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

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

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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

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Outline

  • Introduction
  • Model reduction : effective dynamics
  • Application : eigenvalue estimation

2 / 22

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Diffusion process

SDE on Rn dxs = −∇V(xs)ds + √ 2β−1dws

3 / 22

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Diffusion process

SDE on Rn dxs = −∇V(xs)ds + √ 2β−1dws Generator L = −∇V · ∇ + 1

β∆

3 / 22

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Diffusion process

SDE on Rn dxs = −∇V(xs)ds + √ 2β−1dws Generator L = −∇V · ∇ + 1

β∆

Invariant measure dπ = ρ(x)dx, where ρ(x) = 1 Z e−βV(x) , with Z = ∫

Rn e−βV(x)dx .

3 / 22

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Diffusion process

Hilbert space H = L2(Rn, π), Inner product ⟨f, g⟩π = ∫

Rn f(x) g(x)ρ(x) dx ,

∀f, g ∈ H , = ⇒ ⟨L f, g⟩π = ⟨f, L g⟩π.

4 / 22

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Diffusion process

Hilbert space H = L2(Rn, π), Inner product ⟨f, g⟩π = ∫

Rn f(x) g(x)ρ(x) dx ,

∀f, g ∈ H , = ⇒ ⟨L f, g⟩π = ⟨f, L g⟩π. Eigenvalues of −L : λi ∈ R with 0 = λ0 < λ1 ≤ · · · ≤ λk ≤ · · · , and −Lφi = λiφi , φ0 ≡ 1.

4 / 22

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Diffusion process

Hilbert space H = L2(Rn, π), Inner product ⟨f, g⟩π = ∫

Rn f(x) g(x)ρ(x) dx ,

∀f, g ∈ H , = ⇒ ⟨L f, g⟩π = ⟨f, L g⟩π. Eigenvalues of −L : λi ∈ R with 0 = λ0 < λ1 ≤ · · · ≤ λk ≤ · · · , and −Lφi = λiφi , φ0 ≡ 1. Question : How to estimate λ1, λ2, · · · numerically when n ≫ 1?

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Operators

Semigroup operator (Tsf)(x) = E ( f(xs) | x0 = x ) , f ∈ H , s ≥ 0 .

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Operators

Semigroup operator (Tsf)(x) = E ( f(xs) | x0 = x ) , f ∈ H , s ≥ 0 . Transfer operator (Tτu)(y) = 1 ρ(y) ∫

Rn p(x, y; τ)u(x)ρ(x)dx ,

y ∈ Rn , where p(x, ·; τ) is the p.d.f. at time τ ≥ 0.

5 / 22

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Operators

Semigroup operator (Tsf)(x) = E ( f(xs) | x0 = x ) , f ∈ H , s ≥ 0 . Transfer operator (Tτu)(y) = 1 ρ(y) ∫

Rn p(x, y; τ)u(x)ρ(x)dx ,

y ∈ Rn , where p(x, ·; τ) is the p.d.f. at time τ ≥ 0. Connections : Tτ = Tτ = e−Lτ

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Outline

  • Introduction
  • Model reduction : effective dynamics
  • Application : eigenvalue estimation

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Motivation : Butane

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Motivation : Butane

Question : How to study the dynamics of the dihedral angle ?

7 / 22

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Effective dynamics

dxs = −∇V(xs)ds + √ 2β−1dws Reaction coordinate ξ : Rn → Rm

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Effective dynamics

dxs = −∇V(xs)ds + √ 2β−1dws Reaction coordinate ξ : Rn → Rm z ∈ Rm, f ∈ H, consider Ωz = {x ∈ Rn | ξ(x) = z},

8 / 22

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Effective dynamics

dxs = −∇V(xs)ds + √ 2β−1dws Reaction coordinate ξ : Rn → Rm z ∈ Rm, f ∈ H, consider Ωz = {x ∈ Rn | ξ(x) = z}, Pf(z) = ∫

Ωz

f(x)dµz(x) = Eπ(f(x) | ξ(x) = z) , = 1 Q(z) ∫

Rn ρ(x)f(x)δ

( ξ(x) − z ) dx , Q(z) = ∫

Rn ρ(x)δ

( ξ(x) − z ) dx , ∫

Rm Q(z)dz = 1

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Effective dynamics

Apply Ito’s formula to ξ(xs) = ⇒ dξl(xs) =Lξl(xs)ds + √ 2β−1

n

∑

i=1

∂ξl ∂xi (xs) dwi

s ,

1 ≤ l ≤ m .

9 / 22

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Effective dynamics

Apply Ito’s formula to ξ(xs) = ⇒ dξl(xs) =Lξl(xs)ds + √ 2β−1

n

∑

i=1

∂ξl ∂xi (xs) dwi

s ,

1 ≤ l ≤ m . This motivates the effective dynamics1 dzs = b(zs) ds + √ 2β−1 σ(zs) dws , zs ∈ Rm , with

  • bl(z) =P(Lξl) ,
  • all′(z) =(

σ σT)ll′(z) = P (

n

∑

i=1

∂ξl ∂xi ∂ξl′ ∂xi ) .

  • 1. Legoll and Lelièvre, Nonlinearity, 2010 .

9 / 22

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Effective dynamics

Alternative expressions :

  • bl(z) = lim

s→0+ E

(ξl(xs) − zl s

  • x0 ∼ µz

) ,

  • all′(z) =β

2 lim

s→0+ E

((ξl(xs) − zl)(ξl′(xs) − zl′) s

  • x0 ∼ µz

) .

10 / 22

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Effective dynamics

Alternative expressions :

  • bl(z) = lim

s→0+ E

(ξl(xs) − zl s

  • x0 ∼ µz

) ,

  • all′(z) =β

2 lim

s→0+ E

((ξl(xs) − zl)(ξl′(xs) − zl′) s

  • x0 ∼ µz

) . Infinitesimal generator of zs is

  • L =

m

∑

l=1

  • bl

∂ ∂zl + 1 β

m

∑

l,l′=1

  • all′

∂2 ∂zl∂zl′ , We have f = f ◦ ξ = ⇒ PLf = ( L f )

  • ξ .

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Effective dynamics

Unique invariant measure of zs is dν = Q(z) dz.

  • L is self-adjoint on space
  • H = L2(Rm, ν).

Let 0 = λ0 < λ1 ≤ λ2 ≤ · · · . be the eigenvalues of − L and φi ∈ H be orthonormal eigenfunctions.

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Effective dynamics — Time scales

Proposition 1

For i ≥ 0, we have λi ≤ λi ≤ λi + 1 β ∫

Rn |∇(φi −

φi ◦ ξ)(x)|2ρ(x)dx.

12 / 22

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Effective dynamics — Time scales

Proposition 1

For i ≥ 0, we have λi ≤ λi ≤ λi + 1 β ∫

Rn |∇(φi −

φi ◦ ξ)(x)|2ρ(x)dx. Especially, let ξ(x) = ( φ1(x), φ2(x), · · · , φm(x) ) ∈ Rm , Then λi = λi , 0 ≤ i ≤ m .

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Effective dynamics — Time scales

Proposition 1

For i ≥ 0, we have λi ≤ λi ≤ λi + 1 β ∫

Rn |∇(φi −

φi ◦ ξ)(x)|2ρ(x)dx. Especially, let ξ(x) = ( φ1(x), φ2(x), · · · , φm(x) ) ∈ Rm , Then λi = λi , 0 ≤ i ≤ m . = ⇒ Optimal reaction coordinate function.

12 / 22

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Effective dynamics — Algorithms

Key issue : estimate coefficients in dzs = b(zs) ds + √ 2β−1 σ(zs) dws .

13 / 22

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Effective dynamics — Algorithms

Key issue : estimate coefficients in dzs = b(zs) ds + √ 2β−1 σ(zs) dws .

  • 1. HMM-like, Blue Moon, constrained dynamics, ...
  • bl(z) =P(Lξl) ,
  • all′(z) =(

σ σT)ll′(z) = P (

n

∑

i=1

∂ξl ∂xi ∂ξl′ ∂xi ) .

  • 2. Equation-free, ...
  • bl(z) = lim

s→0+ E

(ξl(xs) − zl s

  • x0 ∼ µz

) ,

  • all′(z) =β

2 lim

s→0+ E

((ξl(xs) − zl)(ξl′(xs) − zl′) s

  • x0 ∼ µz

) .

  • 3. Extended system, TAMD, ...

13 / 22

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Outline

  • Introduction
  • Model reduction : effective dynamics
  • Application : eigenvalue estimation

14 / 22

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Eigenvalue estimation 1

Suppose N basis functions ψl = ψl ◦ ξ, 1 ≤ l ≤ N, are given. We want to apply Galerkin method to solve the eigenvalue problem −Lf = λf , (1) in the subspace span{ψ1, ψ2, · · · , ψN}.

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Eigenvalue estimation 1

Suppose N basis functions ψl = ψl ◦ ξ, 1 ≤ l ≤ N, are given. We want to apply Galerkin method to solve the eigenvalue problem −Lf = λf , (1) in the subspace span{ψ1, ψ2, · · · , ψN}. Write f =

N

∑

i=1

ωiψi, then (1) = ⇒ CX = λSX, with Cll′ = ⟨−Lψl, ψl′⟩π, Sll′ = ⟨ψl, ψl′⟩π , 1 ≤ l, l′ ≤ N ,

1. Noé and Nüske, Multiscale Model. Simul. 2013. Pérez-Hernández, Paul, et al. J. Chem. Phys. 2013.

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Eigenvalue estimation

Using ψl = ψl ◦ ξ and PLψl = ( L ψl )

  • ξ , we have

Sll′ =⟨ψl, ψl′⟩π = ⟨ ψl, ψl′⟩ν , Cll′ =⟨−Lψl, ψl′⟩π = ⟨− L ψl, ψl′⟩ν .

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Eigenvalue estimation

Using ψl = ψl ◦ ξ and PLψl = ( L ψl )

  • ξ , we have

Sll′ =⟨ψl, ψl′⟩π = ⟨ ψl, ψl′⟩ν , Cll′ =⟨−Lψl, ψl′⟩π = ⟨− L ψl, ψl′⟩ν . Since ν is the invariant measure of zs, Sll′ = lim

T→+∞

1 T ∫ T

  • ψl(zs)

ψl′(zs)ds ≈ 1 M − M0

M

∑

i=M0+1

  • ψl(zi∆t)

ψl′(zi∆t) .

16 / 22

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Eigenvalue estimation

Similarly, let τ be a small parameter, Cll′ = − lim

s→0

lim

T→+∞

1 T ∫ T

  • ψl′(zt+s) −

ψl′(zt) s

  • ψl(zt) dt

≈ − 1 2(M − M0)τ

M

∑

i=M0+1

[

  • ψl(zi∆t+τ)

ψl′(zi∆t) + ψl(zi∆t) ψl′(zi∆t+τ) − 2 ψl(zi∆t) ψl′(zi∆t) ] .

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Eigenvalue estimation

Similarly, let τ be a small parameter, Cll′ = − lim

s→0

lim

T→+∞

1 T ∫ T

  • ψl′(zt+s) −

ψl′(zt) s

  • ψl(zt) dt

≈ − 1 2(M − M0)τ

M

∑

i=M0+1

[

  • ψl(zi∆t+τ)

ψl′(zi∆t) + ψl(zi∆t) ψl′(zi∆t+τ) − 2 ψl(zi∆t) ψl′(zi∆t) ] . = ⇒ We can approximate eigenvalues of CX = λSX, and therefore −Lf = λf, by simulating the effective dynamics.

17 / 22

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Eigenvalue estimation — A simple 2D example

Potential V(x) = V1(θ) + 1

ϵ V2(r, θ),

V1(θ) =      [ 1 − 9

π2

( θ − π

3

)2]2 θ > π

3 , 3 5 − 2 5 cos 3θ

− π

3 < θ < π 3

[ 1 − 9

π2

( θ + π

3

)2]2 θ < − π

3 ,

V2(r, θ) = ( r 2 − 1 − 1 1 + 4rθ2 )2 . Dynamics dxs = −∇V(xs)ds + √ 2β−1dws. β = 4.0, ϵ = 0.05, ξ(x) = θ(x) ∈ [−π, π] . 9 Gaussian basis functions. τ = 20∆t.

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Eigenvalue estimation — A simple 2D example

−2 −1 1 2 x1 −2 −1 1 2 x2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

(a) Potential V

−π −2π

3

−π

3 π 3 2π 3

π

−10 −5 5 10

  • b

(b) b

−π −2π

3

−π

3 π 3 2π 3

π

0.4 0.6 0.8 1.0 1.2 1.4

  • σ

(c) σ

500 1000 1500 2000 2500 3000

t −π −2π

3

−π

3 π 3 2π 3

π θ

(d) trajectory

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Eigenvalue estimation — A simple 2D example

L 0.0000 0.0102 0.0438 1.4579

  • L

0.0000 0.0123 0.0430 2.0676 Eff-MC 2.02 × 10−7 0.0154 0.0444 2.3409

Table: First 4 eigenvalues.

−2 −1 1 2 −2 −1 1 2

ϕ D

−2 −1 1 2

ϕ D

1

−2 −1 1 2

ϕ D

2

−2 −1 1 2

ϕ D

3

−0.016 −0.012 −0.008 −0.004 0.000 0.004 0.008 0.012 0.016

Figure: First 4 eigenfunctions of operator e− βV

2 Le βV 2 .

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Conclusions

Summary :

  • 1. Introduction
  • 2. Model reduction : effective dynamics
  • 3. Application : eigenvalue estimation

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Conclusions

Summary :

  • 1. Introduction
  • 2. Model reduction : effective dynamics
  • 3. Application : eigenvalue estimation

Related topics & future work :

  • 1. How to choose reaction coordinate, basis functions
  • 2. Langevin dynamics
  • 3. Algorithms for simulating effective dynamics.

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Thank you !

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