Accelerated Monte Carlo sampling through non-reversibility and factorisation

In the context of the high-dimensionality and multimodality of the potentials encountered in chemical physics, massive efforts have been devoted into the development of efficient simulation methodologies. Since the first works of Molecular dynamics (MD) and Markov-chain Monte Carlo (MCMC) methods, bidimensional particle systems, in particular with hard-core interactions, have played the role of a challenging but inspiring test bed. In spite of their apparent simplicity, they exhibit a rich behavior where topological defects bind or unbind themselves in pair, ruling by doing so the transition phases. Standard detailed-balance (or reversible) Markov chains then often fail at exploring efficiently the related configuration space. I will first present the MCMC simulation strategies known as Event-Chain Monte Carlo and based on rejection-free and non-reversible processes, which were first developed to address this challenge. These methods are based on the breaking of detailed balance in the underlying stochastic sampling processes, upgrading them from discrete Markov chains to piecewise deterministic Markov processes (PDMP). PDMP sampling has now shown performances competing with its MD counterparts in chemical physics and, beyond, with state-of-the-art sampling schemes, e.g. Hamiltonian Monte Carlo, in statistical inference. I will discuss how the generalisation of these methods to most systems was made possible by consistently replacing the time-reversibility symmetry of detailed balance by potential or global symmetries. I will finally address the question of practical implementation, in particular regarding computational complexity and how the factorisation, first used to design the first PDMP for sphere systems, can be used even in standard MCMC schemes to reduce the computational complexity cost, as e.g. in long-range systems.