Importance sampling of rare events in chaotic systems

Abstract

Abstract. 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 MetropolisHastings 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 lowand 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].

Taming chaos to sample rare events: The effect of weak chaos

“…The general ideas sketched above are valid for broad classes of sampling methods, but here we focus on a Metropolis-Hastings setting [14,15], in line with our previous works revised in [7]. In this setting, starting from x ∈ Γ a new state x ∈ Γ is proposed according to g(x |x).…”

“…Explicit expressions for t have been derived for the escape time and finite time Lyapunov exponent (see Ref. [7]) and for the dispersion in spatially extended (diffusive) systems (see Ref. [10]).…”

“…The construction proposed above has been successfully implemented to obtain efficient Monte Carlo methods in different problems [7], including calculations of finitetime Lyapunov exponents in N coupled oscillators (with N up to 1024) [9] and the computation of trajectories with high dispersion in diffusive systems such as the Lorentz gas [10]. Here we focus on violations of the simplifying assumptions and approximations made above.…”

“…The key ingredient is to use information about the chaoticity of the last sampled trajectory to construct a proposal distribution that efficiently finds a new trajectory of interest. First we review our previously proposed [7] approach (in Sec. II), which has been successful in different problems involving strongly chaotic systems [7][8][9][10].…”

“…First we review our previously proposed [7] approach (in Sec. II), which has been successful in different problems involving strongly chaotic systems [7][8][9][10]. We then focus (in Sec.…”