Particle Rare Event Stochastic Simulation P. Del Moral INRIA Centre - PowerPoint PPT Presentation

Introduction An introduction to interacting stochastic algorithms Some rare event models An introduction to continuous time models Particle Rare Event Stochastic Simulation P. Del Moral INRIA Centre Bordeaux-Sud Ouest HIM workshop : Numerical methods in molecular simulation, April 2008

Introduction An introduction to interacting stochastic algorithms Some rare event models An introduction to continuous time models Outline Introduction 1 Some motivations Some stochastic rare event models Stochastic sampling strategies An introduction to interacting stochastic algorithms 2 Mean field particle methods Interacting Markov chain Monte Carlo models (i-MCMC) Some rare event models 3 Boltzmann-Gibbs distribution flows Markov processes with fixed terminal values Multi-splitting rare events excursions Fixed time level set entrances Particle absorption models An introduction to continuous time models 4 Some references 5

Introduction An introduction to interacting stochastic algorithms Some rare event models An introduction to continuous time models Some motivations Some typical rare events Physical/biological/economical stochastic process : atomic/molecular configurations fluctuations, queueing evolutions, communication network, portfolio and financial assets, ... Potential function-Event restrictions : Energy/Hamiltonian potential functions, overflows levels, critical thresholds, epidemic propagations, radiation dispersion, ruin levels. Objectives Compute rare event probabilities. Find the law of the whole random process trajectories evolving in a critical regime � prediction ⊕ control. � Solution : Stochastic genealogical type tree fault model ∼ Branching+interacting evolutionary particle model (Branching on ”more likely” gateways to critical regimes)

Introduction An introduction to interacting stochastic algorithms Some rare event models An introduction to continuous time models Some stochastic rare event models Event restrictions Event restrictions X r.v. ∈ ( E , E ) with µ = Law ( X ) A ∈ E with 0 < µ ( A ) = P ( X ∈ A ) ≃ 10 − p and p >> 1. 1 µ ( A ) 1 A ( x ) µ ( dx ) = P ( X ∈ dx | X ∈ A ) η ( dx ) = Examples R , R d , R {− n ,..., n } 2 , ∪ n ≥ 0 ( R d ) { 0 ,..., n } , . . . E = [ a , ∞ [ , V − 1 ([ a , ∞ [) , { an excursion hits B before C } . . . A = µ = uniform on E finite � combinatorial counting pb First heuristic A n ↓ A � A n +1 -interacting MCMC with local targets ∝ 1 A n ( x ) µ ( dx )

Introduction An introduction to interacting stochastic algorithms Some rare event models An introduction to continuous time models Some stochastic rare event models A pair of more precise examples Non intersecting random walks/connectivity constants : n ) ∈ E := ( Z d × . . . × Z d ) ( X ′ 0 , . . . , X ′ X = � � ( x ′ 0 , . . . , x ′ n ) : ∀ 0 ≤ p < q ≤ n x ′ p � = x ′ A = q 1 ⇒ µ ( A ) = (2 d ) n × # { not ∩ walks with length n } ≃ exp (c n ) Law (( X ′ 0 , . . . , X ′ X ′ p � = X ′ ⇒ η = n ) | ∀ p < q ≤ n q ) Second heuristic ∼ multiplicative structure : � Accept-Reject interacting X ′ -motions

Introduction An introduction to interacting stochastic algorithms Some rare event models An introduction to continuous time models Some stochastic rare event models Random walk confinements/Lyap. exp. and top eigenval. : � p ∈ A ′ � n ) ∈ ( Z d × . . . × Z d ) : ∀ 0 ≤ p ≤ n x ′ ( x ′ 0 , . . . , x ′ A = p ∈ A ′ ) ≃ e − λ ( A ′ ) n ⇒ µ ( A ) = P ( ∀ 0 ≤ p ≤ n X ′ and ⇒ η = Law (( X ′ 0 , . . . , X ′ X ′ p ∈ A ′ ) n ) | ∀ 0 ≤ p ≤ n Same heuristic ∼ multiplicative structure : � Accept-Reject interacting X ′ -motions

Introduction An introduction to interacting stochastic algorithms Some rare event models An introduction to continuous time models Some stochastic rare event models More examples of stochastic rare event models P ( ∩ 0 ≤ p ≤ n { X p ∈ A p } ) , Law (( X p ) 0 ≤ p ≤ n | ∩ 0 ≤ p ≤ n { X p ∈ A p } ) Ex. : Law (( X ′ 0 , . . . , X ′ n ) | ∩ 0 ≤ p < q ≤ n {� X ′ p − X ′ q � ≥ ǫ ) Soft penalization : 1 A n � exp ( − β 1 �∈ A n ) Terminal level set conditioning : P ( V n ( X n ) ≥ a ) & Law (( X 0 , . . . , X n ) | V n ( X n ) ≥ a ) Fixed terminal value : Law π (( X 0 , . . . , X n ) | X n = x n ). Critical excursion behavior : ∪ in excursion space P ( X hits B before C ) & Law ( X | X hits B before C ) Last heuristic : � Interacting X -excursions on gateways levels � B . � interacting X -transitions increasing the potential V n .

Introduction An introduction to interacting stochastic algorithms Some rare event models An introduction to continuous time models Some stochastic rare event models A single (sequential) Feynman-Kac/Boltzmann-Gibbs formulation:     � 1  d P X d η n = G p ( X p ) n Z n  0 ≤ p < n G n =1 An = Law (( X 0 , . . . , X n ) | X 0 ∈ A 0 , . . . , X n ∈ A n ) and Z n P ( X 0 ∈ A 0 , . . . , X n ∈ A n ) = Observation : η n = ”complex nonlinear” transformation of η n − 1         � � G p ( X p )  = G p ( X p )  G n ( X n )   0 ≤ p ≤ n 0 ≤ p ≤ ( n − 1) Same heuristic ∼ multiplicative structure : � (Accept-Reject) G -interacting X -motions [and inversely!]

Introduction An introduction to interacting stochastic algorithms Some rare event models An introduction to continuous time models Stochastic sampling strategies Stochastic modeling Rare event = cascade of intermediate (less) rare events (increasing energies, critical levels, multilevel gateways). η n =Law(process | a series of n intermediate ↓ events) =nonlinear distribution flow with ↑ level of complexity. 1 η 0 → η 1 → . . . → η n − 1 → η n ( dx ) = γ n (1) γ n ( dx ) → . . . Rare event probabilities = normalizing constants γ n (1) = Z n . Interacting stochastic sampling strategy Interacting stoch. algo. = sampling w.r.t. a flow of meas. Mean field particle models (sequential Monte Carlo, population Monte Carlo, particle filters, pruning, spawning, reconfiguration, quantum Monte carlo, go with the winner). Interacting MCMC models (new i-MCMC technology) .

Introduction An introduction to interacting stochastic algorithms Some rare event models An introduction to continuous time models Nonlinear distribution flows η n ∈ P ( E n ) probability measures on ( E n , E n ) ( ↑ complexity). Φ n : P ( E n − 1 ) �→ P ( E n ) η n = Φ n ( η n − 1 ) with Two important transformations Markov transport eq. : M n ( x n − 1 , dx n ) from E n − 1 into E n � ( η n − 1 M n )( dx n ) := η n − 1 ( dx n − 1 ) M n ( x n − 1 , dx n ) E n − 1 Boltzmann-Gibbs transformation : G n : E n → R + 1 Ψ G n ( η n )( dx n ) := η n ( G n ) G n ( x n ) η n ( dx n )

Introduction An introduction to interacting stochastic algorithms Some rare event models An introduction to continuous time models Feynman-Kac distribution flows (Pr´ ediction,Correction)=(Exploration,Selection)=( G n , M n ) Heuristics � particle occupation measures ( ξ i n = i -th walker/individual/particle time=n) N � n := 1 η N δ ξ i n ≃ N ↑∞ η n = Φ n ( η n − 1 ) := Ψ G n − 1 ( η n − 1 ) M n N i =1 Solution : X n Markov ∼ transitions M n   � γ n ( f n )  f n ( X n )  η n ( f n ) = with γ n ( f n ) = E G p ( X p ) γ n (1) 0 ≤ p < n Multiplicative formula � Unbias estimation    � � � η N  = G p ( X p ) η p ( G p ) ≃ N ↑∞ p ( G p ) E 0 ≤ p < n 0 ≤ p < n 0 ≤ p < n

Introduction An introduction to interacting stochastic algorithms Some rare event models An introduction to continuous time models Running example Confinement potential Running example : G n = 1 A (or 1 A n ) : ⇒ γ n (1) P ( ∀ 0 ≤ p < n X p ∈ A ) = P ( X n ∈ dx n | ∀ 0 ≤ p < n X p ∈ A ) η n = Key multiplicative formula � � γ n (1) = η p ( G p ) = P ( X p ∈ A | ∀ 0 ≤ q < p X q ∈ A ) 0 ≤ p < n 0 ≤ p < n Note : η n � = Law of a Markov process with local restrictions to A .

Introduction An introduction to interacting stochastic algorithms Some rare event models An introduction to continuous time models Structural stability properties State space enlargements � same model! X n = ( X ′ n − 1 , X ′ X n = ( X ′ 0 , . . . , X ′ n ) or n ) or excursions Ex.: X n = ( X ′ 0 , . . . , X ′ n )   �  f n ( X ′ 0 , . . . , X ′ G p ( X ′ 0 , . . . , X ′  ⇒ η n ( f n ) ∝ E n ) p ) 0 ≤ p < n Boltzmann-Gibbs’ formulation :     � d η n = 1  d P X G p ( X p ) n Z n  0 ≤ p < n

Introduction An introduction to interacting stochastic algorithms Some rare event models An introduction to continuous time models Structural stability properties Importance sampling distributions � same model! Change of proba. : X n = ( X ′ n − 1 , X ′ n ) � Y n = ( Y ′ n − 1 , Y ′ n )     � �  ∝ E  f n ( X n )  f n ( Y n )  E G p ( X p ) H p ( Y p ) 0 ≤ p < n 0 ≤ p < n = G (1) × G (2) Related weighted meas. G n = G ǫ n n × G 1 − ǫ n = . . . n n n Complexity and Sampling problems Path integration formulae, infinite dimensional state spaces Nonlinear-Nongaussian models Complex probability mass variations