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On Synchronized Fleming-Viot Particle Systems
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Abstract
This article presents a variant of Fleming-Viot particle systems, which are a standard way to approximate the law of a Markov process with killing as well as related quantities. Classical Fleming-Viot particle systems proceed by simulating N trajectories, or particles, according to the dynamics of the underlying process, until one of them is killed. At this killing time, the particle is instantaneously branched on one of the other ones, and so on until a fixed and finite final time T. In our variant, we propose to wait until K particles are killed and then rebranch them independently on the alive ones. Specifically, we focus our attention on the large population limit and the regime where K/N has a given limit when N goes to infinity. In this context, we establish consistency and asymptotic normality results. The variant we propose is motivated by applications in rare event estimation problems.
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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 10m 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.
We apply the adaptive multilevel splitting (AMS) method to the Ceq → Cax transition of alanine dipeptide in vacuum. Some properties of the algorithm are numerically illustrated, such as the unbiasedness of the probability estimator and the robustness of the method with respect to the reaction coordinate. We also calculate the transition time obtained via the probability estimator, using an appropriate ensemble of initial conditions. Finally, we show how the AMS method can be used to compute an approximation of the committor function. © 2019 Wiley Periodicals, Inc.
We introduce the concept of reliability in watermarking as the ability of assessing that a probability of false alarm is very low and below a given significance level. We propose an iterative and self-adapting algorithm which estimates very low probabilities of error. It performs much quicker and more accurately than a classical Monte Carlo estimator. The article finishes with applications to zero-bit watermarking (probability of false alarm, error exponent), and to probabilistic fingerprinting codes (probability of wrongly accusing a given user, code length estimation).
This paper discusses a novel strategy for simulating rare events and an associated Monte Carlo estimation of tail probabilities.
Our method uses a system of interacting particles and exploits a Feynman-Kac representation of that system to analyze their
fluctuations. Our precise analysis of the variance of a standard multilevel splitting algorithm reveals an opportunity for
improvement. This leads to a novel method that relies on adaptive levels and produces, in the limit of an idealized version
of the algorithm, estimates with optimal variance. The motivation for this theoretical work comes from problems occurring
in watermarking and fingerprinting of digital contents, which represents a new field of applications of rare event simulation
techniques. Some numerical results show performance close to the idealized version of our technique for these practical applications.
KeywordsRare event–Sequential importance sampling–Feynman-Kac formula–Metropolis-Hastings–Fingerprinting–Watermarking
We study the existence and asymptotic properties of a conservative branching particle system driven by a diffusion with smooth
coefficients for which birth and death are triggered by contact with a set. Sufficient conditions for the process to be non-explosive
are given. In the Brownian motions case the domain of evolution can be non-smooth, including Lipschitz, with integrable Martin
kernel. The results are valid for an arbitrary number of particles and non-uniform redistribution after branching. Additionally,
with probability one, it is shown that only one ancestry line survives. In special cases, the evolution of the surviving particle
is studied and for a two particle system on a half line we derive explicitly the transition function of a chain representing
the position at successive branching times.
KeywordsFleming–Viot branching–Immortal particle–Martin kernel–Doeblin condition–Jump diffusion process
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
We consider a strong Markov process with killing and prove an approximation
method for the distribution of the process conditioned not to be killed when it
is observed. The method is based on a Fleming-Viot type particle system with
rebirths, whose particles evolve as independent copies of the original strong
Markov process and jump onto each others instead of being killed. Our only
assumption is that the number of rebirths of the Fleming-Viot type system
doesn't explode in finite time almost surely and that the survival probability
of the original process remains positive in finite time. The approximation
method generalizes previous results and comes with a speed of convergence. A
criterion for the non-explosion of the number of rebirths is also provided for
general systems of time and environment dependent diffusion particles. This
includes, but is not limited to, the case of the Fleming-Viot type system of
the approximation method. The proof of the non-explosion criterion uses an
original non-attainability of (0,0) result for pair of non-negative
semi-martingales with positive jumps.
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 consider a branching particle model in which particles move inside a Euclidean domain according to the following rules. The particles move as independent Brownian motions until one of them hits the boundary. This particle is killed but another randomly chosen particle branches into two particles, to keep the population size constant. We prove that the particle population does not approach the boundary simultaneously in a finite time in some Lipschitz domains. This is used to prove a limit theorem for the empirical distribution of the particle family.
We analyze and simulate a two-dimensional Brownian multi-type particle system with death and branching (birth) depending on the position of particles of different types. The system is confined in the two-dimensional box, whose boundaries act as the sink of Brownian particles. The branching rate matches the death rate so that the total number of particles is kept constant. In the case of m types of particles in the rectangular box of size a, b and elongated shape a >> b we observe that the stationary distribution of particles corresponds to the m-th Laplacian eigenfunction. For smaller elongations a > b we find a configurational transition to a new limiting distribution. The ratio a/b for which the transition occurs is related to the value of the m-th eigenvalue of the Laplacian with rectangular boundaries. This work was supported in part by the NSF grant DMS 9322689 and KBN and FWPN grants.
This article first presents a short historical perpective of the importance splitting approach to simulate and estimate rare events, with a detailed description of several variants. We then give an account of recent theoretical results on these algorithms, including a central limit theorem for Adaptive Multilevel Splitting (AMS). Considering the asymptotic variance in the latter, the choice of the importance function, called the reaction coordinate in molecular dynamics, is also discussed. Finally, we briefly mention some worthwhile applications of AMS in various domains.
We investigate the application of the adaptive multilevel splitting algorithm for the estimation of tail probabilities of solutions of stochastic differential equations evaluated at a given time and of associated temporal averages. We introduce a new, very general, and effective family of score functions that is designed for these problems. We illustrate its behavior in a series of numerical experiments. In particular, we demonstrate how it can be used to estimate large deviations rate functionals for the longtime limit of temporal averages.
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.
I Preliminaries.- II Semimartingales and Stochastic Integrals.- III Semimartingales and Decomposable Processes.- IV General Stochastic Integration and Local Times.- V Stochastic Differential Equations.- VI Expansion of Filtrations.- References.
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 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 method is presented for efficiently computing small failure probabilities encountered in seismic risk problems involving
dynamic analysis. It is based on a procedure recently developed by the writers called Subset Simulation in which the central idea is that
a small failure probability can be expressed as a product of larger conditional failure probabilities, thereby turning the problem of
simulating a rare failure event into several problems that involve the conditional simulation of more frequent events. Markov chain Monte
Carlo simulation is used to efficiently generate the conditional samples, which is otherwise a nontrivial task. The original version of
Subset Simulation is improved by allowing greater flexibility for incorporating prior information about the reliability problem so as to
increase the efficiency of the method. The method is an effective simulation procedure for seismic performance assessment of structures
in the context of modern performance-based design. This application is illustrated by considering the failure of linear and nonlinear
hysteretic structures subjected to uncertain earthquake ground motions. Failure analysis is also carried out using the Markov chain samples
generated during Subset Simulation to yield information about the probable scenarios that may occur when the structure fails.
We consider a system of N Brownian particles evolving independently in a domain D. As soon as one particle reaches the boundary it is killed and one of the other particles is chosen uniformly and splits into two independent particles resuming a new cycle of independent motion until the next boundary hit. We prove the hydrodynamic limit for the joint law of the empirical measure process and the average number of visits to the boundary as N approaches infinity.
We consider a system {X1,¼,XN}{\{X_1,\ldots,X_N\}} of N particles in a bounded d-dimensional domain D. During periods in which none of the particles X1,¼,XN{X_1,\ldots,X_N} hit the boundary ¶D{\partial D} , the system behaves like N independent d-dimensional Brownian motions. When one of the particles hits the boundary ¶D{\partial D} , then it instantaneously jumps to the site of one of the remaining N−1 particles with probability (N−1)−1. For the system {X1,¼,XN}{\{X_1,\ldots,X_N\}} , the existence of an invariant measure nmu n{\nu\mskip-12mu \nu} has been demonstrated in Burdzy et al. [Comm Math Phys 214(3):679–703, 2000]. We provide a structural formula for this invariant
measure nmu n{\nu\mskip-12mu \nu} in terms of the invariant measure m of the Markov chain x{\xi} which returns the sites the process X:=(X1,¼,XN){X:=(X_1,\ldots,X_N)} jumps to after hitting the boundary ¶DN{\partial D^N} . In addition, we characterize the asymptotic behavior of the invariant measure m of x{\xi} when N → ∞. Using the methods of the paper, we provide a rigorous proof of the fact that the stationary empirical measure processes
\frac1Nåi=1NdXi{\frac1N\sum_{i=1}^N\delta_{X_i}} converge weakly as N → ∞ to a deterministic constant motion. This motion is concentrated on the probability measure whose density with respect
to the Lebesgue measure is the first eigenfunction of the Dirichlet Laplacian on D. This result can be regarded as a complement to a previous one in Grigorescu and Kang [Stoch Process Appl 110(1):111–143,
2004].
A new simulation approach, called ‘subset simulation’, is proposed to compute small failure probabilities encountered in reliability analysis of engineering systems. The basic idea is to express the failure probability as a product of larger conditional failure probabilities by introducing intermediate failure events. With a proper choice of the conditional events, the conditional failure probabilities can be made sufficiently large so that they can be estimated by means of simulation with a small number of samples. The original problem of calculating a small failure probability, which is computationally demanding, is reduced to calculating a sequence of conditional probabilities, which can be readily and efficiently estimated by means of simulation. The conditional probabilities cannot be estimated efficiently by a standard Monte Carlo procedure, however, and so a Markov chain Monte Carlo simulation (MCS) technique based on the Metropolis algorithm is presented for their estimation. The proposed method is robust to the number of uncertain parameters and efficient in computing small probabilities. The efficiency of the method is demonstrated by calculating the first-excursion probabilities for a linear oscillator subjected to white noise excitation and for a five-story nonlinear hysteretic shear building under uncertain seismic excitation.
- 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
preprint arXiv:1611.00515, 2016.
- 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
preprint arXiv:1709.06771, 2017.