Numerical methods in molecular dynamics and multiscale problems

Transcription

1 Numerical methods in molecular dynamics and multiscale problems Two examples T. Lelièvre CERMICS - Ecole des Ponts ParisTech & MicMac project-team - INRIA Horizon Maths December 2012

2 Introduction The aim of molecular dynamics simulations is to understand the relationships between the macroscopic properties of a molecular system and its atomistic features. In particular, one would like to to evaluate numerically macroscopic quantities from models at the microscopic scale. Some examples of macroscopic quantities: (i) Thermodynamics quantities (averages of some observables over configurations): stress, heat capacity, free energy,... E µ (ϕ(x)). (ii) Dynamical quantities (averages of some observables over trajectories): transition rates, transition path... E(F((X t ) t 0 )).

3 Introduction A real example: Diffusion of adatoms on a surface. (A. Voter). Los Alamos

4 Introduction Many applications in various fields: biology, physics, chemistry, materials science. Molecular dynamics computations consume today a lot of CPU time. A molecular dynamics model amounts essentially in choosing a potential V which associates to a configuration (x 1,...,x N ) = x R 3N an energy V(x 1,...,x N ). (N is large!). In the canonical (NVT) ensemble, configurations are distributed according to the Boltzmann-Gibbs probability measure: dµ(x) = Z 1 exp( βv(x)) dx, where Z = exp( βv(x)) dx is the partition function and β = (k B T) 1 is proportional to the inverse of the temperature.

5 Introduction To sample µ, ergodic dynamics wrt to µ are used. A typical example is the over-damped Langevin (or gradient) dynamics: dx t = V(X t ) dt + 2β 1 dw t. This is a simple gradient descent dynamics, perturbed by noise. To compute dynamical quantities, these are also typically the dynamics of interest. Thus, E µ (ϕ(x)) 1 T T 0 ϕ(x t ) dt and E(F((X t ) t 0 )) 1 N N F((X m t ) t 0 ). m=1

6 Introduction Difficulty: In practice, X t is a metastable process, so that the convergence to equilibrium is very slow. Timescale of the microscopic process: s. TImescale of the macroscopic process: from 10 6 s to 10 2 s. A 2d schematic picture: Xt 1 is a slow variable (a metastable dof) of the system. x 2 V(x 1, x 2 ) X 1 t x 1 t

7 Introduction We will present some algorithms using parallel architectures to circumvent the central numerical difficulty: metastability For computing thermodynamics quantities, there is a clear classification of available methods, and the difficulties are now well understood (in particular for free energy computations, see for example the review [TL, Rousset, Stoltz, 2010]). On the opposite, computing efficiently dynamical quantities remains a challenge.

8 Introduction Outline of the talk: 1. The Parallel Replica dynamics: This is one instance of an algorithm to generate efficiently metastable dynamics. 2. A multiscale parareal algorithm: How to use a reduced description to accelerate dynamical microscopic simulations? Remark: There are many other techniques: splitting techniques (FFS, TIS), hyperdynamics and temperature accelerated dynamics [Voter, Fichthorn], the string method [E, Ren, Vanden-Eijnden], transition path sampling methods [Chandler, Bolhuis, Dellago], milestoning techniques [Elber, Schuette, Vanden-Eijnden], etc...

9 The Parallel Replica Algorithm

10 The Parallel Replica Algorithm The Parallel Replica Algorithm, proposed by A.F. Voter in 1998, is a method to get efficiently a "coarse-grained projection" of a dynamics. Let us consider again the overdamped Langevin dyanmics: dx t = V(X t ) dt + 2β 1 dw t and let assume that we are given a smooth mapping S : R d N which to a configuration in R d associates a state number. Think of a numbering of the wells of the potential V. The aim of the parallel replica dynamics is to generate very efficiently a trajectory (S t ) t 0 which has (almost) the same law as (S(X t )) t 0.

11 The Parallel Replica Algorithm Decorrelation step: run the dynamics on a reference walker...

12 The Parallel Replica Algorithm Decorrelation step:... until it remains trapped for a time τ corr.

13 The Parallel Replica Algorithm Dephasing step: generate new initial conditions in the state.

14 The Parallel Replica Algorithm Dephasing step: generate new initial conditions in the state.

15 The Parallel Replica Algorithm Dephasing step: generate new initial conditions in the state.

16 The Parallel Replica Algorithm Dephasing step: generate new initial conditions in the state.

17 The Parallel Replica Algorithm Dephasing step: generate new initial conditions in the state.

18 The Parallel Replica Algorithm Dephasing step: generate new initial conditions in the state.

19 The Parallel Replica Algorithm Dephasing step: generate new initial conditions in the state.

20 The Parallel Replica Algorithm Parallel step: run independent trajectories in parallel...

21 The Parallel Replica Algorithm Parallel step:... and detect the first transition event.

22 The Parallel Replica Algorithm Parallel step: update the time clock: T simu = T simu + NT.

23 The Parallel Replica Algorithm A new decorrelation step starts...

24 The Parallel Replica Algorithm New decorrelation step

25 The Parallel Replica Algorithm In summary, three steps: Decorrelation step: does the reference walker remain trapped in a set? Dephasing step: prepare many initial conditions in this trapping set. Parallel step: detect the first escaping event. Some numerical simulations...

26 The Parallel Replica Algorithm Question: Why does this work? Why is there no error introduced by the parallel step? This is related to the very general question: why does the state to state dynamics is (almost) Markovian?

27 The quasi-stationary distribution Answer: there exists a probability distribution ν (the QSD) attached to each state, such that: If the process X t remains in the well for a sufficiently long time, then X t is distributed according to ν. If the process starts from ν, then, (i) the exit time is exponentially distributed and (ii) independent from the exit point. This justifies the algorithm.

28 The quasi-stationary distribution Error estimate: Let T W denote the exit time from W for the stochastic process X t. Then, for τ corr C λ 2 λ 1, sup E(f(T W t,x TW ) T W τ corr ) E ν (f(t W,X TW )) f, f L 1 C exp( (λ 2 λ 1 )τ corr ), where λ 2 < λ 1 < 0 are the two first eigenvalues of L = div ( V +β 1 ) with absorbing boundary conditions on W.

29 The Parallel Replica Algorithm: conclusions In summary, We know what τ corr should be, and thus: (i) we can try to estimate it and (ii) we can apply this algorithm to other fields. The algorithm is efficient if the typical time it takes to leave the well is much smaller than the typical time it takes to reach the local equilibrium (QSD). (This is a spectral gap requirement.)

30 A multiscale parareal algorithm: the main idea We would like to describe an algorithm which consists in coupling a macroscopic model with a microscopic model in order to efficiently generate the microscopic dynamics. The macroscopic model is the predictor, and the microscopic model is the corrector. Question 1: obtaining a reduced description. Question 2: using it in a multiscale parareal algorithm.

31 Effective dynamics Recall the original dynamics dx t = V(X t ) dt + 2β 1 dw t. We are given a smooth one dimensional function ξ : R d R. Problem: propose a Markovian dynamics (say on z t R) that approximates the dynamics (ξ(x t )) t 0.

32 Construction of the effective dynamics By Itô, one has dξ(x t ) = ( V ξ+β 1 ξ)(x t ) dt+ 2β 1 ξ(x t ) ξ(x t) ξ(x t ) dw t First attempt: d z t = b(t, z t ) dt + 2β 1 σ(t, z t ) db t with ( ) b(t, z) = E ( V ξ +β 1 ξ)(x t ) ξ(x t ) = z ( ) σ 2 (t, z) = E ξ 2 (X t ) ξ(x t ) = z. Then, for all time t 0, L(ξ(X t )) = L( z t )! But b and σ are untractable numerically...

33 By Itô, one has Construction of the effective dynamics dξ(x t ) = ( V ξ+β 1 ξ)(x t ) dt+ 2β 1 ξ(x t ) ξ(x t) ξ(x t ) dw t The effective dynamics: dz t = b(z t ) dt + 2β 1 σ(z t ) db t with ) b(z) = E µ (( V ξ +β 1 ξ)(x) ξ(x) = z ) σ 2 (z) = E µ ( ξ 2 (X) ξ(x) = z. Related approaches: Mori-Zwanzig and projection operator formalism [E/Vanden-Eijnden,...], asymptotic approaches [Papanicolaou, Freidlin, Pavliotis/Stuart,...].

34 Typically ρ R. The proof [Legoll, TL] is PDE-based. It is inspired by a decomposition of the entropy proposed in [Grunewald/Otto/Villani/Westdickenberg], and entropy estimates. Error analysis: time marginals The effective dynamics is ergodic (detailed balanced) wrt ξ µ. Moreover Under the assumptions ( ξ = Cte for simplicity): (H1) The conditional probability measures µ( ξ(x) = z) satisfy a Logarithmic Sobolev Inequality with constant ρ, (H2) Bounded coupling assumption: V V V L κ, (H3) The probability measure µ satisfies a Logarithmic Sobolev Inequality with constant R, Then, C > 0, t 0, L(ξ(X t )) L(z t ) TV ( ) κ ( ) C min H(L(X 0 ) µ) H(L(X t ) µ), exp( β 1 Rt). ρ

35 A multiscale parareal algorithm The basic algorithm is very much inspired by the parareal algorithm [Lions, Maday Turinici]. Let us present it in a very simple setting: Consider the fine dynamics: ẋ = αx + p T y, ẏ = 1 (qx Ay). ǫ The variable u = (x, y) contains a slow variable x R and a fast variable y R d 1. Associated to that dynamics, we have an expensive fine propagator F t over the time range t. Under appropriate assumption, a natural coarse dynamics (ǫ 0) on x is: Ẋ = (α+p T A 1 q)x. Associated to that dynamics, we have a cheap coarse propagator G t over the time range t.

36 The algorithm Let us now present the algorithm Let u(0) = u 0 be the initial condition. 1. Initialization: a) Compute {X0 n} 0 n N sequentially by using the coarse propagator: X 0 0 = R(u 0 ), X n+1 0 = G t (X n 0). b) Lift the macroscopic approximation to the microscopic level: u 0 0 = u 0 and, for all 1 n N, u n 0 = L(X n 0).

37 2. Assume that, for some k 0, the sequences { uk n } { } 0 n N and X n k are known. To compute these sequences at the 0 n N iteration k + 1, a) For all 0 n N 1, compute (in parallel) using the coarse and the fine-scale propagators X n+1 k = G t (Xk), n u n+1 k = F t (uk). n b) For all 0 n N 1, evaluate the jumps (the difference between the two propagated values) at the macroscopic level: J n+1 k = R(u n+1 k ) X n+1 k. c) Compute { } Xk+1 n sequentially by 0 n N Xk+1 0 = R(u 0 ), X n+1 k+1 = G t(xk+1)+j n n+1 k. d) Compute { u n+1 } k+1 by matching the result of the local 0 n N 1 microscopic computation, u n+1 k, on the corrected macroscopic state X n+1 k+1 : uk+1 0 = u 0 and, for all 0 n N 1, u n+1 n+1 k+1 = P(Xk+1, un+1 k ).

38 Error analysis In the simple setting above, the restriction, lifting and matching operators we use are: R(x, y) = x, L(X) = (X,(A 1 q)x) and P(X,(x, y)) = (X, y). Let us assume that F t and G t are exact. Then the errors on the macroscopic variable and the microscopic variable are: for all k 0, for all k 0, sup Xk n Ru(t n) Cǫ 1+ k/2, 0 n N sup uk n u(t n) Cǫ 1+ k/2, 0 n N

39 Numerical results E N k / x(tn) e N k / u(tn) k = 0 k = 1 k = 2 k = 3 k = 4 k = 5 k = ǫ ǫ Work in progress: extensions to nonlinear problems, stochastic problems,...

40 Conclusion Notice that the numerical methods we have presented are based on some dimension reduction and coarse-graining techniques (through the functions ξ or S). They are easy to parallelize. Two useful mathematical tools to quantify metastability: LSI and QSD.

41 Conclusion A few references: C. Le Bris, T. Lelièvre, M. Luskin and D. Perez, A mathematical formalization of the parallel replica dynamics, Monte Carlo Methods and Applications, 18(2), , F. Legoll and T. Lelièvre, Effective dynamics using conditional expectations, Nonlinearity, 23, , F. Legoll, T. Lelièvre and G. Samaey, A micro-macro parareal algorithm: application to singularly perturbed ordinary differential equations,