Central Limit Theorem for Adaptive Multilevel Splitting Estimators in an Idealized Setting

Bréhier, Charles-Édouard; Goudenège, Ludovic; Tudela, Loïc

doi:10.1007/978-3-319-33507-0_10

Cited by 9 publications

(26 citation statements)

References 15 publications

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“…To analyze how efficient the algorithm is depending on n and k, one can look at the variance; more precisely, in the regime when a and k are fixed and n → +∞, a Central Limit Theorem holds, see [16]: Theorem 3.2. We have the following convergence in law, when n → +∞, and for fixed k and a (such that p = P(X > a) > 0):…”

Section: Theoretical Resultsmentioning

confidence: 99%

“…The fundamental argument in [15] and [16] is the embedding of the estimation problem of p into the problem of estimating the conditional probability p(x) = P(X > a|X > x) for any x ∈ [0, a]; for that purpose, we introduce the estimatorp n,k (x) obtained thanks to Algorithm 1 where in the initialization step the random variables are generated according to L(X|X > x). Then Theorem 3.1 is a consequence of two facts:…”

Section: Theoretical Resultsmentioning

confidence: 99%

“…We now review the results from [16] and [15]. The algorithm 1 defined below is well-posed and yields a family of unbiased estimators under the following condition.…”

Section: The Ideal Casementioning

confidence: 99%

“…In contrast with (16) and (29) which have a rather elementary proof once we have the appropriate regularization properties, the estimate (30) is based on a non trivial interpolation result from [6]. Notice however that those estimates are sub-optimal (because of ε > 0).…”

Section: Estimates Of the Error In The Total Variation Distancementioning

confidence: 99%

“…After a brief review of reduction of variance strategies, especially the Multilevel Splitting approach (see [42]), the author introduces different adaptive versions of the latter and states a series of its properties: the goal is to build a theoretically unbiased estimator in an idealized setting and then to explain and to test numerically how the algorithm should be modified to preserve the unbiasedness property in a more general setting. This note is mainly based on the following joint works: [16] and [15].…”

Section: Introductionmentioning

confidence: 99%

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Recent advances in various fields of numerical probability

et al. 2015

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Abstract. The goal of this paper is to present a series of recent contributions on some various problems of numerical probability. Beginning with the Richardson-Romberg Multilevel Monte-Carlo method which, among other fields of applications, is a very efficient method for the approximation of diffusion processes, we focus on some adaptive multilevel splitting algorithms for rare event simulation. Then, the third part is devoted to the simulation of McKean-Vlasov forward and decoupled forwardbackward stochastic differential equations by some cubature algorithms. Finally, we tackle the problem of the weak error estimation in total variation norm for a general Markov semi-group.

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Section: Theoretical Resultsmentioning

confidence: 99%

Section: Theoretical Resultsmentioning

confidence: 99%

“…We now review the results from [16] and [15]. The algorithm 1 defined below is well-posed and yields a family of unbiased estimators under the following condition.…”

Section: The Ideal Casementioning

confidence: 99%

Section: Estimates Of the Error In The Total Variation Distancementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Recent advances in various fields of numerical probability

et al. 2015

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Computing return times or return periods with rare event algorithms

Lestang¹,

Ragone²,

Bréhier³

et al. 2018

J. Stat. Mech.

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The average time between two occurrences of the same event, referred to as its return time (or return period), is a useful statistical concept for practical applications. For instance insurances or public agency may be interested by the return time of a 10 m flood of the Seine river in Paris. However, due to their scarcity, reliably estimating return times for rare events is very difficult using either observational data or direct numerical simulations. For rare events, an estimator for return times can be built from the extrema of the observable on trajectory blocks. Here, we show that this estimator can be improved to remain accurate for return times of the order of the block size. More importantly, we show that this approach can be generalised to estimate return times from numerical algorithms specifically designed to sample rare events. So far those algorithms often compute probabilities, rather than return times. The approach we propose provides a computationally extremely efficient way to estimate numerically the return times of rare events for a dynamical system, gaining several orders of magnitude of computational costs. We illustrate the method on two kinds of observables, instantaneous and time-averaged, using two different rare event algorithms, for a simple stochastic process, the Ornstein-Uhlenbeck process. As an example of realistic applications to complex systems, we finally discuss extreme values of the drag on an object in a turbulent flow.

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Coupling rare event algorithms with data-based learned committor functions using the analogue Markov chain

Lucente¹,

Rolland²,

Herbert³

et al. 2022

J. Stat. Mech.

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Rare events play a crucial role in many physics, chemistry, and biology phenomena, when they change the structure of the system, for instance in the case of multistability, or when they have a huge impact. Rare event algorithms have been devised to simulate them efficiently, avoiding the computation of long periods of typical fluctuations. We consider here the family of splitting or cloning algorithms, which are versatile and specifically suited for far-from-equilibrium dynamics. To be efficient, these algorithms need to use a smart score function during the selection stage. Committor functions are the optimal score functions. In this work we propose a new approach, based on the analogue Markov chain, for a data-based learning of approximate committor functions. We demonstrate that such learned committor functions are extremely efficient score functions when used with the adaptive multilevel splitting algorithm. We illustrate our approach for a gradient dynamics in a three-well potential, and for the Charney–DeVore model, which is a paradigmatic toy model of multistability for atmospheric dynamics. For these two dynamics, we show that having observed a few transitions is enough to have a very efficient data-based score function for the rare event algorithm. This new approach is promising for use for complex dynamics: the rare events can be simulated with a minimal prior knowledge and the results are much more precise than those obtained with a user-designed score function.

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Central Limit Theorem for Adaptive Multilevel Splitting Estimators in an Idealized Setting

Cited by 9 publications

References 15 publications

Recent advances in various fields of numerical probability

Recent advances in various fields of numerical probability

Computing return times or return periods with rare event algorithms

Coupling rare event algorithms with data-based learned committor functions using the analogue Markov chain

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