Boltzmann-Gibbs Preserving Langevin Integrators

Boltzmann-Gibbs Preserving Langevin Integrators Nawaf Bou-Rabee Applied and Comp. Math., Caltech INI Workshop on Markov-Chain Monte-Carlo Methods March 28, 2008

4000 atom cluster simulation

Governing Equations of 4000 atom cluster

Governing Equations of 4000 atom cluster Consider the Hamiltonian

Governing Equations of 4000 atom cluster Consider the Hamiltonian H(q, p) = 1 2 pt M 1 p + U(q)

Governing Equations of 4000 atom cluster Consider the Hamiltonian H(q, p) = 1 2 pt M 1 p + U(q) Langevin equations; C is symmetric

Governing Equations of 4000 atom cluster Consider the Hamiltonian H(q, p) = 1 2 pt M 1 p + U(q) Langevin equations; C is symmetric dq = H p dt, dp = H q dt cc H p dt + σc1/2 dw.

Scheme

Scheme BG-preserving SVI P 1/2 k = e ch p k h 2 q k+1 = q k + hp 1/2 k, p k+1 = P 1/2 k h 2 U q (q k)+ U q (q k+1). 1 e 2ch β ξ k,

4 3.5 3 2.5 c=8, T=1 2 1.5 1 0.5 0 0 2 4 6 8 10 12 14 16 18 20 4000 atom cluster SVI instantaneous temperature

4 3.5 3 2.5 c=64, T=1 2 1.5 1 0.5 0 0 2 4 6 8 10 12 14 16 18 20 4000 atom cluster SVI instantaneous temperature

4 3.5 3 2.5 c=512, T=1 2 1.5 1 0.5 0 0 2 4 6 8 10 12 14 16 18 20 4000 atom cluster SVI instantaneous temperature

4 3.5 3 2.5 c=4096, T=1 2 1.5 1 0.5 0 0 2 4 6 8 10 12 14 16 18 20 4000 atom cluster SVI instantaneous temperature

4 3.5 3 2.5 c=8, T=1 2 1.5 1 0.5 0 0 2 4 6 8 10 12 14 16 18 20 4000 atom cluster LAMMPS instantaneous temperature

4 3.5 3 2.5 c=16, T=1 2 1.5 1 0.5 0 0 2 4 6 8 10 12 14 16 18 20 4000 atom cluster LAMMPS instantaneous temperature

4 3.5 3 2.5 c=32, T=1 2 1.5 1 0.5 0 0 2 4 6 8 10 12 14 16 18 20 4000 atom cluster LAMMPS instantaneous temperature

Unstable for c>32 4000 atom cluster LAMMPS instantaneous temperature

Langevin Process

Langevin Process Consider the Hamiltonian

Langevin Process Consider the Hamiltonian H(q, p) = 1 2 pt M 1 p + U(q)

Langevin Process Consider the Hamiltonian H(q, p) = 1 2 pt M 1 p + U(q) Langevin equations; C is symmetric

Langevin Process Consider the Hamiltonian H(q, p) = 1 2 pt M 1 p + U(q) Langevin equations; C is symmetric dq = H p dt, dp = H q dt cc H p dt + σc1/2 dw.

Langevin Process Consider the Hamiltonian H(q, p) = 1 2 pt M 1 p + U(q) Langevin equations; C is symmetric dq = H p dt, dp = H q dt cc H p dt + σc1/2 dw. Hamiltonian

Langevin Process Consider the Hamiltonian H(q, p) = 1 2 pt M 1 p + U(q) Langevin equations; C is symmetric dq = H p dt, dp = H q dt cc H p dt + σc1/2 dw. Hamiltonian Dissipative

Langevin Process Consider the Hamiltonian H(q, p) = 1 2 pt M 1 p + U(q) Langevin equations; C is symmetric dq = H p dt, dp = H q dt cc H p dt + σc1/2 dw. Hamiltonian Dissipative Diffusive

Langevin Process Langevin equations dq = H p dt, dp = H q dt cc H p dt + σc1/2 dw.

Langevin Process Langevin equations dq = H p dt, dp = H q preserves BG measure: dt cc H p dt + σc1/2 dw.

Langevin Process Langevin equations dq = H p dt, dp = H q preserves BG measure: dt cc H p dt + σc1/2 dw. dµ = Z 1 π(q, p)dqdp = Z 1 exp ( βh(q, p)) dqdp Z = T Q π(q, p)dqdp

Langevin Process Langevin equations dq = H p dt, dp = H q preserves BG measure: dt cc H p dt + σc1/2 dw. dµ = Z 1 π(q, p)dqdp = Z 1 exp ( βh(q, p)) dqdp Z = T Q π(q, p)dqdp if C is definite equilibrium (q t,p t ) exponentially converges to

Splitting Technique Langevin Equations { dq = H p dt, dp = H q dt cc H p dt + σc1/2 dw.

Splitting Technique Langevin Equations { dq = H p dt, dp = H q Divide and conquer dt cc H p dt + σc1/2 dw.

Splitting Technique Langevin Equations { dq = H p dt, dp = H q Divide and conquer { dq = 0 Ornstein- Uhlenbeck dt cc H p dt + σc1/2 dw. dp = cc H p dt + σc1/2 dw

Splitting Technique Langevin Equations { dq = H p dt, dp = H q Divide and conquer { dq = 0 Ornstein- Uhlenbeck Hamiltonian dt cc H p dt + σc1/2 dw. dp = cc H p dt + σc1/2 dw { dq = H p dt dp = H q dt

Hamiltonian { dq = H p dt dp = H q dt Apply a pth-order accurate symplectic integrator

Hamiltonian { dq = H p dt dp = H q dt Apply a pth-order accurate symplectic integrator θ h : T Q T Q

Hamiltonian { dq = H p dt dp = H q dt Apply a pth-order accurate symplectic integrator θ h : T Q T Q Recall, integrator is symplectic, i.e., θhω =Ω

Hamiltonian { dq = H p dt dp = H q dt

Hamiltonian { dq = H p dt dp = H q dt Using backward error analysis H(θh N one can show that (q, p)) H(q, p) γ(q, p)h p

Hamiltonian { dq = H p dt dp = H q dt Using backward error analysis H(θh N one can show that (q, p)) H(q, p) γ(q, p)h p 0.24 0.2405 energy error of a Q 0.241 0.2415 0.242 symplectic integrator remains bounded for exponentially long times 0 20 40 60 80 100 120 140 160 180 200 t

{ Ornstein- Uhlenbeck dq = 0 dp = cc H p dt + σc1/2 dw A linear SDE

{ Ornstein- Uhlenbeck dq = 0 dp = cc H p dt + σc1/2 dw A linear SDE dp(t) = ccm 1 p(t)dt + σc 1/2 dw (t) p(0) = p,

{ Ornstein- Uhlenbeck dq = 0 dp = cc H p dt + σc1/2 dw A linear SDE dp(t) = ccm 1 p(t)dt + σc 1/2 dw (t) p(0) = p, Solution p(t) = exp( ccm 1 t)p + σ t 0 exp( ccm 1 (t s))c 1/2 dw (s).

{ Ornstein- Uhlenbeck dq = 0 dp = cc H p dt + σc1/2 dw A linear SDE dp(t) = ccm 1 p(t)dt + σc 1/2 dw (t) p(0) = p, Solution p(t) = exp( ccm 1 t)p + σ t 0 exp( ccm 1 (t s))c 1/2 dw (s). Define map ψ h :(q, p) (q, p(h)).

Ornstein-Uhlenbeck Process ψ h :(q, p) (q, p(h)).

Ornstein-Uhlenbeck Process ψ h :(q, p) (q, p(h)).

Ornstein-Uhlenbeck Process ψ h :(q, p) (q, p(h)). Properties:

Ornstein-Uhlenbeck Process ψ h :(q, p) (q, p(h)). Properties: preserves BG measure

Ornstein-Uhlenbeck Process ψ h :(q, p) (q, p(h)). Properties: preserves BG measure however, not even consistent with Langevin SDEs

Divide and Conquer

Divide and Conquer θ h : T Q T Q approximates { dq = H p dt dp = H q dt

Divide and Conquer θ h : T Q T Q approximates ψ h : T Q T Q solves { { dq = H p dt dp = H q dt dq = 0 dp = cc H p dt + σc1/2 dw

Divide and Conquer θ h : T Q T Q approximates ψ h : T Q T Q solves { { dq = H p dt dp = H q dt dq = 0 dp = cc H p dt + σc1/2 dw φ h = θ h ψ h approximates { dq = H p dt, dp = H q dt cc H p dt + σc1/2 dw.

Divide and Conquer θ h : T Q T Q approximates ψ h : T Q T Q solves { { dq = H p dt dp = H q dt dq = 0 dp = cc H p dt + σc1/2 dw φ h = θ h ψ h approximates { dq = H p dt, dp = H q dt cc H p dt + σc1/2 dw. Remark: trivial to extend to manifolds

Composite Map φ h = θ h ψ h Recall mean-squared norm Y (x) ms = ( E [ Y (x) 2]) 1/2, Y : E E

Composite Map φ h = θ h ψ h Recall mean-squared norm Y (x) ms = ( E [ Y (x) 2]) 1/2, Y : E E First-order, mean-square convergent φ k h(q, p) ϕ tk (q, p) ms Ch.

Composite Map φ h = θ h ψ h Nearly preserves BG measure.

Composite Map φ h = θ h ψ h Nearly preserves BG measure. Define deviation from BG as N h : L 2 µ(t Q) R N h (f) = T Q E [ f(φ N h (q, p)) ] dµ T Q f(q, p)dµ

Composite Map φ h = θ h ψ h Nearly preserves BG measure. Define deviation from BG as N h : L 2 µ(t Q) R N h (f) = T Q E [ f(φ N h (q, p)) ] dµ T Q f(q, p)dµ Recall a Markov chain preserves a measure means N h (f) =0, f L 2 µ(t Q)

Composite Map φ h = θ h ψ h Theorem: Forward error of composite map satisfies

Composite Map φ h = θ h ψ h Theorem: Forward error of composite map satisfies (e ) N h (f) = f(q, p) β (H((θ N h ) 1 (q,p)) H(q,p)) 1 dµ. T Q

Composite Map φ h = θ h ψ h Theorem: Forward error of composite map satisfies (e ) N h (f) = f(q, p) β (H((θ N h ) 1 (q,p)) H(q,p)) 1 T Q dµ. change in energy of a symplectic integrator

Composite Map φ h = θ h ψ h Theorem: Forward error of composite map satisfies

Composite Map φ h = θ h ψ h Theorem: Forward error of composite map satisfies (e ) N h (f) = f(q, p) β (H((θ N h ) 1 (q,p)) H(q,p)) 1 dµ. T Q

Composite Map φ h = θ h ψ h Theorem: Forward error of composite map satisfies (e ) N h (f) = f(q, p) β (H((θ N h ) 1 (q,p)) H(q,p)) 1 T Q dµ. Corollary: (by long-time energy preservation) N h (f) e βmhp 1 sup f 1

Composite Map φ h = θ h ψ h Theorem: Forward error of composite map satisfies (e ) N h (f) = f(q, p) β (H((θ N h ) 1 (q,p)) H(q,p)) 1 T Q dµ. Corollary: (by long-time energy preservation) N h (f) e βmhp 1 Corollary: (by exponential convergence to equilibrium) E Ce λeh 0 /(2h) µ µ h TV E + Ce λeh 0 /(2h) E sup f 1 e βmhp 1

Boltzmann-Gibbs Preserving Integrator Define symplectic-energy-momentum integrator as in Kane, Marsden, and Ortiz [1999]

Boltzmann-Gibbs Preserving Integrator Define symplectic-energy-momentum integrator as in Kane, Marsden, and Ortiz [1999] θ hk : T Q T Q

Boltzmann-Gibbs Preserving Integrator Define symplectic-energy-momentum integrator as in Kane, Marsden, and Ortiz [1999] θ hk : T Q T Q Composite map φ hk = ψ hk θ hk

Boltzmann-Gibbs Preserving Integrator Define symplectic-energy-momentum integrator as in Kane, Marsden, and Ortiz [1999] θ hk : T Q T Q Composite map φ hk = ψ hk θ hk Langevin integrator (i.e., mean-squared convergent)

Boltzmann-Gibbs Preserving Integrator Define symplectic-energy-momentum integrator as in Kane, Marsden, and Ortiz [1999] θ hk : T Q T Q Composite map φ hk = ψ hk θ hk Langevin integrator (i.e., mean-squared convergent) exactly preserves BG measure

Preserving Conformal Symplecticity BG- Quasi- Symplecticity BG-SVI X X X LD-SVI X CS-SVI X X LAMMPS X

Cubic Oscillator c=2, T=1

Cubic Oscillator c=4, T=1

Cubic Oscillator c=8, T=1

Rare Event Simulation

For more information see: http://arxiv.org/pdf/0712.4123 http://www.acm.caltech.edu/~nawaf/