On the stability and the applications of interacting particle systems

Transcription

1 1/42 On the stability and the applications of interacting particle systems P. Del Moral INRIA Bordeaux Sud Ouest PDE/probability interactions - CIRM, April 2017 Synthesis joint works with A.N. Bishop, J. Garnier, A. Guionnet, A. Kurtzmann, J. Jacod M. Ledoux, L. Miclo, J. Tugaut, L. Wu

2 2/42 Nonlinear Markov chains Mean field particle methods Some stochastic perturbation tools Feynman-Kac models ( filtering, rare events,... ) A brief review on genetic type models Non commutative models Lyapunov weighted dynamics model Particle Gibbs-Glauber dynamics Many body Feynman-Kac (particle) models Duality formulae + Taylor expansions Branching proc. and Interacting MCMC Continuous time models Feynman-Kac models - Diffusion Monte Carlo Gradient type diffusions Ensemble Kalman-Bucy filter Stability + Unif. prop. chaos

3 3/42 Nonlinear Markov chains Mean field particle methods Some stochastic perturbation tools Feynman-Kac models ( filtering, rare events,... ) Particle Gibbs-Glauber dynamics Continuous time models

4 4/42 Nonlinear distribution flows Probability measures η n on some state spaces S n η n = Φ n (η n 1 )

5 4/42 Nonlinear distribution flows Probability measures η n on some state spaces S n η n = Φ n (η n 1 ) Ex. Feynman-Kac measures, conditional distributions, nonlinear MCMC, discrete time McKean-Vlasov, discrete time approximations of any nonlinear integro-diff eq. of proba (jumps, collisions, diffusions),...

6 4/42 Nonlinear distribution flows Probability measures η n on some state spaces S n η n = Φ n (η n 1 ) Ex. Feynman-Kac measures, conditional distributions, nonlinear MCMC, discrete time McKean-Vlasov, discrete time approximations of any nonlinear integro-diff eq. of proba (jumps, collisions, diffusions),... Continuous time Discrete time on time mesh Drift + Diffusion : Euler type scheme Jumps + Exponential rates : Geometric/Bernoulli type schemes η n = Φ n (η n 1 ) dt 0 nonlinear integro-diff equation

7 5/42 Nonlinear Markov models η n = Φ n (η n 1 ) Always K n,η Markov transitions s.t. Φ n (η n 1 ) = η n 1 K n,ηn 1 = Law ( ) X n = ηn i.e. : P(X n dx n X n 1 ) = K n,ηn 1 (X n 1, dx n )

8 5/42 Nonlinear Markov models η n = Φ n (η n 1 ) Always K n,η Markov transitions s.t. Φ n (η n 1 ) = η n 1 K n,ηn 1 = Law ( ) X n = ηn i.e. : P(X n dx n X n 1 ) = K n,ηn 1 (X n 1, dx n ) Non unique Markov interpretations,...

9 6/42 McKean measures (historical process) P( ( X 0,..., X n ) d(x0,..., x n )) = η 0 (dx 0 ) K 1,η0 (x 0, dx 1 )... K n,ηn 1 (x n 1, dx n )

10 6/42 McKean measures (historical process) P( ( ) X 0,..., X n d(x0,..., x n )) = η 0 (dx 0 ) K 1,η0 (x 0, dx 1 )... K n,ηn 1 (x n 1, dx n ) defined sequentially with η n = Φ n (η n 1 ) = n-th marginal = Law(X n )

11 6/42 McKean measures (historical process) P( ( ) X 0,..., X n d(x0,..., x n )) = η 0 (dx 0 ) K 1,η0 (x 0, dx 1 )... K n,ηn 1 (x n 1, dx n ) defined sequentially with η n = Φ n (η n 1 ) = n-th marginal = Law(X n ) Nb: the historical process is also a nonlinear Markov chain X n := ( X 0,..., X n )

12 6/42 McKean measures (historical process) P( ( ) X 0,..., X n d(x0,..., x n )) = η 0 (dx 0 ) K 1,η0 (x 0, dx 1 )... K n,ηn 1 (x n 1, dx n ) defined sequentially with η n = Φ n (η n 1 ) = n-th marginal = Law(X n ) Nb: the historical process is also a nonlinear Markov chain X n := ( X 0,..., X n ) Law(X n ) = η n = Φ n (η n 1 )

13 /42 McKean measures (historical process) P( ( ) X 0,..., X n d(x0,..., x n )) = η 0 (dx 0 ) K 1,η0 (x 0, dx 1 )... K n,ηn 1 (x n 1, dx n ) defined sequentially with η n = Φ n (η n 1 ) = n-th marginal = Law(X n ) Nb: the historical process is also a nonlinear Markov chain X n := ( X 0,..., X n ) Law(X n ) = η n = Φ n (η n 1 ) = η n 1 K n,ηn 1,...

14 7/42 Nonlinear Markov chains Mean field particle methods Some stochastic perturbation tools Feynman-Kac models ( filtering, rare events,... ) Particle Gibbs-Glauber dynamics Continuous time models

15 8/42 Mean field particle interpretation Markov chain ξ n = (ξ 1 n, ξ 2 n,..., ξ N n ) S N n ξ i n 1 ξ i n K n,η N n 1 (ξ i n 1, dx n ) with 1 N 1 i N δ ξ i n := η N n N η n Same models for the McKean measures ξ i n = (ξi 0,..., ξ i n) mean field particle of X n := ( X 0,..., X n )

16 8/42 Mean field particle interpretation Markov chain ξ n = (ξ 1 n, ξ 2 n,..., ξ N n ) S N n ξ i n 1 ξ i n K n,η N n 1 (ξ i n 1, dx n ) with 1 N 1 i N δ ξ i n := η N n N η n Same models for the McKean measures ξ i n = (ξi 0,..., ξ i n) mean field particle of X n := ( X 0,..., X n ) Interpretations: Stochastic adaptive grid, stochastic linearization, microscopic particle model, Sequential Monte Carlo method,...

17 9/42 Nonlinear Markov chains Mean field particle methods Some stochastic perturbation tools Feynman-Kac models ( filtering, rare events,... ) Particle Gibbs-Glauber dynamics Continuous time models

18 10/42 General fluctuation theorem local fluctuation random fields η N n Φ n ( η N n 1 ) = 1 N V N n

19 10/42 General fluctuation theorem local fluctuation random fields η N n Φ n ( η N n 1 ) = 1 N V N n f /Multivariate/Functional/Donsker theorems (dp+guionnet [99],Miclo[00],Ledoux [00],Jacod[02],...,Wu[11,13]) ( V N n ) n 0 N (V n ) n 0 with independent centered Gaussian fields V n s.t. E ( [V n (f )] 2) = η n 1 [ Kn,ηn 1 [f K n,ηn 1 (f )] 2]

20 11/42 Intuitive picture nonlinear sg : η n = Φ n(η n 1) = Φ p,n(η p) = η n Local fluctuations transport formulation : η 0 η 1 = Φ 1(η 0) η 2 = Φ 0,2(η 0) Φ 0,n(η 0) η0 N Φ 1(η0 N ) Φ 0,2(η0 N ) Φ 0,n(η0 N ) η1 N Φ 2(η1 N ) Φ 1,n(η1 N ) η2 N Φ 2,n(η2 N ). ηn 1 N Φ n(ηn 1) N ηn N

21 11/42 Intuitive picture nonlinear sg : η n = Φ n(η n 1) = Φ p,n(η p) = η n Local fluctuations transport formulation : η 0 η 1 = Φ 1(η 0) η 2 = Φ 0,2(η 0) Φ 0,n(η 0) η0 N Φ 1(η0 N ) Φ 0,2(η0 N ) Φ 0,n(η0 N ) η1 N Φ 2(η1 N ) Φ 1,n(η1 N ) η2 N Φ 2,n(η2 N ). ηn 1 N Φ n(ηn 1) N ηn N Local fluctuation errors η N n = Φ n(η N n 1) + 1 N V N n

22 12/42 Key decomposition formulae For any time-window ηn N η n = ηn N Φ n p,n(ηn p) N + Φ n p,n(ηn p) N Φ n p,n(η n p) }{{}}{{} e c 2 p then c = c 1 + c 2 e c 1 p / N p = (2c) 1 log N = e c 1 p / N = e c 2 p = More refined analysis ( 1 N ) c2 /(c 1 +c 2 ) η N n η n = n q=0 [Φ q,n(η N q ) Φ q,n(φ q(η N q 1))]

23 12/42 Key decomposition formulae For any time-window ηn N η n = ηn N Φ n p,n(ηn p) N + Φ n p,n(ηn p) N Φ n p,n(η n p) }{{}}{{} e c 2 p then c = c 1 + c 2 e c 1 p / N p = (2c) 1 log N = e c 1 p / N = e c 2 p = More refined analysis ( 1 N ) c2 /(c 1 +c 2 ) η N n η n = = n q=0 n q=0 [Φ q,n(η N q ) Φ q,n(φ q(η N q 1))] ( [Φ q,n Φ q(ηq 1) N + 1 ) ( ) ] Vn N Φ q,n Φ q(ηq 1) N N Semigroup/Stability/Uniform propagation of chaos/... dp+guionnet CRAS [99], and IHP [00], Miclo [00], [02], [03], Rousset [05],...

24 13/42 Another tool I N q := q-multi-indexes with values η (N,q) n (f ) := m(ξ n ) q := 1 (N) q c I N q δ ( ξ c(1) n ),...,ξn c(q)

25 13/42 Another tool I N q := q-multi-indexes with values η (N,q) n (f ) := m(ξ n ) q := 1 (N) q c I N q δ ( ξ c(1) n ),...,ξn c(q) combinatorial/transport/coalescent tree formulae between m(ξ n ) q and m(ξ n ) q and m(ξ n ) q m(ξ n ) q tv (q 1) 2 /N

26 13/42 Another tool I N q := q-multi-indexes with values η (N,q) n (f ) := m(ξ n ) q := 1 (N) q c I N q δ ( ξ c(1) n ),...,ξn c(q) combinatorial/transport/coalescent tree formulae between m(ξ n ) q and m(ξ n ) q and m(ξ n ) q m(ξ n ) q tv (q 1) 2 /N Why these objects?

27 13/42 Another tool I N q := q-multi-indexes with values η (N,q) n (f ) := m(ξ n ) q := 1 (N) q c I N q δ ( ξ c(1) n ),...,ξn c(q) combinatorial/transport/coalescent tree formulae between m(ξ n ) q and m(ξ n ) q and m(ξ n ) q m(ξ n ) q tv (q 1) 2 /N Why these objects? Propagation of chaos = Bias E ( f (ξn, 1..., ξn) q ) ( ) ξ n 1 = E η n (N,q) (f ) ξ n 1 = η (N,q) n 1 K q m(ξ (f ) n 1)

28 Another tool I N q := q-multi-indexes with values η (N,q) n (f ) := m(ξ n ) q := 1 (N) q c I N q δ ( ξ c(1) n ),...,ξn c(q) combinatorial/transport/coalescent tree formulae between m(ξ n ) q and m(ξ n ) q and m(ξ n ) q m(ξ n ) q tv (q 1) 2 /N Why these objects? Propagation of chaos = Bias E ( f (ξn, 1..., ξn) q ) ( ) ξ n 1 = E η n (N,q) (f ) ξ n 1 = η (N,q) n 1 K q m(ξ (f ) n 1) Non asymp. Taylor exp.+ Uniform propagation of chaos q-particles 13/42 dp Springer [04], dp+peters SIAM [06], dp+patras+rubenthaler AoAP [06], dp+miclo SAA[07] Chan+dp+Jasra Arxiv [14]

29 14/42 Nonlinear Markov chains Mean field particle methods Some stochastic perturbation tools Feynman-Kac models ( filtering, rare events,... ) A brief review on genetic type models Non commutative models Lyapunov weighted dynamics model Particle Gibbs-Glauber dynamics Continuous time models

30 15/42 Feynman-Kac measures with Q n (F n ) = Γ n (F n )/Γ n (1) and η n (f n ) = γ n (f n )/γ n (1) Γ n (F n ) = E (F n (X n ) Z n (X )) and γ n (f n ) = E (f n (X n ) Z n (X )) with the historical process X n = (X 0,..., X n ) and Z n (X ) = G k (X k ) 0 k<n

31 15/42 Feynman-Kac measures with Q n (F n ) = Γ n (F n )/Γ n (1) and η n (f n ) = γ n (f n )/γ n (1) Γ n (F n ) = E (F n (X n ) Z n (X )) and γ n (f n ) = E (f n (X n ) Z n (X )) with the historical process X n = (X 0,..., X n ) and Z n (X ) = G k (X k ) 0 k<n Mean field particle Genetic/Branching process ξ n = (ξ i n) i η n η N n = 1 δ N ξ i n 1 i N }{{} :=m(ξ n)

32 15/42 Feynman-Kac measures with Q n (F n ) = Γ n (F n )/Γ n (1) and η n (f n ) = γ n (f n )/γ n (1) Γ n (F n ) = E (F n (X n ) Z n (X )) and γ n (f n ) = E (f n (X n ) Z n (X )) with the historical process X n = (X 0,..., X n ) and Z n (X ) = G k (X k ) 0 k<n Mean field particle Genetic/Branching process ξ n = (ξ i n) i η n η N n = 1 δ N ξ i n 1 i N }{{} :=m(ξ n) path-space = 1 δ N (ξ i 0,n,ξ1,n i,...,ξi n,n ) 1 i N }{{} =Law ancestral line := X n Q n

33 16/42 Graphical illustration : η n η N n := 1 N 1 i N δ ξ i n

34 16/42 Graphical illustration : η n η N n := 1 N 1 i N δ ξ i n

35 16/42 Graphical illustration : η n η N n := 1 N 1 i N δ ξ i n

36 16/42 Graphical illustration : η n η N n := 1 N 1 i N δ ξ i n

37 16/42 Graphical illustration : η n η N n := 1 N 1 i N δ ξ i n

38 16/42 Graphical illustration : η n η N n := 1 N 1 i N δ ξ i n

39 16/42 Graphical illustration : η n η N n := 1 N 1 i N δ ξ i n

40 16/42 Graphical illustration : η n η N n := 1 N 1 i N δ ξ i n

41 16/42 Graphical illustration : η n η N n := 1 N 1 i N δ ξ i n

42 16/42 Graphical illustration : η n η N n := 1 N 1 i N δ ξ i n

43 16/42 Graphical illustration : η n η N n := 1 N 1 i N δ ξ i n

44 16/42 Graphical illustration : η n η N n := 1 N 1 i N δ ξ i n

45 16/42 Graphical illustration : η n η N n := 1 N 1 i N δ ξ i n

46 16/42 Graphical illustration : η n η N n := 1 N 1 i N δ ξ i n

47 16/42 Graphical illustration : η n η N n := 1 N 1 i N δ ξ i n

48 16/42 Graphical illustration : η n η N n := 1 N 1 i N δ ξ i n

49 16/42 Graphical illustration : η n η N n := 1 N 1 i N δ ξ i n

50 16/42 Graphical illustration : η n η N n := 1 N 1 i N δ ξ i n

51 16/42 Graphical illustration : η n η N n := 1 N 1 i N δ ξ i n

52 17/42 Equivalent particle algorithms Sequential Monte Carlo Sampling Resampling Particle Filters Prediction Updating Genetic Algorithms Mutation Selection Evolutionary Population Exploration Branching-selection Diffusion Monte Carlo Free evolutions Absorption Quantum Monte Carlo Walkers motions Reconfiguration Sampling Algorithms Transition proposals Accept-reject-recycle

53 17/42 Equivalent particle algorithms Sequential Monte Carlo Sampling Resampling Particle Filters Prediction Updating Genetic Algorithms Mutation Selection Evolutionary Population Exploration Branching-selection Diffusion Monte Carlo Free evolutions Absorption Quantum Monte Carlo Walkers motions Reconfiguration Sampling Algorithms Transition proposals Accept-reject-recycle Many other lively buzzwords : bootstrapping, spawning, cloning, pruning, replenish, splitting, enrichment, go with the winner, look-ahead, weighted dynamics, quantum teleportation, Heuristic style algo Particle Feynman-Kac models Convergence analysis : Uniform propagation of chaos, concentration, stability, CLT, LDP, MDP, Empirical processes,...

54 18/42 Feynman-Kac measures Unnormalized measures γ n (f n ) = η n (f n ) η p (G p ) 0 p<n

55 18/42 Feynman-Kac measures Unnormalized measures γ n (f n ) = η n (f n ) η p (G p ) 0 p<n Backward Markov chain formula G n 1 (x n 1 )M n (x n 1, dx n ) = H n (x n 1, x n ) λ n (dx n ) Q n (d(x 0,..., x n )) = η n (dx n ) M q+1,ηq (x q+1, dx q ) }{{} 0 q<n η q(dx q) H q+1(x q,x q+1)

56 18/42 Feynman-Kac measures Unnormalized measures γ n (f n ) = η n (f n ) η p (G p ) 0 p<n Backward Markov chain formula G n 1 (x n 1 )M n (x n 1, dx n ) = H n (x n 1, x n ) λ n (dx n ) Q n (d(x 0,..., x n )) = η n (dx n ) M q+1,ηq (x q+1, dx q ) }{{} 0 q<n η q(dx q) H q+1(x q,x q+1) Application domains : Any conditional distributions, Bayesian prior/posterior, nonlinear filtering, Boltzmann-Gibbs measures, branching processes, Twisted importance sampling,...

57 19/42 Non commutative models with G n (x n ) R d d s.t. u S d 1 := { u = 1} we have G n (x) u > 0 f n (x 0,..., x n ) R d and 0 p n A p = A 0 A 1... A n Γ n (f n ).u 0 := E f n (X 0,..., X n ) = E f n (X 0,..., X n ) 0 p<n 0 p<n G p (X p ).u 0 G p (X p ) X n = (X n, U n ) ( E n S d 1) and G n (X n ) = G n (X n ).U n and the walk on the sphere model U n+1 = G n(x n ).U n G n (X n ).U n

58 20/42 of P n (ϕ)(x) = E x (ϕ(y n )) with Y n+1 = F n (Y n, W n ) First variational equation Jac (Y n+1 ) = G n (Y n, W n ) Jac (Y n ) with Gn i,j = x j Fn i = G p (X p ) with X n = (Y n, W n ) 0 p n P n (ϕ)(x).u 0 = E f n (X n ) }{{} = (ϕ)(y n).u n 0 p<n G p (X p ) }{{} = G p(x p).u p FK model w.r.t. Y n weighted with the directional Lyap. exp. 0 p n G p (X p ) = Jac (Y n ).u 0 = 0 p n Jac (Y p ).u 0 Jac (Y p 1 ).u 0

59 1/42 Related Feynman-Kac model X n = (X n, X n+1 ) and G n (X n ) = Jac (X n+1 ) α / Jac (X n ) α Feynman-Kac model = The Lyapunov weighted dynamics model dq n = 1 Z n Jac (X n ) α dp n α < 0 dq n favors low Lyapunov trajectories α > 0 dq n favors high Lyapunov trajectories Hyper-references : T Laffargue, K.D. Nguyen Thu Lam, J. Kurchan, J. Tailleur LDP of Lyapunov exp. (2013) S. Tanese Nicola, J. Kurchan. Metastable states, transitions, basins and borders at finite temperature J. Stat. Phys. (2004). J. Tailleur, S. Tanese Nicola, J. Kurchan. Kramers equations an supersymmetry J. Stat. Phys. (2006). J. Tailleur, J. Kurchan. Probing rare physical trajectories with Lyapunov weighted dynamics, Nature Physics (2007) Genealogical particle analysis of aare events (joint work with J. Garnier)AAP (2005).

60 22/42 Particle Feynman-Kac measures Unbiased particle unnormalized measures (dp MPRF (96)) γn N (f n ) := ηn N (f n ) ηp N (G p ) 0 p<n

61 22/42 Particle Feynman-Kac measures Unbiased particle unnormalized measures (dp MPRF (96)) γn N (f n ) := ηn N (f n ) ηp N (G p ) 0 p<n Particle backward model (dp+doucet+singh HAL (09), M2AN (10)) Q N n (d(x 0,..., x n )) := ηn N (dx n ) M q+1,η N q (x q+1, dx q ) }{{} 0 q<n ηq N (dxq) Hq+1(xq,xq+1)

62 23/42 A first theorem Theorem [dp+kohn+patras, Arxiv (14), CRAS (15)+IHP (16)] Law ( X n (ξ k,k ) 0 k n ) = Law of backward ancestral line X n given all pop.

63 3/42 A first theorem Theorem [dp+kohn+patras, Arxiv (14), CRAS (15)+IHP (16)] Law ( X n (ξ k,k ) 0 k n ) = Law of backward ancestral line X n given all pop. := ηn N (dx n ) M n,η N n 1 (x n, dx n 1 ) M 1,η N 0 (x 1, dx 0 ) with the occupation measures η N k = m(ξ k,k ) := 1 N 1 i N δ ξ i k,k

64 24/42 Nonlinear Markov chains Mean field particle methods Some stochastic perturbation tools Feynman-Kac models ( filtering, rare events,... ) Particle Gibbs-Glauber dynamics Many body Feynman-Kac (particle) models Duality formulae + Taylor expansions Branching proc. and Interacting MCMC Continuous time models

65 25/42 Many body FK measures π n and F n (ξ n ) := m(ξ n )(f n ) and G n (ξ n ) := m(ξ n )(G n ) Z n (ξ) := G k (ξ k ) Z n = η k (G k ) 0 k<n 0 k<n

66 25/42 Many body FK measures π n and F n (ξ n ) := m(ξ n )(f n ) and G n (ξ n ) := m(ξ n )(G n ) Z n (ξ) := G k (ξ k ) Z n = η k (G k ) 0 k<n 0 k<n Unbiasedness property in terms of many-body FK measures π n (F n ) : E (F n (ξ n ) Z n (ξ))

67 5/42 Many body FK measures π n and F n (ξ n ) := m(ξ n )(f n ) and G n (ξ n ) := m(ξ n )(G n ) Z n (ξ) := G k (ξ k ) Z n = η k (G k ) 0 k<n 0 k<n Unbiasedness property in terms of many-body FK measures π n (F n ) : E (F n (ξ n ) Z n (ξ)) = E (f n (X n ) Z n (X )) η n (f n ) (Note: the definition of π n if for any symmetric functions F n (ξ n ) )

68 5/42 Many body FK measures π n and F n (ξ n ) := m(ξ n )(f n ) and G n (ξ n ) := m(ξ n )(G n ) Z n (ξ) := G k (ξ k ) Z n = η k (G k ) 0 k<n 0 k<n Unbiasedness property in terms of many-body FK measures π n (F n ) : E (F n (ξ n ) Z n (ξ)) = E (f n (X n ) Z n (X )) η n (f n ) (Note: the definition of π n if for any symmetric functions F n (ξ n ) ) Unbiasedness property in terms of ancestral lines ( ) E (f n (X n ) Z n (ξ)) = E f n (X n) Z n (ξ) = E (f n (X n ) Z n (X ))

69 26/42 Second Extended Many body FK measures π n Definition: (Extended) Many-body FK measures π n (F n ) : E (F n (X n, ξ n ) Z n (ξ)) with historical process ξ n = (ξ 0,..., ξ n ) (ancestral lines evolution).

70 26/42 Second Extended Many body FK measures π n Definition: (Extended) Many-body FK measures π n (F n ) : E (F n (X n, ξ n ) Z n (ξ)) with historical process ξ n = (ξ 0,..., ξ n ) (ancestral lines evolution). Particle Gibbs-Glauber dynamics (PGD) Target Law π (X n, ξ n )

71 26/42 Second Extended Many body FK measures π n Definition: (Extended) Many-body FK measures π n (F n ) : E (F n (X n, ξ n ) Z n (ξ)) with historical process ξ n = (ξ 0,..., ξ n ) (ancestral lines evolution). Particle Gibbs-Glauber dynamics (PGD) Target Law π (X n, ξ n ) } { } { X n = z Xn = z (X n ξ n = x) Xn = z ξ n = x ξ n = x ξ n = x (ξ n X n = z)

72 26/42 Second Extended Many body FK measures π n Definition: (Extended) Many-body FK measures π n (F n ) : E (F n (X n, ξ n ) Z n (ξ)) with historical process ξ n = (ξ 0,..., ξ n ) (ancestral lines evolution). Particle Gibbs-Glauber dynamics (PGD) Target Law π (X n, ξ n ) } { } { X n = z Xn = z (X n ξ n = x) Xn = z ξ n = x ξ n = x ξ n = x (ξ n X n = z) Important observation: X n X n is a Markov process on path space

73 Duality formula To sample the second PGD transition. 27/42

74 27/42 Duality formula To sample the second PGD transition. Theo.- Duality [dp+kohn+patras, Arxiv (14), CRAS (15)+IHP (16)] F n E (F n (X n, ξ n ) Z n (ξ)) = E ( F n (X n, ξ n ) Z n(x ) ) with the dual process: ξ n defined as ξ n but with a given frozen ancestral line X n

75 27/42 Duality formula To sample the second PGD transition. Theo.- Duality [dp+kohn+patras, Arxiv (14), CRAS (15)+IHP (16)] F n E (F n (X n, ξ n ) Z n (ξ)) = E ( F n (X n, ξ n ) Z n(x ) ) with the dual process: ξ n defined as ξ n but with a given frozen ancestral line X n Corollary - Backward duality Same duality PGD when (X n, ξ n ) is replaced by (X n, ξ n,n ) with the population historical process ξ n,n = (ξ k,k ) 0 k n

76 28/42 Taylor expansions X n X n Markov process on path space with η n -reversible transition K n n = FIXED length/duration of the ancestral trajectories N = number of particles used in the genealogical trees

77 28/42 Taylor expansions X n X n Markov process on path space with η n -reversible transition K n n = FIXED length/duration of the ancestral trajectories N = number of particles used in the genealogical trees Theo. [dp+kohn+patras, Arxiv (14), CRAS (15)+IHP (16)] K n (x,.) = η n + 1 k l 1 N k d (k) K n (x,.) + 1 N l+1 l+1 K n (x,.) ( l 1) Derivatives coalescent/colored trees.

78 Taylor expansions X n X n Markov process on path space with η n -reversible transition K n n = FIXED length/duration of the ancestral trajectories N = number of particles used in the genealogical trees Theo. [dp+kohn+patras, Arxiv (14), CRAS (15)+IHP (16)] K n (x,.) = η n + 1 k l 1 N k d (k) K n (x,.) + 1 N l+1 l+1 K n (x,.) ( l 1) Derivatives coalescent/colored trees. d (k) K n (cnk 2 ) k and (l+1) K n (cn(l + 1) 2 ) l+1 28/42 as soon as N > cn(l + 1) 2, for some c <.

79 A couple of direct corollaries 29/42

80 29/42 A couple of direct corollaries Cor. 1: m 1 and osc(f ) 1 K m n (f )(x) η n (f ) β (K m n ) (cn/n) m ( min (N,m) 0 )

81 29/42 A couple of direct corollaries Cor. 1: m 1 and osc(f ) 1 K m n (f )(x) η n (f ) β (K m n ) (cn/n) m ( min (N,m) 0 ) Cor. 2: m 1, p 1 and f 1 ] m(f Km n (f ) η n (f ) Lp(η n) [d N m (1) K n ) Lp(η n) (cn/n)m+1

82 30/42 Some comments Spatial branching processes varying pop. size N n First moments FK-models FK-particle = fixed pop.size

83 30/42 Some comments Spatial branching processes varying pop. size N n First moments FK-models FK-particle = fixed pop.size Particle/Absorption models = FK models with potential G [0, 1] η n = Law(X c n T killing > n) η = quasi-invariant meas.

84 30/42 Some comments Spatial branching processes varying pop. size N n First moments FK-models FK-particle = fixed pop.size Particle/Absorption models = FK models with potential G [0, 1] η n = Law(X c n T killing > n) η = quasi-invariant meas. Mutations = MCMC with target η n e βnv (x) λ(dx) and branching rates/selection weights e (βn+1 βn)v (x) β n = interacting MCMC/SMC/...

85 30/42 Some comments Spatial branching processes varying pop. size N n First moments FK-models FK-particle = fixed pop.size Particle/Absorption models = FK models with potential G [0, 1] η n = Law(X c n T killing > n) η = quasi-invariant meas. Mutations = MCMC with target η n e βnv (x) λ(dx) and branching rates/selection weights e (βn+1 βn)v (x) β n = interacting MCMC/SMC/... Mutations = MCMC with target η n 1 An (x) λ(dx) and branching rates/selection weights 1 An+1 A n = interacting MCMC/SMC/Multilevel splitting...

86 31/42 Nonlinear Markov chains Mean field particle methods Some stochastic perturbation tools Feynman-Kac models ( filtering, rare events,... ) Particle Gibbs-Glauber dynamics Continuous time models Feynman-Kac models - Diffusion Monte Carlo Gradient type diffusions Ensemble Kalman-Bucy filter Stability + Unif. prop. chaos

87 32/42 (weak sense) : infinitesimal generators L t,η t η t (f ) = η t L t,ηt (f ) := η t (dx) L t,ηt (f )(x) L t,η regular familly X t ( McKean measure) Ex. jump-diff. S L t,η (f )(x) = a t (x, η) f (x) σ2 t (x, η) f (x) +λ t (x, η) (f (y) f (x)) K t,η (x, dy)

88 32/42 (weak sense) : infinitesimal generators L t,η t η t (f ) = η t L t,ηt (f ) := η t (dx) L t,ηt (f )(x) L t,η regular familly X t ( McKean measure) Ex. jump-diff. S L t,η (f )(x) = a t (x, η) f (x) σ2 t (x, η) f (x) +λ t (x, η) (f (y) f (x)) K t,η (x, dy) Mean field particle model ξt i with generator L t,η N t with ηt N = N 1 1 i N δ ξ i t

89 2/42 (weak sense) : infinitesimal generators L t,η t η t (f ) = η t L t,ηt (f ) := η t (dx) L t,ηt (f )(x) L t,η regular familly X t ( McKean measure) Ex. jump-diff. S L t,η (f )(x) = a t (x, η) f (x) σ2 t (x, η) f (x) +λ t (x, η) (f (y) f (x)) K t,η (x, dy) Mean field particle model ξ i t with generator L t,η N t with η N t = N 1 Local perturbation eq. 1 i N dη N t (f ) = η N t L t,η N t (f ) dt + 1 N dm N t (f ) δ ξ i t

90 33/42 FKS-semigroups γ t = γ s Q s,t Normalization ( { t } ) Q s,t (f )(x) = E f (X t ) exp V u (X u )du X s = x s η t (f ) := γ t (f )/γ t (1) Jump-diffusion particles with L t,η (f )(x) = L X (f )(x) + V (x) (f (y) f (x)) η t (dy) Denormalization ( t ) γ t (f ) = η t (f ) exp η s (V s )ds 0

91 34/42 FKS-Stochastic analysis Particle estimations ( [ t ]) E f (X t ) exp W (X s )ds 0 [ t = η t (f ) exp unbias N 0 [ t ηt N (f ) exp 0 ] η s (W )ds ] ηs N (W )ds

92 34/42 FKS-Stochastic analysis Particle estimations ( [ t ]) E f (X t ) exp W (X s )ds 0 [ t = η t (f ) exp unbias N 0 [ t ηt N (f ) exp 0 ] η s (W )ds ] ηs N (W )ds Ground states of Schrodinger op. : ( DMC, QMC) (v.p. λ ground state h (L µ-reversible)) [ t ] lim N,t ηn t h dµ et exp ηs N (W )ds e λt 0

93 34/42 FKS-Stochastic analysis Particle estimations ( [ t ]) E f (X t ) exp W (X s )ds 0 [ t = η t (f ) exp unbias N 0 [ t ηt N (f ) exp 0 ] η s (W )ds ] ηs N (W )ds Ground states of Schrodinger op. : ( DMC, QMC) (v.p. λ ground state h (L µ-reversible)) [ t ] lim N,t ηn t h dµ et exp ηs N (W )ds e λt 0 Asymptotic theory discrete time models. Some refs: dp+miclo [00,02,03,07] Mathias Rousset, Tony Lelièvre, and Gabriel Stoltz MATHERIALS/MICMAC INRIA team.

94 35/42 Gradient diff. - Ex: V (x) := x 4 4 x 2 2 and F (x) := α x 2 2 dx t = [ V + F η t ] ( ) X t dt + σdbt Applications : micro-macro-dilute physical models N = N 2 = particles algorithm?

95 35/42 Gradient diff. - Ex: V (x) := x 4 4 x 2 2 and F (x) := α x 2 2 dx t = [ V + F η t ] ( ) X t dt + σdbt Applications : micro-macro-dilute physical models N = N 2 = particles algorithm? V convex pioneering work of Malrieux [01]+Bakry-Emery tools.

96 35/42 Gradient diff. - Ex: V (x) := x 4 4 x 2 2 and F (x) := α x 2 2 dx t = [ V + F η t ] ( ) X t dt + σdbt Applications : micro-macro-dilute physical models N = N 2 = particles algorithm? V convex pioneering work of Malrieux [01]+Bakry-Emery tools. V and F symmetric + constant first moment (a.k.a. Law(X 0 ) symmetric or stationnary or? ) and α := inf F > sup V > 0 = F (x) := F (x) α x 2 2 and V (x) := V (x) + α x 2 2 convex case Carillo-McCann-Villani [03]

97 35/42 Gradient diff. - Ex: V (x) := x 4 4 x 2 2 and F (x) := α x 2 2 dx t = [ V + F η t ] ( ) X t dt + σdbt Applications : micro-macro-dilute physical models N = N 2 = particles algorithm? V convex pioneering work of Malrieux [01]+Bakry-Emery tools. V and F symmetric + constant first moment (a.k.a. Law(X 0 ) symmetric or stationnary or? ) and α := inf F > sup V > 0 = F (x) := F (x) α x 2 2 and V (x) := V (x) + α x 2 2 convex case Carillo-McCann-Villani [03] V non convex and σ too small non uniqueness invariant measures pionnering work by Tugaut [10], σ large enough and α > 1 Uniform propagation of chaos dp+tugaut [13].

98 36/42 Kalman-Bucy filter Linear+Gaussian filtering problem { dxt = A X t dt + R 1/2 dw t dy t = C X t dt + Σ 1/2 dv t F t := σ(y s, s t). Optimal L 2 -filter = Kalman-Bucy filter X t := E(X t F t ) and P t := E ( (X t E(X t F t )) (X t E(X t F t )) )

99 36/42 Kalman-Bucy filter Linear+Gaussian filtering problem { dxt = A X t dt + R 1/2 dw t dy t = C X t dt + Σ 1/2 dv t F t := σ(y s, s t). Optimal L 2 -filter = Kalman-Bucy filter X t := E(X t F t ) and P t := E ( (X t E(X t F t )) (X t E(X t F t )) ) d X t = A X t dt + P t C Σ 1 ( dy t C X ) t dt with the Riccati equation t P t = Ricc(P t ) := AP t + P t A P t SP t + R with S := C ΣC

100 37/42 Nonlinear Kalman-Bucy diffusion [ )] dx t = A X t dt + R 1/2 dw t +P ηt C Σ 1 dy t (CX t dt + Σ 1/2 dv t with the covariance matrices P ηt = η t [(e η t (e))(e η t (e)) ] with e(x) := x. E ( X t F t ) = Xt and P ηt = P t. Ensemble Kalman-Bucy filter = Interacting particles with empirical covariance p t := P η N t P t and m t = η N t (e) E(X t F t )

101 38/42 Nonlinear models Extended Kalman-Bucy-filters d X t = A( X t ) dt + P t C Σ 1 [ dy t C X ] t dt with the stochastic Riccati equation: t P t = A( X t )P t + P t A( X t ) + R P t SP t

102 38/42 Nonlinear models Extended Kalman-Bucy-filters d X t = A( X t ) dt + P t C Σ 1 [ dy t C X ] t dt with the stochastic Riccati equation: t P t = A( X t )P t + P t A( X t ) + R P t SP t McKean-Vlasov interpretation dx t = A ( X t, E[X t G t ] ) dt + R 1/2 dw t [ )] +P ηt C R 1 2 dy t (CX t dt + Σ 1/2 dv t with the drift function A(x, m) := A [m] + A [m] (x m).

103 39/42 Some illustrations Langevin type signal processes R = σ 2 Id and (A, A) = ( V, 2 V) Non quadratic potential (q R r, Q 1, Q 2 0) V(x) = 1 2 Q 1x, x + q, x Q 2x, x 3/2 Interacting diffusion gradient flows V(x) = U 1 (x i ) + 1 i r 1 i j r U 2 (x i, x j ) for some convex confining potential U i : R i [0, [

104 40/42 Regularity conditions Full observation S = s Id and λ A := sup x R r 1 λ max ( A(x) + A(x) ) < 0 A(x) A(y) κ A x y Examples: Langevin signal-diffusion ( ) (λ A, κ A ) = β 2 1 λ min (Q 1 ), 2λ 3/2 max(q 2 ). more generally 2 V v Id Lipschitz condition

105 41/42 Stability theorem (X t, Z t ) = McKean-Vlasov starting at (X 0, Z 0 ) Theo [dp-kurtzmann-tugaut] When λ A is sufficiently large we have W 2 (Law(X t ), Law(Z t )) c exp [ t λ] for some λ > 0. more explicit description in terms of (R, S, κ A ).

106 2/42 Propagation of chaos and P N t := Law(m t, p t ) P t := Law( X t, P t ) Q N t := Law(ξ 1 t ) Q t := Law(X t ) Theo SIAM Control & Opt [16] [dp-kurtzmann-tugaut] When λ A is sufficiently large, β ]0, 1/2] s.t. sup t 0 ( ) ( ) W 2 P N t, P t sup W 2 Q N t, Q t c N β t 0 Linear models unif. propagation of chaos with β = 1/2 for any stable drift matrix. Arxiv 1 (Stability KBF)+ A.N. Bishop Arxiv 2 (Perturbation KBF)+ A.N. Bishop Arxiv 3 (Unif. EnKBF)+ J. Tugaut. Arxiv 4 (Stability EKBF)+ A. Kurtzmann, J. Tugaut.