Statistique d'ordre | EPFL Graph Search

En statistiques, la statistique d'ordre de rang k d'un échantillon statistique est égal à la k-ième plus petite valeur. Associée aux statistiques de rang, la statistique d'ordre fait partie des outils fondamentaux de la statistique non paramétrique et de l'inférence statistique. Deux cas importants de la statistique d'ordre sont les statistiques du minimum et du maximum, et dans une moindre mesure la médiane de l'échantillon ainsi que les différents quantiles. Quand on emploie la théorie des probabilités pour analyser les statistiques d'ordre d'un échantillon issu d'une loi de probabilité continue, la fonction de répartition est employée pour ramener l'analyse au cas de la statistique d'ordre sur une loi uniforme continue Notation et exemples Soit une expérience conduisant à l'observation d'un échantillon de 4 nombres, prenant les valeurs suivantes : :6, 9, 3, 8, que l'on note selon la convention : :x_1=6;\ \ x_2=9;\ \ x_3=3;\ \ x_4

Numerical investigation of particle-laden thermally driven turbulent flows in enclosure

Riccardo Puragliesi

Nowadays, the global increase of energy demand and the necessity to satisfy high safety standards have led engineers and scientists to focus their efforts in order to understand and describe fundamental phenomena that are crucial for a correct design of the new generation nuclear power plants. In this framework, the present thesis aims at providing a first insight of the mechanisms of deposition of aerosol particles inside a closed geometry where relatively strong currents are present due to turbulent natural convective flows. Direct Numerical Simulations were conducted coupling high-order pseudo-spectral code with a Lagrangian particle tracker. Laminar flows were computed in two and three dimensions in order to benchmark the code with published reference data. A parametric study was performed for three different aerosol micro-size particle diameters and two super-critical Rayleigh numbers in a square cavity. An extended analysis of the turbulent flows is provided in terms of first and second order statistics, time-averaged momentum and energy budgets, and moreover, important terms appearing in the transport equations of turbulent kinetic energy and temperature variance are also briefly discussed. Furthermore, the evolution in time of particle concentration for the three different diameters is considered. The text provides information about the deposition velocity, the deposition patterns on the cavity surfaces, the influence of lift and thermophoretic forces and the fractal dimension. The same size dependent parametric study for the three different sets of micro-size particles was carried out in a fully three-dimensional closed cubic cavity for one super-critical Rayleigh number. A detailed investigation of the turbulence was performed by means of statistical quantities, signal processing and conditional averaging, in order to get a general view of the complexity of the flow and its characteristics. Further on, the sedimentation process is studied in the same way as for the two dimensional case. Finally a simple theoretical deposition model is provided in order to interpret the numerical results for the aerosol phase.

EPFL2010

Large Eddy Simulation of Particle Dynamics Inside a Differentially Heated Cavity

Christoph Bosshard

In nuclear safety, most severe accident scenarios lead to the presence of fission products in aerosol form in the closed containment atmosphere. It is important to understand the particle depletion process to estimate the risk of a release of radioactivity to the environment should a containment break occur. As a model for the containment, we use the three-dimensional differentially heated cavity (DHC) problem. DHC is a cubical box with a hot wall and a cold wall on vertical opposite sides. On the other walls of the cube we have adiabatic boundary conditions. For the velocity field the no-slip boundary condition is valid. The flow of the air in the cavity is described by the Boussinesq equations. Complex flow patterns develop and the flow characteristics depend on the non-dimensional Rayleigh and Prandtl numbers. The predominant flow type in the DHC is a turbulent natural convection flow. This work aims at reaching Rayleigh numbers and turbulent levels as high as possible given the available computational resources. The method used to simulate the turbulent flow is the large eddy simulation (LES) where the dynamics of the large eddies is resolved by the computational grid and the small eddies are modelled by the introduction of subgrid scale quantities using a filter function. Numerically, the LES equations are discretized by the spectral element method. Particle trajectories are computed using the Lagrangian particle tracking method, including the relevant forces (drag, gravity, thermophoresis). Four different particle sets with each set containing one million particles and diameters of 10 μm, 15 μm, 25 μm and 35 μm are simulated. The complexity and the size of the large three-dimensional problem requires the use of massively parallel supercomputers. Spectral element methods are naturally suitable for parallelisation by distributing the elements among the processors. For the Lagrangian particle tracking we use a method where equal numbers of particles are assigned to every processor. The flow field is broadcast and every particle processor tracks the assigned particles, a procedure which leads to a perfect load balancing. Simulation results for the flow field and particle sizes from 15 μm to 35 μm at a Rayleigh number of 109 are compared to previous results from a direct numerical simulation. First order statistics of the LES flow fields are in very good agreement with the direct numerical simulation while the agreement of second order moments is fair. Also the turbulent structures associated to the maximum of turbulent kinetic energy production are correctly reproduced. Particle statistics in the LES and the direct numerical simulation were similar and the settling rates practically identical. Contrary to previous particle simulations in LES, it was found that no model was necessary for the influence of the unresolved flow scales on the particle motions. This can be explained, because the important settling mechanism is through gravity and particle deposition at the walls by turbophoresis is negligible.

EPFL2012

Distributed top-k aggregation queries at large

Sebastian Michel

Top-k query processing is a fundamental building block for efficient ranking in a large number of applications. Efficiency is a central issue, especially for distributed settings, when the data is spread across different nodes in a network. This paper introduces novel optimization methods for top-k aggregation queries in such distributed environments. The optimizations can be applied to all algorithms that fall into the frameworks of the prior TPUT and KLEE methods. The optimizations address three degrees of freedom: 1) hierarchically grouping input lists into top-k operator trees and optimizing the tree structure, 2) computing data-adaptive scan depths for different input sources, and 3) data-adaptive sampling of a small subset of input sources in scenarios with hundreds or thousands of query-relevant network nodes. All optimizations are based on a statistical cost model that utilizes local synopses, e.g., in the form of histograms, efficiently computed convolutions, and estimators based on order statistics. The paper presents comprehensive experiments, with three different real-life datasets and using the ns-2 network simulator for a packet-level simulation of a large Internet-style network.