Model reduction of diffusion process along reaction coordinate and - PowerPoint PPT Presentation

Model reduction of diffusion process along reaction coordinate and related issues Wei Zhang Freie Universität Berlin joint work with Carsten Hartmann and Christof Schütte Stochastic Dynamics Out of Equilibrium, Apr. 20, IHP , Paris 1 / 22

Outline • Introduction • Model reduction : effective dynamics • Application : eigenvalue estimation 2 / 22

Diffusion process SDE on R n √ 2 β − 1 dw s dx s = −∇ V ( x s ) ds + 3 / 22

Diffusion process SDE on R n √ 2 β − 1 dw s dx s = −∇ V ( x s ) ds + L = −∇ V · ∇ + 1 Generator β ∆ 3 / 22

Diffusion process SDE on R n √ 2 β − 1 dw s dx s = −∇ V ( x s ) ds + L = −∇ V · ∇ + 1 Generator β ∆ Invariant measure d π = ρ ( x ) dx , where ∫ ρ ( x ) = 1 Z e − β V ( x ) , R n e − β V ( x ) dx . with Z = 3 / 22

Diffusion process H = L 2 ( R n , π ) , Hilbert space ∫ ⟨ f , g ⟩ π = R n f ( x ) g ( x ) ρ ( x ) dx , ∀ f , g ∈ H , Inner product = ⇒ ⟨L f , g ⟩ π = ⟨ f , L g ⟩ π . 4 / 22

Diffusion process H = L 2 ( R n , π ) , Hilbert space ∫ ⟨ f , g ⟩ π = R n f ( x ) g ( x ) ρ ( x ) dx , ∀ f , g ∈ H , Inner product = ⇒ ⟨L f , g ⟩ π = ⟨ f , L g ⟩ π . Eigenvalues of −L : λ i ∈ R with 0 = λ 0 < λ 1 ≤ · · · ≤ λ k ≤ · · · , and −L φ i = λ i φ i , φ 0 ≡ 1. 4 / 22

Diffusion process H = L 2 ( R n , π ) , Hilbert space ∫ ⟨ f , g ⟩ π = R n f ( x ) g ( x ) ρ ( x ) dx , ∀ f , g ∈ H , Inner product = ⇒ ⟨L f , g ⟩ π = ⟨ f , L g ⟩ π . Eigenvalues of −L : λ i ∈ R with 0 = λ 0 < λ 1 ≤ · · · ≤ λ k ≤ · · · , and −L φ i = λ i φ i , φ 0 ≡ 1. Question : How to estimate λ 1 , λ 2 , · · · numerically when n ≫ 1? 4 / 22

Operators Semigroup operator ( ) ( T s f )( x ) = E f ( x s ) | x 0 = x , f ∈ H , s ≥ 0 . 5 / 22

Operators Semigroup operator ( ) ( T s f )( x ) = E f ( x s ) | x 0 = x , f ∈ H , s ≥ 0 . Transfer operator ∫ 1 y ∈ R n , ( T τ u )( y ) = R n p ( x , y ; τ ) u ( x ) ρ ( x ) dx , ρ ( y ) where p ( x , · ; τ ) is the p.d.f. at time τ ≥ 0. 5 / 22

Operators Semigroup operator ( ) ( T s f )( x ) = E f ( x s ) | x 0 = x , f ∈ H , s ≥ 0 . Transfer operator ∫ 1 y ∈ R n , ( T τ u )( y ) = R n p ( x , y ; τ ) u ( x ) ρ ( x ) dx , ρ ( y ) where p ( x , · ; τ ) is the p.d.f. at time τ ≥ 0. Connections : T τ = T τ = e −L τ 5 / 22

Outline • Introduction • Model reduction : effective dynamics • Application : eigenvalue estimation 6 / 22

Motivation : Butane 7 / 22

Motivation : Butane Question : How to study the dynamics of the dihedral angle ? 7 / 22

Effective dynamics √ dx s = −∇ V ( x s ) ds + 2 β − 1 dw s ξ : R n → R m Reaction coordinate 8 / 22

Effective dynamics √ dx s = −∇ V ( x s ) ds + 2 β − 1 dw s ξ : R n → R m Reaction coordinate Ω z = { x ∈ R n | ξ ( x ) = z } , z ∈ R m , f ∈ H , consider 8 / 22

Effective dynamics √ dx s = −∇ V ( x s ) ds + 2 β − 1 dw s ξ : R n → R m Reaction coordinate Ω z = { x ∈ R n | ξ ( x ) = z } , z ∈ R m , f ∈ H , consider ∫ P f ( z ) = f ( x ) d µ z ( x ) = E π ( f ( x ) | ξ ( x ) = z ) , Ω z ∫ ( ) 1 = R n ρ ( x ) f ( x ) δ ξ ( x ) − z dx , Q ( z ) ∫ ∫ ( ) Q ( z ) = R n ρ ( x ) δ ξ ( x ) − z dx , R m Q ( z ) dz = 1 8 / 22

Effective dynamics Apply Ito’s formula to ξ ( x s ) = ⇒ n √ ∑ ∂ξ l ( x s ) dw i 2 β − 1 d ξ l ( x s ) = L ξ l ( x s ) ds + s , 1 ≤ l ≤ m . ∂ x i i = 1 9 / 22

Effective dynamics Apply Ito’s formula to ξ ( x s ) = ⇒ n √ ∑ ∂ξ l ( x s ) dw i 2 β − 1 d ξ l ( x s ) = L ξ l ( x s ) ds + s , 1 ≤ l ≤ m . ∂ x i i = 1 This motivates the effective dynamics 1 √ z s ∈ R m , dz s = � 2 β − 1 � b ( z s ) ds + σ ( z s ) dw s , � b l ( z ) = P ( L ξ l ) , ( n ) ∑ with ∂ξ l ∂ξ l ′ σ T ) ll ′ ( z ) = P � a ll ′ ( z ) =( � σ � . ∂ x i ∂ x i i = 1 1. Legoll and Lelièvre, Nonlinearity, 2010 . 9 / 22

Effective dynamics Alternative expressions : � ( ξ l ( x s ) − z l ) � � b l ( z ) = lim s → 0 + E � x 0 ∼ µ z , s � ( ( ξ l ( x s ) − z l )( ξ l ′ ( x s ) − z l ′ ) ) a ll ′ ( z ) = β � � 2 lim s → 0 + E � x 0 ∼ µ z . s 10 / 22

Effective dynamics Alternative expressions : � ( ξ l ( x s ) − z l ) � � b l ( z ) = lim s → 0 + E � x 0 ∼ µ z , s � ( ( ξ l ( x s ) − z l )( ξ l ′ ( x s ) − z l ′ ) ) a ll ′ ( z ) = β � � 2 lim s → 0 + E � x 0 ∼ µ z . s Infinitesimal generator of z s is m m ∑ ∑ ∂ 2 ∂ + 1 � � L = � b l a ll ′ ∂ z l ∂ z l ′ , ∂ z l β l = 1 l , l ′ = 1 We have ( � ) f = � L � f ◦ ξ = ⇒ P L f = f ◦ ξ . 10 / 22

Effective dynamics Unique invariant measure of z s is d ν = Q ( z ) dz . � � H = L 2 ( R m , ν ) . L is self-adjoint on space Let 0 = � λ 0 < � λ 1 ≤ � λ 2 ≤ · · · . be the eigenvalues of − � φ i ∈ � L and � H be orthonormal eigenfunctions. 11 / 22

Effective dynamics — Time scales Proposition 1 For i ≥ 0 , we have ∫ λ i ≤ λ i + 1 λ i ≤ � φ i ◦ ξ )( x ) | 2 ρ ( x ) dx . R n |∇ ( φ i − � β 12 / 22

Effective dynamics — Time scales Proposition 1 For i ≥ 0 , we have ∫ λ i ≤ λ i + 1 λ i ≤ � φ i ◦ ξ )( x ) | 2 ρ ( x ) dx . R n |∇ ( φ i − � β Especially, let ( ) ∈ R m , ξ ( x ) = φ 1 ( x ) , φ 2 ( x ) , · · · , φ m ( x ) Then � λ i = λ i , 0 ≤ i ≤ m . 12 / 22

Effective dynamics — Time scales Proposition 1 For i ≥ 0 , we have ∫ λ i ≤ λ i + 1 λ i ≤ � φ i ◦ ξ )( x ) | 2 ρ ( x ) dx . R n |∇ ( φ i − � β Especially, let ( ) ∈ R m , ξ ( x ) = φ 1 ( x ) , φ 2 ( x ) , · · · , φ m ( x ) Then � λ i = λ i , 0 ≤ i ≤ m . = ⇒ Optimal reaction coordinate function. 12 / 22

Effective dynamics — Algorithms Key issue : estimate coefficients in √ dz s = � 2 β − 1 � b ( z s ) ds + σ ( z s ) dw s . 13 / 22

Effective dynamics — Algorithms Key issue : estimate coefficients in √ dz s = � 2 β − 1 � b ( z s ) ds + σ ( z s ) dw s . 1. HMM-like, Blue Moon, constrained dynamics, ... � b l ( z ) = P ( L ξ l ) , ( n ) ∑ ∂ξ l ∂ξ l ′ σ T ) ll ′ ( z ) = P � a ll ′ ( z ) =( � σ � . ∂ x i ∂ x i i = 1 2. Equation-free, ... � ( ξ l ( x s ) − z l ) � � b l ( z ) = lim � x 0 ∼ µ z s → 0 + E , s � ( ( ξ l ( x s ) − z l )( ξ l ′ ( x s ) − z l ′ ) ) a ll ′ ( z ) = β � � � x 0 ∼ µ z 2 lim s → 0 + E . s 3. Extended system, TAMD, ... 13 / 22

Outline • Introduction • Model reduction : effective dynamics • Application : eigenvalue estimation 14 / 22

Eigenvalue estimation 1 Suppose N basis functions ψ l = � ψ l ◦ ξ , 1 ≤ l ≤ N , are given. We want to apply Galerkin method to solve the eigenvalue problem −L f = λ f , (1) in the subspace span { ψ 1 , ψ 2 , · · · , ψ N } . 15 / 22

Eigenvalue estimation 1 Suppose N basis functions ψ l = � ψ l ◦ ξ , 1 ≤ l ≤ N , are given. We want to apply Galerkin method to solve the eigenvalue problem −L f = λ f , (1) in the subspace span { ψ 1 , ψ 2 , · · · , ψ N } . ∑ N Write f = ω i ψ i , i = 1 then (1) = ⇒ CX = λ SX , with 1 ≤ l , l ′ ≤ N , C ll ′ = ⟨−L ψ l , ψ l ′ ⟩ π , S ll ′ = ⟨ ψ l , ψ l ′ ⟩ π , 1. Noé and Nüske, Multiscale Model. Simul. 2013. Pérez-Hernández, Paul, et al. J. Chem. Phys. 2013. 15 / 22

Eigenvalue estimation ( � ) Using ψ l = � L � ψ l ◦ ξ and P L ψ l = ◦ ξ , we have ψ l S ll ′ = ⟨ ψ l , ψ l ′ ⟩ π = ⟨ � ψ l , � ψ l ′ ⟩ ν , C ll ′ = ⟨−L ψ l , ψ l ′ ⟩ π = ⟨− � L � ψ l , � ψ l ′ ⟩ ν . 16 / 22

Eigenvalue estimation ( � ) Using ψ l = � L � ψ l ◦ ξ and P L ψ l = ◦ ξ , we have ψ l S ll ′ = ⟨ ψ l , ψ l ′ ⟩ π = ⟨ � ψ l , � ψ l ′ ⟩ ν , C ll ′ = ⟨−L ψ l , ψ l ′ ⟩ π = ⟨− � L � ψ l , � ψ l ′ ⟩ ν . Since ν is the invariant measure of z s , ∫ T 1 ψ l ( z s ) � � S ll ′ = lim ψ l ′ ( z s ) ds T T → + ∞ 0 M ∑ 1 ψ l ( z i ∆ t ) � � ≈ ψ l ′ ( z i ∆ t ) . M − M 0 i = M 0 + 1 16 / 22

Eigenvalue estimation Similarly, let τ be a small parameter, ∫ T ψ l ′ ( z t + s ) − � � 1 ψ l ′ ( z t ) � C ll ′ = − lim ψ l ( z t ) dt lim T s s → 0 T → + ∞ 0 M [ ∑ 1 ψ l ( z i ∆ t + τ ) � � ≈ − ψ l ′ ( z i ∆ t ) 2 ( M − M 0 ) τ i = M 0 + 1 ] + � ψ l ( z i ∆ t ) � ψ l ′ ( z i ∆ t + τ ) − 2 � ψ l ( z i ∆ t ) � ψ l ′ ( z i ∆ t ) . 17 / 22

Eigenvalue estimation Similarly, let τ be a small parameter, ∫ T ψ l ′ ( z t + s ) − � � 1 ψ l ′ ( z t ) � C ll ′ = − lim ψ l ( z t ) dt lim T s s → 0 T → + ∞ 0 M [ ∑ 1 ψ l ( z i ∆ t + τ ) � � ≈ − ψ l ′ ( z i ∆ t ) 2 ( M − M 0 ) τ i = M 0 + 1 ] + � ψ l ( z i ∆ t ) � ψ l ′ ( z i ∆ t + τ ) − 2 � ψ l ( z i ∆ t ) � ψ l ′ ( z i ∆ t ) . = ⇒ We can approximate eigenvalues of CX = λ SX , and therefore −L f = λ f , by simulating the effective dynamics. 17 / 22