Introduction Framework and basic properties

Dieudonne Universite

Description

Introduction Framework and basic properties Case of general densities More precise results for smooth densities Application to Random Placement of Non-Overlapping Balls The problem of hard spheres Abstract probabilistic setting Geometric Analysis of Metropolis Algorithm on Bounded Domain L. Michel (joint work with P. Diaconis and G. Lebeau ) Laboratoire J.-A. Dieudonne Universite de Nice L. Michel (joint work with P. Diaconis and G. Lebeau ) Geometric Analysis of Metropolis Algorithm on Bounded Domain

• metropolis algorithm


• spheres abstract


• probabilistic setting


• markov kernel


• phase transition studies


• borel measure

Sujets

• Université
• Metropolis?Hastings algorithm
• Markov kernel
• Borel measure

Informations

• Publié par: pefav
• Nombre de lectures: 17
• Langue: English

Extrait

Introduction Framework and basic properties Case of general densities More precise results for smooth densities Application to Random Placement of Non-Overlapping Balls

Geometric

(joint

The problem of hard spheres Abstract probabilistic setting

Analysis of Metropolis Bounded Domain

work

with

L. Michel P. Diaconis and

G.

Laboratoire J.-A. Dieudonne ´ Universite´deNice

L. Michel (joint work with P. Diaconis and G. Lebeau )

Algorithm

Lebeau

)

on

Geometric Analysis of Metropolis Algorithm on Bounded Domain

Introduction Framework and basic properties Case of general densities More precise results for smooth densities Application to Random Placement of Non-Overlapping Balls

The problem of hard spheres Abstract probabilistic setting

Letµ=ρ(x)dxbe a probability measure on [a,b] and letfbe a regular function on [a,b]. We want to compute numerically the quantityI= b 1 −a R b a f(x)dµ(x). Standard ”deterministic” method consist to divide [a,b] into Ninterval and to approximateIby P k N =1 A k whereA k is the area corresponding to thekth interval. Probabilist approach: let (x n ) n∈N be a sequence of numbers in [a,b] such thatx n is choosen at random with respect toµ. Then, the quantity N 1 P N n=1 f(x n ) provides a good approximation ofI. A priori, ”choose a point at random with respect toµ” is not a simpler problem than ”computeI”. The Metropolis Algorithm provides an efficient procedure to sample fromµ.

L. Michel (joint work with P. Diaconis and G. Lebeau )

Geometric Analysis of Metropolis Algorithm on Bounded Domain

Introduction Framework and basic properties Case of general densities More precise results for smooth densities Application to Random Placement of Non-Overlapping Balls

The problem of hard spheres

The problem of hard spheres Abstract probabilistic setting

Consider a fixed box inR d ,B=]−1,1[ d  consider the. We problem of placement ofNballs of radius >0 with centers inB under the condition thatthe balls do not overlap. We denote O N, ⊂B N  Wethe set of all possible configurations. endoweO N, with the Lebesgue measuredL.

Problem: Build a sample of pointsX 1 , . . . ,X r ∈ O N, with respect todL.

distributed uniformly

This problem occurs in statistical physics in phase transition studies. It can be formulated in a more abstract setting.

L. Michel (joint work with P. Diaconis and G. Lebeau )

Geometric Analysis of Metropolis Algorithm on Bounded Domain