Rare Event Simulation for Stochastic Dynamics in Continuous Time

Abstract

Large deviations for additive path functionals of stochastic dynamics and related numerical approaches have attracted significant recent research interest. We focus on the question of convergence properties for cloning algorithms in continuous time, and establish connections to the literature of particle filters and sequential Monte Carlo methods. This enables us to derive rigorous convergence bounds for cloning algorithms which we report in this paper, with details of proofs given in a further publication. The tilted generator characterizing the large deviation rate function can be associated to non-linear processes which give rise to several representations of the dynamics and additional freedom for associated numerical approximations. We discuss these choices in detail, and combine insights from the filtering literature and cloning algorithms to compare different approaches and improve efficiency.

Approximating the Cumulant Generating Function of Triangles in the Erdös–Rényi Random Graph

“…If β 2 > 0 the probability of finding triangles is enhanced with respect to the Erdös-Rényi case, while it is decreased if β 2 < 0. In the limiting case β 2 → −∞ the edge-triangle model (2) gives zero probability to graphs containing triangles.…”

“…where • n denotes expectation w.r.t. the measure ν n defined in (2). We shall be interested in the behavior of very large graphs which mathematically is described by the (thermodynamic) limit n → ∞.…”

“…A standard tool for estimating the probability of rare events in the Erdös-Rényi random graph is the importance sampling technique, see for instance [6]. Here we follow an approach based on population dynamics, called "cloning" [1,2,11,[19][20][21]30]. In this section we adapt the method to a purely geometric problem, by introducing a dynamics for the graph construction.…”

“…Since the increment of T (x) in going from size i to size i + 1 requires i 2 operations (being the number of triangles in the smaller graph known from the previous step), the overall cost for evaluating the total number of triangles in the final graph of size n will be n i=2 i 2 = O(n 3 ). Of course, to reach a prescribed level of accuracy, we have to choose the size of the cloning family as an increasing function of n. This has been discussed in [1,2]. The procedure is further illustrated with the example of size n = 3 in the Appendix.…”

Sampling rare events across dynamical phase transitions

“…An extension of such technique has led to a more efficient version of the cloning algorithm in continuous time which significantly improves the large deviation function estimators in the long time and large number of clones limit [92]. In addition, some rigorous bounds on convergence properties of the algorithm have been established [93,94].…”