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Molecular Simulation: modeling, algorithms and mathematical analysis

Final Report Summary - MSMATH (Molecular Simulation: modeling, algorithms and mathematical analysis)

The ERC project MSMath aims at studying various algorithms and models which are used in molecular simulation. Molecular dynamics consists in simulating on a computer the evolution of a molecular system. The simulation of matter at the atomic scale is used as a complement to experiments in many fields: molecular biology, physical chemistry, materials science, etc. Such simulations consume huge computer resources. Examples include the study of the change of conformations of large molecules for biological applications (think of ligand-protein binding problems, with applications to drug design) or the diffusion of an atom on a substrate in materials science. Even though molecular dynamics is nowadays used in many fields of science and consumes a large part of the computational power of large scale computing centers, very few mathematicians are working on those algorithms, compared for example to computational fluid dynamics. One output of the MSMath project was to contribute to the development of a community of young researchers working on this subject.

The main numerical challenge in molecular dynamics comes from the time and length scales: to be able to infer quantities of interest at the macroscopic scale, one has to simulate very large systems over very long timescales. This implies that naive algorithms do not give satisfactory results in reasonable computational time. The MSMath project brought important contributions in three directions.

The sampling of high-dimensional multimodal measures is crucial in computational statistical physics, as well as in many other fields (uncertainty quantification, Bayesian inference, etc). Many enhanced sampling techniques have been proposed or analyzed in the MSMath project: free energy adaptive biasing techniques, non-reversible perturbations, piecewise deterministic Markov processes, ergodic properties of discretized Feynman-Kac semigroups, Metropolis Hastings algorithms to sample probability measures supported on submanifolds... In particular, we made a lot of progress on the understanding of adaptive biasing potential methods, which are sampling techniques where the walker is forced to visit new regions as time goes: by using tools from stochastic approximation theory and the study of self avoiding random walks, new insights on the convergence of these methods have been obtained.

One central question of the project was the efficient sampling of metastable dynamics. Indeed, the stochastic processes used to describe the evolution of atomic systems typically remain trapped for very long times in some regions of the phase space called metastable states. It is crucial to simulate efficiently the transitions from one metastable state to another, in order to get insights on the properties of the system over macroscopic timescales. Sampling trajectories is of course more complicated than sampling measures in finite dimensional space, and requires dedicated methods. One of the major output of the MSMath project was to bring important contributions to the mathematical understanding of the exit event from a metastable state, and to the efficient sampling of metastable trajectories by improving existing algorithms or devising new numerical techniques. In particular, the development of the quasi-stationary distribution approach to study the exit event enabled us to draw a rigorous link between the (overdamped) Langevin dynamics and kinetic Monte Carlo models, namely between continuous state space Markov models, and discrete state space Markov models parameterized by the Arrhenius-Eyring-Kramers laws.

Coarse-graining is central to reducing dimensionality in molecular dynamics. We worked on the analysis of the quality of model reduction techniques, and their use for computational purposes. In particular, we obtained new error estimates for the Mori-Zwanzig projection applied to non-reversible dynamics.

In the MSMath project, mathematical tools from the analysis of partial differential equations and probability theory have been used or developed to quantify the metastability of stochastic processes. Many of these results gave rise to new or improved numerical techniques, which have been implemented in softwares used by practitioners (NAMD, Tripoli) and tested on large scale test cases, in collaboration with chemists or physicists.