Introduction to asymptotic techniques for stochastic systems with multiple time-scales

Transcription

1 Introduction to asymptotic techniques for stochastic systems with multiple time-scales Eric Vanden-Eijnden Courant Institute

2 Motivating examples Consider the ODE {Ẋ = Y 3 + sin(πt) + cos( 2πt) X() = x Ẏ = ε 1 (Y X) Y () = y. If ε 1, Y is very fast and it will adjust rapidly to the current value of X, i.e. we will have Y = X + O(ε) at all times. Then the equation for X reduces to (1) Ẋ = X 3 + sin(πt) + cos( 2πt). (2)

3 #)& # ")& "!)&!!!)&!"! " # $ % & ' ( The solution of (1) when ε =.5 and we took X t= = 2, Y t= = 1. X is shown in blue, and Y in green. Also shown in red is the solution of the limiting equation (2).

4 In contrast, consider the SDE {Ẋ = Y 3 + sin(πt) + cos( 2πt), X t= = x Ẏ = ε 1 (Y X) + ε 1/2 Ẇ, Y t= = y. The equation for Y at fixed X = x defines an Ornstein-Uhlenbeck process whose equilibrium probability density is ρ(y x) = e (y x)2 π. The limiting equation is obtained by averaging the right hand-side of the equation for X in (3) with respect to this density: (3) Ẋ = X X + sin(πt) + cos( 2πt), X = x. (4) Note the new term 3 2 α2 X due to the noise in (3).

5 % $ # "!!"!#!$! " # $ % & ' ( The solution of (3) with X t= = 2, Y t= = 1 when ε = 1 3. X is shown in blue, and Y in green. Also shown in red is the solution of the limiting equation (4). Notice how noisy Y is.

6 Singular perturbations techniques for Markov processes Consider the system {Ẋ = f(x, Y ), Xt= = x R n Ẏ = ε 1 g(x, Y ), Y t= = y R m. Assume that the equation for Y at fixed X = x defines an Markov process which is ergodic for every x with respect to the probability distribution dµ x (y) If F (x) = f(x, y)dµ x (y) exists R m then in the limit as ε the evolution for X solution of (5) is governed by (5) Ẋ = F (X) X() = x. (6)

7 Derivation. Let L be the infinitesimal generator of the Markov process generated by (5) 1( lim Ex,y φ(x t, Y t ) φ(x, y) ) = (Lφ)(x, y) (7) t + t Then L = L + ε 1 L 1, and u(x, y, t) = E x,y φ(y, X) satisfies the backward Kolmogorov equation u t = L u + ε 1 L 1 u, u t= = φ (8) Look for a solution in the form of so that lim ε u = u. u = u + εu 1 + O(ε 2 ) (9)

8 Inserting (9) into (8) and equating equal powers in ε leads to the hierarchy of equations L 1 u =, L 1 u 1 = u t L u, L 1 u 2 = (1) The first equation tells that u belong to the null-space of L 1. Assuming that for every x, L 1 is the generator of an ergodic Markov process with equilibrium distribution µ x (y), this null-space is spanned by functions constant in y, i.e. u = u (x, t). Since the null-space of L 1 is non-trivial, the next equations each requires a solvability condition, namely that their right hand-side belongs to the range of L 1.

9 To see what this solvability condition actually is, take the expectation of both sides of the second equation in (1) with respect to dµ x (x). This gives ( u ) = dµ x (y) R t L u (11) m Explicitly, (11) is where u t = F (x) xu (12) F (x) = R m f(x, y)dµ x (y) (13) (12) is the backward Kolmogorov equation of (6).

10 Remark: Computing the expectation with respect to µ x (y) in practice. We have (e L1t φ)(x, y) φ(x, y)dµ x (y) as t R m In other words, if u(x, y, t) satisfies u t = L 1u, u t= = φ so that formally u(x, y, t) = (e L1t φ)(x, y), we have lim u(x, y, t) = φ(x, y)dµ x (y) t R m But since u(x, y, t) = E y φ(y x ) where it follows that Ẏ x = g(x, Y x ) t ) φ(x, y)dµ x (y) = lim E y φ(y x R t m 1 = lim T T T φ(y x t )dt (14)

11 Example: the Lorenz 96 (L96) model L96 consists of K slow variables X k coupled to J K fast variables Y j,k whose evolution is governed by Ẋ k = X k 1 (X k 2 X k+1 ) X k + F x + h J x Y j,k J j=1 (15) Ẏ j,k = 1 ( ) Yj+1,k (Y j+2,k Y j 1,k ) Y j,k + h y X k. ε We will study (15) with F x = 1, h x =.8, h y = 1, K = 9, J = 8, and two values of ε: ε = 1/128 and ε = 1/124.

12 Typical time-series of the slow (black line) and fast (grey line) modes; K = 9, J = 8, ε = 1/128. The subplot displays a typical snapshot of the slow and fast modes at a given time.

13 PDF of the slow variable; K = 9, J = 8, black line: ε = 1/16, grey line: ε = 1/124. The insensitivity in ε of the PDFs indicates that the slow variables have already converged close to their limiting behavior when ε = 1/16.

14 ACFs of the slow (thick line) and fast (thin line) variables; ε = 1/128, grey line: ε = 1/124. The subplot is the zoom-in of the main graph which shows the transient decay of the ACFs of the fast modes becoming faster as ε is decreased: this is the only signature in the ACFs of the fact that the Y j,k s are faster.

15 Typical PDFs of the coupling term (h x /J) J j=1 Z j,k(x) for various values of x. These PDFs are robust against variations in the initial conditions indicating that the dynamics of the fast modes conditional on the slow ones being fixed is ergodic.

16 Comparison between the ACFs and PDFs; black line: limiting dynamics; full grey line: ε = 1/128. Also shown in dashed grey are the corresponding ACF and PDF produced by the truncated dynamics where the coupling of the slow modes X k with the fast ones, Y j,k, is artificially switched off.

17 Black points: scatterplot of the forcing F (x) in the limiting dynamics. Grey points: scatter plot of the bare coupling term (h x /J) J j=1 Y j,k(x) when ε = 1/128.

18 Diffusive time-scales Suppose that F (x) = f(x, y)dµ x (y) =. (16) R m Then the limiting equation on the O(1) time-scale is trivial, Ẋ =, and the interesting dynamics arises on the diffusive time-scale O(ε 1 ). Consider then {Ẋ = ε 1 f(x, Y ), X t= = x R n Ẏ = ε 2 g(x, Y ), Y t= = y R m. (17) Proceeding as before we arrive at the following limiting equation when ε : where and Ẋ = b(x) + σ(x) Ẇ t, (18) ( Lφ)(x) b(x) x φ(x) (σσt )(x) : x x φ(x) ( = dt dµ x (y)f(x, y) x Ey f(x, Yt x ) x φ(x) ) (19) R m Ẏ x = g(x, Y x )

19 Derivation. The backward Kolmogorov equation for u(x, y, t) = E x,y f(x) is now u t = ε 1 L u + ε 2 L 1 u. Inserting the expansion u = u + εu 1 + ε 2 u 2 + O(ε 2 ) (we will have to go one order in ε higher than before) in this equation now gives L 1 u =, L 1 u 1 = L u, L 1 u 2 = u t L u 1, L 1 u 3 = The first equation tells that u (x, y, t) = u (x, t). (2) The solvability condition for the second equation is satisfied by assumption because of (16). Therefore this equation can be formally solved as u 1 = L 1 1 L 2u. Inserting this expression in the third equation in (2) and considering the solvability condition for this equation, we obtain the limiting equation for u : u t = Lu, (21) where L = dµ x (y)l L 1 1 L. R m

20 To see what this equation is explicitly, notice that L 1 1 g(y) is the steady state solution of v t = L 1v + g(y). The solution of this equation with the initial condition v(y, ) = can be represented by Feynman-Kac formula as Therefore L 1 1 g(y) = E y g(yt x )dt, L in the limiting backward Kolmogorov equa- and the operator tion (21) is (19). v(y, t) = E y t g(y x s )ds,

21 Example: the Lorenz 96 (L96) model Consider Ẋ k = ε (X k 1 (X k 2 X k+1 ) + X k ) + h J x ( ) Yj,k+1 Y j,k 1 J j=1 Ẏ j,k = 1 ( ) Yj+1,k (Y j+2,k Y j 1,k ) Y j,k + F y + hy X k, ε (22)

22 ACFs and PDFs (subplot) of the slow variable X k evolving under (22). Grey line: ε = 1/256; full black line: ε = 1/128; dashed black line: ε = 1/128 with time rescaled as t 2t. The near perfect match confirms that evolution of X k converges to some limiting dynamics on the O(1/ε) time-scale.

23 Coupled truncated Burgers-Hopf (TBH). I. Stable periodic orbits Ẋ 1 = 1 ε b 1X 2 Y 1 + ay 1 (R 2 (X1 2 + X2)) 2 bx 2 (α + (X1 2 + X2)), 2 Ẋ 2 = 1 ε b 2X 1 Y 1 + ax 2 (R 2 (X1 2 + X2)) 2 + bx 1 (α + (X1 2 + X2)), 2 Ẏ k = Re ik Up Uq ε b 3δ 1,k X 1 X 2, p+q+k= Ż k = Im ik Up Uq, 2 p+q+k= where U k = Y k + iz k. Truncated system: stable periodic orbit (limit cycle) (X 1 (t), X 2 (t)) = R (cos ωt, sin ωt) with frequency ω = b(α + R 2 ).

24 NB: Truncated Burgers (Majda & Timofeyev, 2) Fourier-Galerkin truncation of the inviscid Burgers-Hopf equation, u t (u2 ) x = : U k = ik 2 U p U q, k < Λ k+p+q= p, q Λ Features common with many complex systems. In particular: Display deterministic chaos. Ergodic on E = k U k 2. Scaling law for the correlation functions with t k O(k 1 ). Here: Used as a model for unresolved modes. Couple truncated Burgers with two resolved variables.

25 Ẋ 1 = 1 ε b 1X 2 Y 1 + ay 1 (R 2 (X1 2 + X2 2 )) bx 2(α + (X1 2 + X2 2 )), Ẋ 2 = 1 ε b 2X 1 Y 1 + ax 2 (R 2 (X1 2 + X2)) 2 + bx 1 (α + (X1 2 + X2)), 2 Ẏ k = Re ik Up Uq ε b 3δ 1,k X 1 X 2, p+q+k= Ż k = Im ik 2 p+q+k= U p U q, Limiting SDEs: Ẋ 1 = b 1 b 2 X 1 + N 1 X 1 X σ 1X 2 Ẇ (t) + ax 1 (R 2 (X X2 2 )) bx 2(α + (X X2 2 )), Ẋ 2 = b 1 b 2 X 2 + N 2 X 2 X σ 2X 1 Ẇ (t) + ax 2 (R 2 (X X2 2 )) + bx 1(α + X X2 2 )),

26 (a) (b) 1 1 x x Contour plots of the joint probability density for the climate variables X 1 and X 2 ; (a) deterministic system with 12 variables; (b) limit SDE. There only remains a ghost of the limit cycle.

27 3 2.5 (a) DNS (b) Reduced model PDF of x PDF of x x x (c) DNS (d) Reduced model PDF of x PDF of x x x 2 Marginal PDFs of X 1 and X 2 for the simulations of the full deterministic system with 12 variables (DNS) and the limit SDE.

28 corr of x 1 (a) t 8 (c) corr of x 2 (b) t 8 (d) cross corr of x 1, x t 8 K 2 (t) t 8 Two-point statistics for X 1 and X 2 ; solid lines - deterministic system with 12 degrees of freedom; dashed lines - limit SDE; (a), (b) correlation functions of X 1 and X 2, respectively; (c) cross-correlation function of X 1 and X 2 ; (d) normalized correlation of energy, K 2 (t) = X 2 2 (t)x2 2 () X X 2 (t)x 2 () 2

29 Coupled truncated Burgers-Hopf (TBH). II. Multiple equilibria Ẋ = 1 ε b 1Y 1 Z 1 + λ(1 αx 2 )x, Ẏ k = Re ik 2 p+q+k= U p U q + 1 ε b 2δ 1,k XZ k, Ż k = Im ik 2 p+q+k= u pu q + 1 ε b 3δ 1,k XY k, Limiting SDE: Ẋ = γx + σẇ (t) + λ(1 αx 2 )X

30 .6 (a) (b) (c).6.6 λ = 1.2 λ =.5 λ=.15 PDF of x x PDF of X for the simulations of the deterministic model (solid lines) and the limit SDE (dashed lines) in three regimes, λ = 1.2,.5,.15.

31 1 (a) 1 (b) 1 (c) corr of x.5 λ =1.2.5 λ =.5.5 λ = t (d) 1.2 (e) 1.2 (f) K(t) λ =1.2 λ = λ = t Two-point statistics of X in three regimes, λ = 1.2,.5,.15 for the deterministic equations (solid lines) and limit SDE (dashed lines) (a), (b), (c) correlation function of x; (d), (e), (f) correlation of energy, K(t), in x.

32 Generalizations Let Z t S be the sample path of a continuous-time Markov process with generator L = L + ε 1 L 1 and assume that L 1 has several ergodic components indexed by x S with equilibrium distribution µ x (z) Then, as ε there exists a limiting on S with generator L = E µx L Similar results available on diffusive time-scales. Can be applied to SDEs, Markov chains (i.e. discrete state-space), deterministic systems (periodic or chaotic), etc.

33 Consider Other situations with limiting dynamics? {Ẋt = f(x t, Y t ), Ẏ t = g(x t, Y t ), (23) and denote by ϕ(x [,t] ) the solution of the equation for Y at time t assuming that X t is known on [, t] Observe that ϕ(x [,t] ) is a functional of {X s, s [, t]}. Then X t satisfies the closed equation Ẋ t = f(x t, ϕ(x [,t] )) (24) In general, X t is not Markov! It becomes Markov when Y t is faster, or...?

34 Other possibility: Weak coupling The system Ẋ t = 1 N Ẏ n t N n=1 = g(x t, Yt n), f(x t, Y n t ), (all the Y n t coupled only via X t ) may have a limit behavior as N which is the same as the limit behavior as Ẋ t = f(x t, Y t ), when ε. Ẏ t = 1 ε g(x t, Y t ),

35 Asymptotic techniques for singularly perturbed Markov processes: R. Z. Khasminsky, On Stochastic Processes Defined by Differential Equations with a Small Parameter, Theory Prob. Applications, 11: , R. Z. Khasminsky, A Limit Theorem for the Solutions of Differential Equations with Random Right-Hand Sides, Theory Prob. Applications, 11:39 46, T. G. Kurtz, Semigroups of conditioned shifts and approximations of Markov processes, Annals of Probability, 3: , G. Papanicolaou, Some probabilistic problems and methods in singular perturbations, Rocky Mountain J. Math, 6: , M. I. Freidlin and A. D. Wentzell, Random perturbations of dynamical systems, 2nd edition, Springer-Verlag, Effective dynamics in L96: E. N. Lorenz, Predictability A problem partly solved, pp in: ECMWF Seminar Proceedings on Predictability, Reading, United Kingdom, ECMWF, C. Rödenbeck, C. Beck, H. Kantz, Dynamical systems with time scale-separation: averaging, stochastic modeling, and central limit theorems, pp in: Stochastic Climate Models, P. Imkeller, J.-S. von Storch eds., Progress in Probability 49, Birkhäuser Verlag, Basel, 21. I. Fatkullin and E. Vanden-Eijnden, A computational strategy for multiscale systems with applications to Lorenz 96 model, J. Comp. Phys. 2:65 638, 24.

36 (Coupled) TBH: A. J. Majda and I. Timofeyev, Remarkable statistical behavior for truncated Burgers- Hopf dynamics, Proc. Nat. Acad. Sci. USA, 97: , 2. A. J. Majda and I. Timofeyev, Statistical mechanics for truncations of the Burgers- Hopf equation: a model for intrinsic stochastic behavior with scaling, Milan Journal of Mathematics, 7(1):39 96, 22. A. J. Majda, I. Timofeyev, and E. Vanden-Eijnden, Systematic strategies for stochastic mode reduction in climate. J. Atmos. Sci., 6(14): , 23. A. J. Majda, I. Timofeyev, and E. Vanden-Eijnden, Stochastic models for selected slow variables in large deterministic systems, Nonlinearity, 19: , 26.