arXiv subjects

Frédéric Cérou

Publications and source records attributed to Frédéric Cérou.

10 recordsLinked to original sources

arXiv2024

Adaptive Reduced Multilevel Splitting

This paper considers the classical problem of sampling with Monte Carlo methods a target rare event distribution defined by a score function that is very expensive to compute. We assume we can build using evaluations of the true score, an approximate surrogate score certified with error bounds. This work proposes a fully adaptive algorithm to sequentially sample surrogate rare event distributions with increasing target levels. An essential contribution consists in sampling at each iteration the surrogate rare event at a critical level corresponding to a specific cost. This cost is related to importance sampling for a target for a given budget. The critical level is calculated solely from the reduced score and its error bound From a practical point of view, sampling the proposal sequence is performed by extending the framework of the popular adaptive multilevel splitting algorithm to the use of score approximations. Numerical experiments evaluate the proposed importance sampling algorithm in terms of computational complexity versus squared error. In particular, we investigate the performance of the algorithm when simulating rare events related to the solution of a parametric PDE, which is approximated by a reduced basis.

stat.CO↗

arXiv2023

Chilled Sampling for Uncertainty Quantification: A Motivation From A Meteorological Inverse Problem

Atmospheric motion vectors (AMVs) extracted from satellite imagery are the only wind observations with good global coverage. They are important features for feeding numerical weather prediction (NWP) models. Several Bayesian models have been proposed to estimate AMVs. Although critical for correct assimilation into NWP models, very few methods provide a thorough characterization of the estimation errors. The difficulty of estimating errors stems from the specificity of the posterior distribution, which is both very high dimensional, and highly ill-conditioned due to a singular likelihood. Motivated by this difficult inverse problem, this work studies the evaluation of the (expected) estimation errors using gradient-based Markov Chain Monte Carlo (MCMC) algorithms. The main contribution is to propose a general strategy, called here chilling, which amounts to sampling a local approximation of the posterior distribution in the neighborhood of a point estimate. From a theoretical point of view, we show that under regularity assumptions, the family of chilled posterior distributions converges in distribution as temperature decreases to an optimal Gaussian approximation at a point estimate given by the Maximum A Posteriori, also known as the Laplace approximation. Chilled sampling therefore provides access to this approximation generally out of reach in such high-dimensional nonlinear contexts. From an empirical perspective, we evaluate the proposed approach based on some quantitative Bayesian criteria. Our numerical simulations are performed on synthetic and real meteorological data. They reveal that not only the proposed chilling exhibits a significant gain in terms of accuracy of the point estimates and of their associated expected errors, but also a substantial acceleration in the convergence speed of the MCMC algorithms.

stat.ME↗

arXiv2022

Fluctuations of Rare Event Simulation with Monte Carlo Splitting in the Small Noise Asymptotics

Diffusion processes with small noise conditioned to reach a target set are considered. The AMS algorithm is a Monte Carlo method that is used to sample such rare events by iteratively simulating clones of the process and selecting trajectories that have reached the highest value of a so-called importance function. In this paper, the large sample size relative variance of the AMS small probability estimator is considered. The main result is a large deviations logarithmic equivalent of the latter in the small noise asymptotics, which is rigorously derived. It is given as a maximisation problem explicit in terms of the quasi-potential cost function associated with the underlying small noise large deviations. Necessary and sufficient geometric conditions ensuring the vanishing of the obtained quantity ('weak' asymptotic efficiency) are provided. Interpretations and practical consequences are discussed.

math.NA↗

arXiv2022

Entropy minimizing distributions are worst-case optimal importance proposals

Importance sampling of target probability distributions belonging to a given convex class is considered. Motivated by previous results, the cost of importance sampling is quantified using the relative entropy of the target with respect to proposal distributions. Using a reference measure as a reference for cost, we prove under some general conditions that the worst-case optimal proposal is precisely given by the distribution minimizing entropy with respect to the reference within the considered convex class of distributions. The latter conditions are in particular satisfied when the convex class is defined using a push-forward map defining atomless conditional measures. Applications in which the optimal proposal is Gibbsian and can be practically sampled using Monte Carlo methods are discussed.

math.NA↗

arXiv2019

On Synchronized Fleming-Viot Particle Systems

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 $(N-1)$ 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 $(N-K)$ 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.

math.PR↗

arXiv2018

On the Asymptotic Normality of Adaptive Multilevel Splitting

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.

math.PR↗

arXiv2017

A Central Limit Theorem for Fleming-Viot Particle Systems with Hard Killing

Fleming-Viot type particle systems represent a classical way to approximate the distribution of a Markov process with killing, given that it is still alive at a final deterministic time. In this context, each particle evolves independently according to the law of the underlying Markov process until its killing, and then branches instantaneously on another randomly chosen particle. While the consistency of this algorithm in the large population limit has been recently studied in several articles, our purpose here is to prove Central Limit Theorems under very general assumptions. For this, we only suppose that the particle system does not explode in finite time, and that the jump and killing times have atomless distributions. In particular, this includes the case of elliptic diffusions with hard killing.

math.PR↗

arXiv2013

New Insights Into Approximate Bayesian Computation

Approximate Bayesian Computation (ABC for short) is a family of computational techniques which offer an almost automated solution in situations where evaluation of the posterior likelihood is computationally prohibitive, or whenever suitable likelihoods are not available. In the present paper, we analyze the procedure from the point of view of k-nearest neighbor theory and explore the statistical properties of its outputs. We discuss in particular some asymptotic features of the genuine conditional density estimate associated with ABC, which is an interesting hybrid between a k-nearest neighbor and a kernel method.

math.ST↗

arXiv2012

On the length of one-dimensional reactive paths

Motivated by some numerical observations on molecular dynamics simulations, we analyze metastable trajectories in a very simplecsetting, namely paths generated by a one-dimensional overdamped Langevin equation for a double well potential. More precisely, we are interested in so-called reactive paths, namely trajectories which leave definitely one well and reach the other one. The aim of this paper is to precisely analyze the distribution of the lengths of reactive paths in the limit of small temperature, and to compare the theoretical results to numerical results obtained by a Monte Carlo method, namely the multi-level splitting approach.

math.PR↗

arXiv2010

A multiple replica approach to simulate reactive trajectories

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.

cond-mat.stat-mech↗