Accelerated sampling schemes for high dimensional systems
In this thesis we discuss accelerated sampling schemes for high dimensional systems, for example molecular dynamics (MD). The development of these methods is fundamental to the eﬀective study of a large class of problems, for which traditional methods converge slowly to the system’s underlying invariant probability distribution. Due to the complexity of the landscape deﬁned by an energy function (or, in statistical models, the log likelihood of the target probability density), the exploration of the probability distribution is severely restricted. This can have detrimental eﬀects on the conclusions drawn from numerical experiments when potentially important states and solutions are absent in the examination of the results as a consequence of poor sampling. The aim of accelerated sampling schemes is to enhance the exploration of the invariant measure by improving the rate of convergence to it. In this work, we ﬁrst focus our attention on numerical methods based on canonical sampling by studying Langevin dynamics, for which the convergence is accelerated by extending the phase-space. We introduce a scheme based on simulated tempering which makes temperature into a dynamical variable and allows switching the temperature up or down during the exploration in such a way that the target probability distribution can be easily obtained from the extended distribution. We show that this scheme is optimal when operated in the inﬁnite switch limit. We discuss the limitations of this method and demonstrate the excellent exploratory properties of it for a moderately complicated biomolecule, alanine-12. Next, we derive a novel approach to constant pressure simulation that forms the basis for a family of pure Langevin barostats. We demonstrate the excellent numerical performance of Lie-Trotter splitting schemes for these systems and the superior accuracy and precision of the simultaneous temperature and pressure control in comparison to currently available schemes. The scientiﬁc importance of this method lies in the ability to control the simulation and to make better predictions for applications in both materials modelling and drug design. We demonstrate this method in simulations of state transitions in crystalline materials using the “Mercedes Benz” potential. In a ﬁnal contribution, we extend the inﬁnite switch schemes to incorporate a general class of collective variables. In particular this allows for tempering in both temperature and pressure when combined with our new barostat. We conclude this thesis by presenting a numerical study of the computational prospects of these methods.