Estimating Accuracy in Classical Molecular Simulation

Transcription

1 Estimating Accuracy in Classical Molecular Simulation University of Illinois Urbana-Champaign Department of Computer Science Institute for Mathematics and its Applications July 2007

2 Acknowledgements Ben Leimkuhler (Edinburgh) Brian Laird (Kansas) Ruslan Davidchack (Leicester) Nana Arizumi (Illinois) S. Bond and B. Leimkuhler Acta Numerica 16, 2007.

3 Motivation Computation of Averages 1.7 Instantaneous Temperature time

4 Motivation Computation of Averages # Trials Instantaneous Temperature

5 Motivation Convergence of Averages Temperature Inverse Stepsize (1/dt)

6 Motivation Divergence of Trajectories norm deviation time

7 Goal What is the error in an average from a MD trajectory? Error = A numerical A exact Any estimate must account for two factors: Error Statistical Error + Truncation Error Asymptotic Bound: Error C 1 1 t + C 2 t p Talk will focus on truncation error. Paper on statistical error: E. Cancès, F. Castella, P. Chartier, E. Faou, C. Le Bris, F. Legoll, G. Turinici, 04, 05.

8 System of Equations Newton s Equations: Force = mass acceleration q = p/m and ṗ = U(q) q = position, m = mass, p = momenta First Order System ż = F (z), where F : R n R n Exact Solution Map z (t) = Φ t (t 0, z 0 )

9 Ergodic Time Average: 1 t A time = lim A (z (τ)) dτ t t 0 Ensemble Average: A ensemble = Ergodicity Ω A time = A ensemble A (z) ρ (z) dz (a.e.)

10 Liouville Equation Continuity Equation for the Probability Density: Generalized Liouville Equation: ρ t + (ρ F ) = 0 or Dρ Dt + ρ F = 0 D ln ρ Dt = F

11 Example: Nosé-Hoover Nosé-Hoover Vector Field Invariant Distribution ρ exp dq dt dp dt dξ dt { 1 k B T = M 1 p = U (q) ξ µ p = p T M 1 p gk B T ( )} 1 2 pt M 1 p + U (q) + ξ2 2µ

12 Error Analysis First Order System ż = F (z) Forward Error: Is the numerical trajectory close to the exact trajectory? z t (t) z (t) C t p Backward Error: Is the numerical trajectory actually an exact trajectory, but for a different problem? Method of Modified Equations F t (z) F (z) C t p

13 Error Analysis Ergodicity: Exact trajectories are sensitive (chaotic) to perturbations in the initial conditions Large Forward Error. Statistics: Thermodynamic properties (averages) are not a function of the details of the initial conditions Small Backward Error.

14 Backward Error Analysis: Modified Equations Given a pth-order numerical method, Ψ, we can always construct a modified vector field, F t, such that the numerical method provides a qth-order approximation to the flow of the modified system. If the numerical method and vector field are time-reversible (symplectic/hamiltonian), the modified vector field will be time-reversible (symplectic/hamiltonian). Unfortunately, even if the vector field is analytic, the modified vector field does not converge as q. Fortunately, it is still useful as a truncated series.

15 Big Picture Flow Map Vector Field Ensemble Φ t Ψ t F F t ρ ρ t

16 Example: Verlet Hamiltonian Verlet Splitting H (q, p) = 1 2 pt M 1 p + U (q) p n+1/2 = p n t 2 U (qn ) q n+1 = q n + tm 1 p n+1/2 p n+1 = p n+1/2 t 2 U ( q n+1) H 1 = 1 2 pt M 1 p, H 2 = U (q)

17 Example: Verlet Strang Splitting exp ( tl) = exp Modified Equations ( exp tl [r] t Solve for L [r] t ( ) ( ) t t 2 L 2 exp ( tl 1 ) exp 2 L 2 + O [ t 3] L 1 = M 1 p q ) = exp L = L 1 + L 2 L 2 = q U (q) p ( ) ( ) t t 2 L 2 exp ( tl 1 ) exp 2 L 2 + O [ t r+1] using Baker-Campbell-Hausdorff formula

18 Example: Verlet Original Hamiltonian: Modified Hamiltonian: H 2, t (q, p) = H (q, p) + t2 12 H (q, p) = 1 2 pt M 1 p + U (q) Verlet conserves H 2, t to 4th order accuracy! ( p T M 1 U M 1 p 1 ) 2 UT M 1 U

19 Example: Generalized Leapfrog Generalized Leapfrog p n+1/2 = p n t 2 qh (q n, p n+1/2) q n+1/2 = q n + t 2 ph (q n, p n+1/2) p n+1 = p n+1/2 t 2 qh q n+1 = q n+1/2 + t 2 ph Generalized Leapfrog Modified Hamiltonian: H t = H + t2 24 (q n+1, p n+1/2) (q n+1, p n+1/2) ( 2Hqj q k H pj H pk + 2H qj p k H pj H qk H pj p k H qj H qk )

20 Liouville Equation for Modified Vector Field Modified Equations dz dt = F t (z) where F t = F + t p G Modified Liouville Equation Weighting factor D Dt ρ t = ρ t F t ω t := ρ t /ρ, assuming ρ, ρ t > 0 implies D Dt ln (ω t) = t p ( G + G ln ρ)

21 Averages Truncation Error Estimate A Num A Exact A (q, p) ρ t dγ A (q, p) ρ dγ Γ Γ A Num 1/ω t Num A/ω t Num 1/ω t Num Reweighted Averages A Exact = A/ω t Num 1/ω t Num + O [ t r ]

22 Example: Nosé-Poincaré Hamiltonian: ( 1 H (q, p, s, π s ) = s 2 s 2 pt M 1 p ) + U (q) + π2 s 2µ + g k T ln s E 0 Nosé-Poincaré Modified Hamiltonian: H t = H NP + t2 12 s ( πs µ s pt M 1 U 1 2 UT M 1 U + 1 s 2 pt M 1 U M 1 p 1 ( 1 2 µ s 2 pt M 1 p ) 2 g k T + 2 g k T π2 s µ 2 )

23 Example: Modified marginal distribution: ρ t (q, p) dp dq = 1 δ [ ] H t (q, s, p, p s ) E 0 d p dq dps ds, C s p s = 1 [ ( )] s H N HN 0 + t 2 G d p dq dp s ds. C s p s δ Change of variables, integrating ρ = 1 e N f η 0 C gk BT + h 2 p s η G(q, 1 eη, p, p s ) dp s. η=η 0 η 0 = 1 ( ) H(q, p) + p2 s g k B T 2 µ + h2 G(q, e η 0, p, p s ) HN 0, More mathematical manipulations { [ ρ = ρ c C exp t2 2p j p k U qj q k ( ) Uq 2 j 1 p 2 2 ]} j gk B T, 24k B T m j m k m j µ m j j j j,k

24 Example: Weighting Factor: { t 2 ω t exp 24 k B T Reweighted Averages: [ 2p T M 1 U (q) M 1 p U (q) T M 1 U (q) 1 µ A Exact A/ω t Num 1/ω t Num ( ) ]} 2 p T M 1 p g k B T Hybrid Monte Carlo: J. Izaguirre and S. Hampton, J. Comput. Phys. 200, E. Akhmatskaya and S. Reich, LNCSE 49, Time correlation functions: R. D. Skeel, Preprint, 2007.

25 Numerical Experiment: System: 256 Particle Gas Lennard-Jones Potential T = 1.5ɛ/k, ρ = 0.95r 3 0, t = 20r 0 m/ɛ Method: Nosé-Poincaré (Symplectic, Time-Reversible) t = 0.012r 0 m/ɛ to r0 m/ɛ Reference: Bond, Laird, and Leimkuhler J. Comput. Phys S. Bond and B. Leimkuhler Acta Numerica 16, 2007.

26 Numerical Experiment: Extended Energy Conservation:

27 Numerical Experiment: Improved Estimator 1.52 Standard Reweighted 1.51 Temperature Inverse Stepsize (1/dt)

28 Numerical Experiment: Improved Estimator Error Standard 2nd Order Reference Reweighted 4th Order Reference 10 3 Error in Temperature Inverse Stepsize (1/dt)

29 Future Directions: Further testing with more systems and averages Extensions to reduce computational cost Other ensembles and numerical methods

30 Stabilizing Hard Sphere Algorithms: 10 0 PSA CVA PCVA MCVA Deviation in Energy Time S. Bond and B. Leimkuhler, SIAM J. Sci. Comput, 2007, In press.