Points and Sparse Grids for Mixture Distributions

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.