arXiv subjects

Clément Walter

Publications and source records attributed to Clément Walter.

4 recordsLinked to original sources

arXiv2017

Shape and fission instabilities of ferrofluids in non-uniform magnetic fields

We study static distributions of ferrofluid submitted to non-uniform magnetic fields. We show how the normal-field instability is modified in the presence of a weak magnetic field gradient. Then we consider a ferrofluid droplet and show how the gradient affects its shape. A rich phase transitions phenomenology is found. We also investigate the creation of droplets by successive splits when a magnet is vertically approached from below and derive theoretical expressions which are solved numerically to obtain the number of droplets and their aspect ratio as function of the field configuration. A quantitative comparison is performed with previous experimental results, as well as with our own experiments, and yields good agreement with the theoretical modeling.

physics.flu-dyn↗

arXiv2015

Point Process-based Monte Carlo estimation

This paper addresses the issue of estimating the expectation of a real-valued random variable of the form $X = g(\mathbf{U})$ where $g$ is a deterministic function and $\mathbf{U}$ can be a random finite- or infinite-dimensional vector. Using recent results on rare event simulation, we propose a unified framework for dealing with both probability and mean estimation for such random variables, \emph{i.e.} linking algorithms such as Tootsie Pop Algorithm (TPA) or Last Particle Algorithm with nested sampling. Especially, it extends nested sampling as follows: first the random variable $X$ does not need to be bounded any more: it gives the principle of an ideal estimator with an infinite number of terms that is unbiased and always better than a classical Monte Carlo estimator -- in particular it has a finite variance as soon as there exists $k \in \mathbb{R} > 1$ such that $\operatorname{E}[X^k] < \infty$. Moreover we address the issue of nested sampling termination and show that a random truncation of the sum can preserve unbiasedness while increasing the variance only by a factor up to 2 compared to the ideal case. We also build an unbiased estimator with fixed computational budget which supports a Central Limit Theorem and discuss parallel implementation of nested sampling, which can dramatically reduce its computational cost. Finally we extensively study the case where $X$ is heavy-tailed.

cs.CE↗

arXiv2015

Rare Event Simulation and Splitting for Discontinuous Random Variables

Multilevel Splitting methods, also called Sequential Monte-Carlo or \emph{Subset Simulation}, are widely used methods for estimating extreme probabilities of the form $P[S(\mathbf{U}) > q]$ where $S$ is a deterministic real-valued function and $\mathbf{U}$ can be a random finite- or infinite-dimensional vector. Very often, $X := S(\mathbf{U})$ is supposed to be a continuous random variable and a lot of theoretical results on the statistical behaviour of the estimator are now derived with this hypothesis. However, as soon as some threshold effect appears in $S$ and/or $\mathbf{U}$ is discrete or mixed discrete/continuous this assumption does not hold any more and the estimator is not consistent. In this paper, we study the impact of discontinuities in the \emph{cdf} of $X$ and present three unbiased \emph{corrected} estimators to handle them. These estimators do not require to know in advance if $X$ is actually discontinuous or not and become all equal if $X$ is continuous. Especially, one of them has the same statistical properties in any case. Efficiency is shown on a 2-D diffusive process as well as on the \emph{Boolean SATisfiability problem} (SAT).

stat.CO↗

arXiv2014

Moving Particles: a parallel optimal Multilevel Splitting method with application in quantiles estimation and meta-model based algorithms

Considering the issue of estimating small probabilities p, ie. measuring a rare domain F = {x | g(x) > q} with respect to the distribution of a random vector X, Multilevel Splitting strategies (also called Subset Simulation) aim at writing F as an intersection of less rare events (nested subsets) such that their measures are conditionally easily computable. However the definition of an appropriate sequence of nested subsets remains an open issue. We introduce here a new approach to Multilevel Splitting methods in terms of a move of particles in the input space. This allows us to derive two main results: (1) the number of samples required to get a realisation of X in F is drastically reduced, following a Poisson law with parameter log 1/p (to be compared with 1/p for naive Monte-Carlo); and (2) we get a parallel optimal Multilevel Splitting algorithm where there is indeed no subset to define any more. We also apply result (1) in quantile estimation producing a new parallel algorithm and derive a new strategy for the construction of first Design Of Experiments in meta-model based algorithms.

stat.CO↗