How mathematics can help to understand stochastic algorithms? Two examples.

Sampling problems are ubiquitous: Bayesian inference, computational statistical physics, uncertainty quantification, reliability analysis are examples of scientific fields which require advanced Monte Carlo methods to sample measures, in high or even infinite dimension. In this talk, I will present two recent works demonstrating the interest of partial differential equations approaches to study advanced stochastic algorithms: accelerated dynamics techniques (used in molecular dynamics to sample paths) and optimal scaling for Metropolis Hastings algorithms (used in computational statistics to sample high-dimensional measures). The mathematical analysis is based respectively on spectral analysis (semi-classical analysis for Witten boundary Laplacians) and on functional inequalities (entropy estimates for nonlinear partial differential equations).

References:
D. Aristoff and T. Lelièvre, Mathematical analysis of Temperature Accelerated Dynamics, arxiv:1305.6569
B. Jourdain, T. Lelièvre and B. Miasojedow, Optimal scaling for the transient phase of Metropolis Hastings algorithms: the longtime behavior, arxiv:1212.5517 to appear in Bernoulli
B. Jourdain, T. Lelièvre and B. Miasojedow, Optimal scaling for the transient phase of Metropolis Hastings algorithms: the mean-field limit, arxiv:1210.7639
C. Le Bris, T. Lelièvre, M. Luskin and D. Perez, A mathematical formalization of the parallel replica dynamics, Monte Carlo Methods and Applications, 18(2), 119-146, 2012
T. Lelièvre and F. Nier, Low temperature asymptotics for Quasi-Stationary Distributions in a bounded domain, arxiv:1309.3898