  Assume interest is in sampling from a probability distribution $\mu$ defined
on $(\mathsf{Z},\mathscr{Z})$. We develop a framework to construct sampling
algorithms taking full advantage of numerical integrators of ODEs, say
$\psi\colon\mathsf{Z}\rightarrow\mathsf{Z}$ for one integration step, to
explore $\mu$ efficiently and robustly. The popular Hybrid/Hamiltonian Monte
Carlo (HMC) algorithm [Duane, 1987], [Neal, 2011] and its derivatives are
example of such a use of numerical integrators. However we show how the
potential of integrators can be exploited beyond current ideas and HMC sampling
in order to take into account aspects of the geometry of the target
distribution. A key idea is the notion of integrator snippet, a fragment of the
orbit of an ODE numerical integrator $\psi$, and its associate probability
distribution $\bar{\mu}$, which takes the form of a mixture of distributions
derived from $\mu$ and $\psi$. Exploiting properties of mixtures we show how
samples from $\bar{\mu}$ can be used to estimate expectations with respect to
$\mu$. We focus here primarily on Sequential Monte Carlo (SMC) algorithms, but
the approach can be used in the context of Markov chain Monte Carlo algorithms
as discussed at the end of the manuscript. We illustrate performance of these
new algorithms through numerical experimentation and provide preliminary
theoretical results supporting observed performance.
