Capturing rare events with the heterogeneous multiscale method David Kelly Eric Vanden-Eijnden Courant Institute New York University New York NY www.dtbkelly.com October 1, 2016 SIAM MPE 16 , Philadelphia, PE David Kelly (CIMS) HMM rare events October 1, 2016 1 / 19
Fast-slow systems Fast slow SDEs: dX ε = f ( X ε , Y ε ) dt dY ε = ε − 1 g ( X ε , Y ε ) + ε − 1 / 2 σ ( X ε , Y ε ) dW dt dt where ε ≪ 1. Let Y x be ‘virtual fast process’ with frozen x : dY x = g ( x , Y x ) + σ ( x , Y x ) dW dt dt Assume that Y x has an ergodic invariant measure µ x and is sufficiently mixing. David Kelly (CIMS) HMM rare events October 1, 2016 2 / 19
Averaging The slow variables satisfy an averaging principle dX X ε → a . s . X where dt = F ( X ) � and F ( x ) = f ( x , y ) µ x ( dy ). David Kelly (CIMS) HMM rare events October 1, 2016 3 / 19
A simple metastable example Suppose µ > 0 and dX ε = Y ε − ( X ε ) 3 dt dY ε = θ ε ( µ X ε − Y ε ) dt + σ √ ε dW 3 . Symmetric double-well This has averaged equation dX dt = µ X − X potential w/equilibria at ±√ µ and saddle at origin. When ε ≪ 1, the long time behavior of X ε will be qualitatively different to the averaged system. The system exhibits hopping between wells due to fluctuations from the average. David Kelly (CIMS) HMM rare events October 1, 2016 4 / 19
The central limit theorem describes small fluctuations about the average. If we let Z ε = ε − 1 / 2 ( X ε − X ) then one can show Z ε → w Z where dZ = B 0 ( X ) Zdt + η ( X ) dV where V is a std Brownian motion and � B 0 ( x ) = ∇ x f ( x , y ) µ x ( dy ) � ∞ � ∇ y E y (˜ + f ( x , Y x ( τ ))) ∇ x b ( x , y ) µ x ( dy ) d τ 0 � ∞ E ˜ f ( x , Y x ( τ ))˜ η ( x ) η T ( x ) = f ( x , Y x (0))) T d τ 0 where ˜ f ( x , y ) = f ( x , y ) − F ( x ). David Kelly (CIMS) HMM rare events October 1, 2016 5 / 19
Suppose X ε satisfies a large deviations principle : ε → 0 ε log P ( X ε ∈ Γ) = − inf lim γ ∈ Γ S [0 , T ] ( γ ) for a set Γ of continuous paths γ : [0 , T ] → R d in the slow state space. A large deviation principle quantifies many important features of O (1) fluctuations in metastable systems. David Kelly (CIMS) HMM rare events October 1, 2016 6 / 19
For instance , suppose that D ⊂ R d is open w/ smooth boundary ∂ D , and x ∗ is an asymptotically stable equilibrium for the averaged system dX dt = F ( X ). Define the transition time τ ε = inf { t > 0 : X ε / ∈ D } . Define the quasi-potential V ( x , y ) = inf γ (0)= x ,γ ( T )= y S [0 , T ] ( γ ) inf T > 0 Then the mean first passage/exit time is given by ε → 0 ε log E τ ε = inf lim y ∈ ∂ D V ( x , y ) David Kelly (CIMS) HMM rare events October 1, 2016 7 / 19
For FS systems, Varadhan’s Lemma (reverse) tells us the following: Let u ( t , x ) = lim ε → 0 ε log E x exp( εϕ ( X ε ( t ))). If u satisfies the Hamilton-Jacobi equation ∂ t u = H ( x , ∇ u ) , u (0 , x ) = ϕ for suitable class of ϕ , then X ε satisfies an LDP with rate function � T S [0 , T ] ( γ ) = L ( γ ( s ) , ˙ γ ( s )) ds 0 where L is the Lagrangian associated with the Hamiltonian H L ( x , β ) = sup ( θ · β − H ( x , θ )) . θ Moral of the story : we can identify LDPs via the associated HJ equation. David Kelly (CIMS) HMM rare events October 1, 2016 8 / 19
Heterogeneous multi scale method for FS systems A simple numerical scheme for the slow variables x ε n ≈ X ε ( n ∆ t ) when ε ≪ 1: � ( n +1)∆ t x ε n +1 = x ε f ( x ε n , Y ε n + n ( s )) ds x ε n ∆ t Then approximate the integral by simulating the virtual fast process on mesh size δ t ≪ ∆ t � ( n +1)∆ t N − 1 � f ( x ε n , Y ε f ( x ε n , y ε n ( s )) ds ≈ n , j ) δ t x ε n ∆ t j =0 where N δ t = ∆ t and (for instance) is given by Euler-Maruyama √ y ε n , j +1 = y ε n , j + ε − 1 g ( x ε n , y ε n , j ) δ t + ε − 1 / 2 σ ( x ε n , y ε n , j ) δ t ξ n , j for j = 0 , . . . , N − 1. David Kelly (CIMS) HMM rare events October 1, 2016 9 / 19
Speeding up the method The key observation of HMM is that one does not need the virtual process Y ε x over the whole window [ n ∆ t , ( n + 1)∆ t ), but only over a fraction of it [ n ∆ t , ( n + 1 /λ )∆ t ] for some λ ≥ 1. By the ergodic theorem � ( n +1)∆ t � ( n +1 /λ )∆ t 1 n ) ≈ λ f ( x ε n , Y ε n ( s )) ds ≈ F ( x ε f ( x ε n , Y ε n ( s )) ds x ε x ε ∆ t ∆ t n ∆ t n ∆ t provided that ∆ t /ε and ∆ t / ( ελ ) are larger than the mixing time for Y x . David Kelly (CIMS) HMM rare events October 1, 2016 10 / 19
HMM summary The update x ε n �→ x ε n +1 works in two steps 1 - Micro step : Compute an approximation F n ,λ ( x ε n ) of the integral � ( n +1 /λ )∆ t λ f ( x ε n , Y ε n ( s )) ds x ε ∆ t n ∆ t by simulating the virtual fast process Y ε n over the window x ε [ n ∆ t , ( n + 1 /λ )∆ t ). Requires δ t ≪ ∆ t , δ t ≪ ε and ∆ t / ( ελ ) larger than mixing time. 2 - Macro step : x ε n +1 = x ε n + F n ,λ ( x ε n )∆ t David Kelly (CIMS) HMM rare events October 1, 2016 11 / 19
We know that HMM is consistent with the averaging principle . That is, as ε → 0 the sequence x ε n defined by HMM converges to x n +1 = x n + F ( x n )∆ t which is a consistent numerical method for the averaged equation dX dt = F ( X ). What about fluctuations? 1 - Let z ε n = ε − 1 / 2 ( x ε n − x n ). Does z ε n converge to a numerical scheme for Z as ε → 0? 2 - Let u n ,λ ( x ) = lim ε → 0 ε log E x exp( ε − 1 ϕ ( x ε n )). Is u n ,λ a numerical method for the true HJ equation? David Kelly (CIMS) HMM rare events October 1, 2016 12 / 19
HMM Fluctuations are inflated by λ As ε → 0, z ε n converges to z n , which is a numerical scheme for the SDE √ dZ λ = B 0 ( X ) Z λ dt + λη ( X ) dV . Moreover, we find that u λ, n ( x ) is a numerical method for the HJ equation ∂ t u λ = 1 λ H ( x , λ ∇ u λ ) where H is the true Hamiltonian for X ε .In particular, the quasi-potential is V λ ( x , y ) = λ − 1 V ( x , y ). It follows that mean first passage times will shrink � 1 � ελ V ( x ∗ , ∂ D ) E τ ε ≍ exp David Kelly (CIMS) HMM rare events October 1, 2016 13 / 19
Why the inflation? In the HMM approximation, with λ ∈ Z , we are essentially replacing � ( n +1 /λ )∆ t � ( n +1)∆ t f ( x , Y ε f ( x , Y ε x ( s )) ds + · · · + x ( s )) ds n ∆ t ( n +( λ − 1) /λ )∆ t with � ( n +1 /λ )∆ t � ( n +1 /λ )∆ t f ( x , Y ε f ( x , Y ε x ( s )) ds + · · · + x ( s )) ds n ∆ t n ∆ t ie. Replace sum of λ weakly correlated random variables with λ × first random variable. Clearly this inflates the variance. David Kelly (CIMS) HMM rare events October 1, 2016 14 / 19
Parallel HMM There is a simple way to fix the problem. The update x ε n �→ x ε n +1 works in two steps 1 - λ parallel micro steps : Compute an approximation F n ,λ ( x ε n ) of the integral � ( n +1 /λ )∆ t λ 1 � f ( x ε n , Y ε n , k ( s )) ds x ε ∆ t n ∆ t k =1 by simulating λ independent copies of the virtual fast processes Y ε n , k for x ε k = 1 , . . . , λ over the window [ n ∆ t , ( n + 1 /λ )∆ t ). 2 - Macro step : x ε n +1 = x ε n + F n ,λ ( x ε n )∆ t David Kelly (CIMS) HMM rare events October 1, 2016 15 / 19
Parallel HMM • Since the virtual fast processes are independent, they can be simulated in parallel. This is a kind of parallel-in-time method. • We can show that this method is in fact consistent with X ε at both the level of small fluctuations and large deviations. David Kelly (CIMS) HMM rare events October 1, 2016 16 / 19
Small fluctuations example I Suppose µ < 1 and dX ε = Y ε − X ε dt dY ε = θ ε ( µ X ε − Y ε ) dt + σ √ ε dW This has averaged equation dX dt = ( µ − 1) X . 0.9 0.8 5 6 =1 0.7 6 =5 4 6 =10 0.6 HMM PHMM 0.5 3 var 0.4 2 0.3 1 0.2 0.1 0 1 2 3 4 5 0 -6 -4 -2 0 2 4 6 λ X David Kelly (CIMS) HMM rare events October 1, 2016 17 / 19
Large deviations example Suppose µ > 0 and dX ε = Y ε − ( X ε ) 3 dt dY ε = θ ε ( µ X ε − Y ε ) dt + σ √ ε dW 3 . This has averaged equation dX dt = µ X − X 50 HMM 40 PHMM 30 LDP τ 20 10 0 1 2 3 4 5 λ Figure: Mean first passage time David Kelly (CIMS) HMM rare events October 1, 2016 18 / 19
References D. Kelly, E. Vanden-Eijnden. Capturing rare events with the heterogeneous multiscale method. arXiv (2016). All my slides are on my website (www.dtbkelly.com) Thank you ! David Kelly (CIMS) HMM rare events October 1, 2016 19 / 19
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