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On the Asymptotic Normality of Adaptive Multilevel Splitting
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Abstract
Adaptive Multilevel Splitting (AMS for short) is a generic Monte Carlo method for Markov processes that simulates rare events and estimates associated probabilities. Despite its practical efficiency, there are almost no theoretical results on the convergence of this algorithm. The purpose of this paper is to prove both consistency and asymptotic normality results in a general setting. This is done by associating to the original Markov process a level-indexed process, also called a stochastic wave, and by showing that AMS can then be seen as a Fleming-Viot type particle system. This being done, we can finally apply general results on Fleming-Viot particle systems that we have recently obtained.
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In the Monte Carlo simulation of particle transport, and especially for shielding applications, variance reduction techniques are widely used to help simulate realisations of rare events and reduce the relative errors on the estimated scores for a given computation time. Adaptive Multilevel Splitting (AMS) is one of these variance reduction techniques that has recently appeared in the literature. In the present paper, we propose an alternative version of the AMS algorithm, adapted for the first time to the field of particle transport. Within this context, it can be used to build an unbiased estimator of any quantity associated with particle tracks, such as flux, reaction rates or even non-Boltzmann tallies like pulse-height tallies and other spectra. Furthermore, the efficiency of the AMS algorithm is shown not to be very sensitive to variations of its input parameters, which makes it capable of significant variance reduction without requiring extended user effort.
We introduce a generalization of the Adaptive Multilevel Splitting algorithm
in the discrete time dynamic setting, namely when it is applied to sample rare
events associated with paths of Markov chains. By interpreting the algorithm as
a sequential sampler in path space, we are able to build an estimator of the
rare event probability (and of any non-normalized quantity associated with this
event) which is unbiased, whatever the choice of the importance function and
the number of replicas. This has practical consequences on the use of this
algorithm, which are illustrated through various numerical experiments.
The Adaptive Multilevel Splitting (AMS) algorithm is a powerful and versatile method for the simulation of rare events. It is based on an interacting (via a mutation-selection procedure) system of replicas, and depends on two integer parameters: n ∈ N * the size of the system and the number k ∈ {1,. .. , n − 1} of the replicas that are eliminated and resampled at each iteration. In an idealized setting, we analyze the performance of this algorithm in terms of a Large Deviations Principle when n goes to infinity, for the estimation of the (small) probability P(X > a) where a is a given threshold and X is real-valued random variable. The proof uses the technique introduced in [BLR15]: in order to study the log-Laplace transform, we rely on an auxiliary functional equation. Such Large Deviations Principle results are potentially useful to study the algorithm beyond the idealized setting, in particular to compute rare transitions probabilities for complex high-dimensional stochastic processes.
The Adaptive Multilevel Splitting algorithm is a very powerful and versatile
method to estimate rare events probabilities. It is an iterative procedure on
an interacting particle system, where at each step, the k less well-adapted
particles among n are killed while k new better adapted particles are
resampled according to a conditional law. We analyze the algorithm in the
idealized setting of an exact resampling and prove that the estimator of the
rare event probability is unbiased whatever k. We also obtain a precise
asymptotic expansion for the variance of the estimator and the cost of the
algorithm in the large n limit, for a fixed k.
Let X be a random vector with distribution μ on ℝ
d
and Φ be a mapping from ℝ
d
to ℝ. That mapping acts as a black box, e.g., the result from some computer experiments for which no analytical expression
is available. This paper presents an efficient algorithm to estimate a tail probability given a quantile or a quantile given
a tail probability. The algorithm improves upon existing multilevel splitting methods and can be analyzed using Poisson process
tools that lead to exact description of the distribution of the estimated probabilities and quantiles. The performance of
the algorithm is demonstrated in a problem related to digital watermarking.
KeywordsMonte Carlo simulation–Rare event–Metropolis-Hastings–Watermarking
A method to generate reactive trajectories, namely equilibrium trajectories
leaving a metastable state and ending in another one is proposed. The algorithm
is based on simulating in parallel many copies of the system, and selecting the
replicas which have reached the highest values along a chosen one-dimensional
reaction coordinate. This reaction coordinate does not need to precisely
describe all the metastabilities of the system for the method to give reliable
results. An extension of the algorithm to compute transition times from one
metastable state to another one is also presented. We demonstrate the interest
of the method on two simple cases: a one-dimensional two-well potential and a
two-dimensional potential exhibiting two channels to pass from one metastable
state to another one.
We present in this article a genetic type interacting particle systems algorithm and a genealogical model for estimating a class of rare events arising in physics and network analysis. We represent the distribution of a Markov process hitting a rare target in terms of a Feynman–Kac model in path space. We show how these branching particle models described in previous works can be used to estimate the probability of the corresponding rare events as well as the distribution of the process in this regime.
Transition path theory (TPT) has been recently introduced as a theoretical framework to describe the reaction pathways of rare events between long lived states in complex systems. TPT gives detailed statistical information about the reactive trajectories involved in these rare events, which are beyond the realm of transition state theory or transition path sampling. In this paper the TPT approach is outlined, its distinction from other approaches is discussed, and, most importantly, the main insights and objects provided by TPT are illustrated in detail via a series of low dimensional test problems.
This new, thoroughly revised and expanded 3rd edition of a classic gives a comprehensive coverage of modern probability in a single book. It is a truly modern text, providing not only classical results but also material that will be important for future research. Much has been added to the previous edition, including eight entirely new chapters on subjects like random measures, Malliavin calculus, multivariate arrays, and stochastic differential geometry. Apart from important improvements and revisions, some of the earlier chapters have been entirely rewritten. To help the reader, the material has been grouped together into ten major areas, each arguably indispensable to any serious graduate student and researcher, regardless of their specialization.
Each chapter is largely self-contained and includes plenty of exercises, making the book ideal for self-study and for designing graduate-level courses and seminars in different areas and at different levels. Extensive notes and a detailed bibliography make it easy to go beyond the presented material if desired.
From the reviews of the first edition:
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Adaptive Multilevel Summation (AMS) is a rare event sampling method that requires minimal parameter tuning and that allows unbiased sampling of transition pathways of a given rare event. Previous simulation studies have verified the efficiency and accuracy of AMS in the calculation of transition times for simple systems in both Monte Carlo and molecular dynamics (MD) simulations. Now, AMS is applied for the first time to a MD simulation of protein-ligand dissociation, representing a leap in complexity from the previous test cases. Of interest is the dissociation rate, which is typically too low to be accessible to conventional MD. The present study joins other recent efforts to develop advanced sampling techniques in MD to calculate dissociation rates, which are gaining importance in the pharmaceutical field as indicators of drug efficacy. The system investigated here, benzamidine bound to trypsin, is an example common to many of these efforts. The AMS estimate of the dissociation rate was found to be (2.6 ± 2.4) × 10(2) s(-1) , which compares well with the experimental value.
The behavior of chains of very many molecules is investigated by solving a restricted random walk problem on a cubic lattice in three dimensions and a square lattice in two dimensions. In the Monte Carlo calculation a large number of chains are generated at random, subject to the restrictions of no crossing or doubling back, to give the average extension of the chain 〈R2〉Av as a function of N, the number of links in the chain. A system of weights is used in order that all possible allowed chains are counted equally. Results for the true random walk problem without weights are obtained also.
The estimation of rare event probability is a crucial issue in areas such as reliability, telecommunications, aircraft management. In complex systems, analytical study is out of question and one has to use Monte Carlo methods. When rare is really rare, which means a probability less than 10^−9, naive Monte Carlo becomes unreasonable. A widespread technique consists in multilevel splitting, but this method requires enough knowledge about the system to decide where to put the levels at hand. This is unfortunately not always possible. In this paper, we propose an adaptive algorithm to cope with this problem: the estimation is asymptotically consistent, costs just a little bit more than classical multilevel splitting and has the same efficiency in terms of asymptotic variance. In the one dimensional case, we prove rigorously the a.s. convergence and the asymptotic normality of our estimator, with the same variance as with other algorithms that use fixed crossing levels. In our proofs we mainly use tools from the theory of empirical processes, which seems to be quite new in the field of rare events.
A guiding principle in the efficient estimation of rare-event probabilities by Monte Carlo is that importance sampling based on the change of measure suggested by a large deviations analysis can reduce variance by many orders of magnitude. In a variety of settings, this approach has led to estimators that are optimal in an asymptotic sense. We give examples, however, in which importance sampling estimators based on a large deviations change of measure have provably poor performance. The estimators can have variance that decreases at a slower rate than a naive estimator, variance that increases with the rarity of the event, and even infinite variance. For each example, we provide an alternative estimator with provably efficient performance. A common feature of our examples is that they allow more than one way for a rare event to occur; our alternative estimators give explicit weight to lower probability paths neglected by leading-term asymptotics.
Transition states are defined as points in configuration space with the highest probability that trajectories passing through them are reactive (i.e., form transition paths between reactants and products). In the high-friction (diffusive) limit of Langevin dynamics, the resulting ensemble of transition states is shown to coincide with the separatrix formed by points of equal commitment (or splitting) probabilities for reaching the product and reactant regions. Transition states according to the new criterion can be identified directly from equilibrium trajectories, or indirectly by calculating probability densities in the equilibrium and transition-path ensembles using umbrella and transition-path sampling, respectively. An algorithm is proposed to calculate rate coefficients from the transition-path and equilibrium ensembles by estimating the frequency of transitions between reactants and products.
Central Limit Theorem for Adaptive Multilevel Splitting Estimators in an Idealized Setting
- C.-E Bréhier
- L Goudenège
- L Tuleda
C.-E. Bréhier, L. Goudenège, and L. Tuleda. Central Limit Theorem
for Adaptive Multilevel Splitting Estimators in an Idealized Setting. In
Proceedings of MCQMC, 2014.
- F Cérou
- B Delyon
- A Guyader
- M Rousset
F. Cérou, B. Delyon, A. Guyader, and M. Rousset.
A Central
Limit Theorem for Fleming-Viot Particle Systems with Hard Killing.
arXiv:1709.06771, 2017.
- F Cérou
- B Delyon
- A Guyader
- M Rousset
F. Cérou, B. Delyon, A. Guyader, and M. Rousset.
A Central
Limit Theorem for Fleming-Viot Particle Systems with Soft Killing.
arXiv:1611.00515, 2017.