Importance sampling of rare events in chaotic systems
1 Max Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany
2 DTU Compute, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark
3 Department of Physics and Center of Physics, University of Minho, 4710-057 Braga, Portugal
4 Physics Engineering Department, Engineering Faculty of the University of Porto, 4200-465 Porto, Portugal
5 School of Mathematics and Statistics, University of Sydney, 2006 NSW, Sydney, Australia
Received: 23 January 2017
Received in final form: 4 May 2017
Published online: 4 October 2017
Finding and sampling rare trajectories in dynamical systems is a difficult computational task underlying numerous problems and applications. In this paper we show how to construct Metropolis-Hastings Monte-Carlo methods that can efficiently sample rare trajectories in the (extremely rough) phase space of chaotic systems. As examples of our general framework we compute the distribution of finite-time Lyapunov exponents (in different chaotic maps) and the distribution of escape times (in transient-chaos problems). Our methods sample exponentially rare states in polynomial number of samples (in both low- and high-dimensional systems). An open-source software that implements our algorithms and reproduces our results can be found in reference [J. Leitao, A library to sample chaotic systems, 2017, https://github.com/jorgecarleitao/chaospp].
Key words: Statistical and Nonlinear Physics
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