Particle filter efficiency under limited communication

Sequential Monte Carlo methods are typically not straightforward to implement on parallel architectures. This is because standard resampling schemes involve communication between all particles. The α-sequential Monte Carlo method was proposed recently as a potential solution to this which limits communication between particles. This limited communication is controlled through a sequence of stochastic matrices known as α-matrices. We study the influence of the communication structure on the convergence and stability properties of the resulting algorithms. In particular, we quantitatively show that the mixing properties of the α-matrices play an important role in the stability properties of the algorithm. Moreover, we prove that one can ensure good mixing properties by using randomized communication structures where each particle only communicates with a few neighbouring particles. The resulting algorithms converge at the usual Monte Carlo rate. This leads to efficient versions of distributed sequential Monte Carlo.

Application of the interacting particle system method to piecewise deterministic Markov processes used in reliability

Variance reduction methods are often needed for the reliability assessment of complex industrial systems, we focus on one variance reduction method in a given context, that is, the interacting particle system (IPS) method used on piecewise deterministic Markov processes (PDMPs) for reliability assessment. The PDMPs are a very large class of processes which benefit from high modeling capacities, they can model almost any Markovian phenomenon that does not include diffusion. In reliability assessment, the PDMPs modeling industrial systems generally involve low jump rates and jump kernels favoring one safe arrival, we call such model a "concentrated PDMP." Used on such concentrated PDMPs, the IPS is inefficient and does not always provide a variance reduction. Indeed, the efficiency of the IPS method relies on simulating many different trajectories during its propagation steps, but unfortunately, concentrated PDMPs are likely to generate the same deterministic trajectories over and over. We propose an adaptation of the IPS method called IPS+M that reduces this phenomenon. The IPS+M consists in modifying the propagation steps of the IPS, by conditioning the propagation to avoid generating the same trajectories multiple times. We prove that, compared to the IPS, the IPS+M method always provides an estimator with a lower variance. We also carry out simulations on two-components systems that confirm these results.

Adaptive multilevel splitting: Historical perspective and recent results

This article first presents a short historical perpective of the importance splitting approach to simulate and estimate rare events, with a detailed description of several variants. We then give an account of recent theoretical results on these algorithms, including a central limit theorem for Adaptive Multilevel Splitting (AMS). Considering the asymptotic variance in the latter, the choice of the importance function, called the reaction coordinate in molecular dynamics, is also discussed. Finally, we briefly mention some worthwhile applications of AMS in various domains.