  We study numerical methods for sampling probability measures in high
dimension where the underlying model is only approximately identified with a
gradient system. Extended stochastic dynamical methods are discussed which have
application to multiscale models, nonequilibrium molecular dynamics, and
Bayesian sampling techniques arising in emerging machine learning applications.
In addition to providing a more comprehensive discussion of the foundations of
these methods, we propose a new numerical method for the adaptive
Langevin/stochastic gradient Nos\'{e}--Hoover thermostat that achieves a
dramatic improvement in numerical efficiency over the most popular stochastic
gradient methods reported in the literature. We also demonstrate that the newly
established method inherits a superconvergence property (fourth order
convergence to the invariant measure for configurational quantities) recently
demonstrated in the setting of Langevin dynamics. Our findings are verified by
numerical experiments.
