Molecular models and data analytics problems give rise to gargantuan systems of stochastic differential equations (SDEs) whose paths ergodically sample multimodal probability distributions. An important challenge for the numerical analyst (or the chemist, or the physicist, or the engineer, or the data scientist) is the design of efficient numerical methods to generate these paths. For SDEs, the numerical perspective is just maturing, with important new methods (and, even more important, new procedures for their construction and analysis) becoming available. One of the interesting ideas is to design stochastic schemes with close attention to the error in invariant measures. Another is to use negative feedback loop controls to regulate a noisy gradient or even the discretisation error itself. To illustrate our approach, I will discuss several different examples including (i) efficient schemes for constrained stochastic dynamics improving accuracy and stability in bio-MD [1,2], and (ii) methods for Bayesian sampling for machine learning applications [3]. If time permits, I will also describe some very recent work on parallel sampling algorithms [4].

[1] B. Leimkuhler and C. Matthews, Rational construction of stochastic-numerical methods for molecular sampling, Applied Mathematics Research Express, 2013.

[2] B. Leimkuhler and C. Matthews, Efficient molecular dynamics using geodesic integration and solvent-solute splitting, Proceedings of the Royal Society A, 2016.

[3] X. Shang, Z. Zhu, B. Leimkuhler and A. Storkey, Covariance-controlled adaptive Langevin thermostat for large-scale Bayesian sampling, Neural and Information Processing Systems (NIPS) 2015.

[4] B. Leimkuhler, C. Matthews and J. Weare, Ensemble preconditioning for Markov chain Monte Carlo simulation, Arxiv: https://arxiv.org/abs/1607.03954

Bio:

Ben Leimkuhler is the Chair of Applied Mathematics at the University of Edinburgh. He received his PhD from the University of Illinois under the direction of C.W. Gear and worked in Helsinki and then at the Universities of Kansas and Leicester before moving to Scotland in 2006. He is well known for his work on the foundations of algorithms for dynamical modelling and sampling, much of it surveyed in two books: Leimkuhler and Reich, Simulating Hamiltonian Dynamics, CUP, 2005 (deterministic methods), Leimkuhler and Matthews, Molecular Dynamics, Springer, 2015 (mostly stochastic methods). He is also a Faculty Fellow of the nascent Alan Turing Institute, a major centre for research in “data science” and the intellectual hub of the London Knowledge Quarter.