Boltzmann-Gibbs Preserving Langevin Integrators

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

2 4000 atom cluster simulation

3 Governing Equations of 4000 atom cluster

4 Governing Equations of 4000 atom cluster Consider the Hamiltonian

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

6 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

7 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.

8 Scheme

9 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,

10 c=8, T= atom cluster SVI instantaneous temperature

11 c=64, T= atom cluster SVI instantaneous temperature

12 c=512, T= atom cluster SVI instantaneous temperature

13 c=4096, T= atom cluster SVI instantaneous temperature

14 c=8, T= atom cluster LAMMPS instantaneous temperature

15 c=16, T= atom cluster LAMMPS instantaneous temperature

16 c=32, T= atom cluster LAMMPS instantaneous temperature

17 Unstable for c> atom cluster LAMMPS instantaneous temperature

18 Langevin Process

19 Langevin Process Consider the Hamiltonian

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

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

22 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.

23 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

24 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

25 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

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

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

28 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

29 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

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

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

32 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

33 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

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

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

36 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

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

38 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

39 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 energy error of a Q symplectic integrator remains bounded for exponentially long times t

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

41 { 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,

42 { 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).

43 { 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)).

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

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

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

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

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

49 Divide and Conquer

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

51 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

52 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.

53 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

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

55 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.

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

57 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

58 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)

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

60 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

61 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

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

63 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

64 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

65 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

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

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

68 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

69 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)

70 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

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

72 Cubic Oscillator c=2, T=1

73 Cubic Oscillator c=4, T=1

74 Cubic Oscillator c=8, T=1

75 Rare Event Simulation

76 For more information see:

Introduction to Hamiltonian Monte Carlo Method

Introduction to Hamiltonian Monte Carlo Method Mingwei Tang Department of Statistics University of Washington mingwt@uw.edu November 14, 2017 1 Hamiltonian System Notation: q R d : position vector, p R