Minimum Action Method for the Study of Rare Events

Transcription

1 Minimum Action Method for the Study of Rare Events WEINAN E Princeton University and Institute for Advanced Study WEIQING REN Institute for Advanced Study AND ERIC VANDEN-EIJNDEN Courant Institute and Institute for Advanced Study Abstract The least action principle from the Wentzell-Freidlin theory of large deviations is exploited as a numerical tool for finding the optimal dynamical paths in spatially extended systems driven by a small noise. The action is discretized and a preconditioned BFGS method is used to optimize the discrete action. Applications are presented for thermally activated reversal in the Ginzburg-Landau model in one and two dimensions, and for noise induced excursion events in the Brusselator taken as an example of nongradient system arising in chemistry. In the Ginzburg-Landau model, the reversal proceeds via interesting nucleation events, followed by propagation of domain walls. The issue of nucleation versus propagation is discussed and the scaling for the number of nucleation events as a function of the reversal time and other material parameters is computed. Good agreement is found with the numerical results. In the Brusselator, whose deterministic dynamics has a single stable equilibrium state, the presence of noise is shown to induce large excursions by which the system cycles out of this equilibrium state. Introduction Consider a stochastic ODE or PDE perturbed by a small noise: ẋ = b(x)+ ε Ẇ u t = L(u)+ εη(x, t) Here ε, Ẇ and η are respectively temporal and spatio-temporal white-noises. In the absence of noise, the system evolves to its equilibrium states and stays there indefinitely. The presence of noise changes that picture over long time scales. The system may hop between metastable states, make excursions out of these states, etc. Communications on Pure and Applied Mathematics, Vol. 000, (003) c 003 John Wiley & Sons, Inc. CCC /98/

2 W. E, W. REN, E. VANDEN-EIJNDEN At a first sight one might think that the influence of noise happens at such a long time scale that it is rarely of practical importance. That this view is incorrect may be understood by noticing that most physical processes will not happen at zero temperature when thermal noise is absent. In fact nature presents naturally a very wide range of temporal scales. The atomistic vibration that gives rise to thermal noise happens on the time-scale of femtoseconds (0 5 seconds), and the processes of interest to our daily lives are mostly on the order of seconds or longer, leaving plenty of room for rare events caused by thermal noise to make their appearance. Chemical reactions, nucleations, and conformational changes of biomolecules are all examples of such rare events. Traditionally the method of choice for a quantitative understanding of the effect of noise has been the Monte Carlo method or direct simulation of the Langevin equation. When the noise is small, which is the case of interest here, these methods become prohibitively expensive, due to the presence of two disparate time-scales: the time-scale of the deterministic dynamics and the time-scale between the rare events caused by the noise. Noticing this difficulty, alternative theories and numerical methods have been proposed. The most notable analytical work is the theory of Wentzell and Freidlin [] which gives an estimate on the probability of the paths in terms of an action functional. The most probable path is given by the one that minimizes the action functional. The Wentzell- Freidlin action is an analog of the Onsager-Machlup action [9, 7], and it can be used as a numerical tool in which optimal trajectories between the initial and final states in the system are computed by optimizing the action functional. We shall call methods of this type minimum action methods, and the optimal path the minimal action path (MAP). In this paper, we develop a minimum action method for spatially extended systems described by PDEs. While the idea of using Wentzell- Freidlin action or variation thereof as numerical tools is not new in the context of ODEs (see e.g. [, 9]), the main challenge in spatially extended systems is that the action functional and its derivative typically involve high order spatial derivatives, giving rise to highly ill-conditioned numerical problems. Therefore a good choice of preconditioner is required in order to achieve an acceptable convergence rate. It should be pointed out that if the system happens to be a gradient system, then over an infinite time interval, the minimal action path becomes a minimal energy path (MEP), which is a heteroclinic orbit that connects two local minima of the potential. Efficient numerical methods

3 MINIMUM ACTION METHOD 3 have been developed for finding the MEP as well as the transition rates. Most notable examples include the nudged elastic band method (NEB) [3] and the string method [3, 4, ]. These methods are typically more efficient than methods based on minimizing the action functional. Yet the minimum action method has the advantage of being applicable to both gradient and non-gradient systems. It can also be applied to studying events on finite-time intervals which have exponentially small probability in small domains, but become ubiquitous when the size of the system is large. This paper is organized as follows. In section, we briefly review the Wentzell-Freidlin theory. In section 3 we present the numerical method for the example of one-dimensional Ginzburg-Landau model. Results on two-dimensional Ginzburg-Landau model are presented in section 4. Nongradient systems are discussed in section 5 with the example of the Brusselator. Some conclusions are drawn in section 6. The appendix contains a discussion on the limited memory BFGS method, which is used in the minimum action method. Wentzell-Freidlin Theory and the Least Action Principle We consider a random process X ε (t) in R n defined by the stochastic differential equation (SDE): (.) Ẋ ε = b(x ε )+ εσ Ẇ, X ε (0) = x 0. Here W (t) is a Wiener process in R n. As ε 0, the trajectory X ε (t) converges in probability to the solution x(t) of the unperturbed equation (.) ẋ = b(x), x(0) = x 0 on every finite time interval. In the path space, the probability distribution of X ε (t) is concentrated in a neighborhood of x(t). Any other event that does not include x(t) and its neighborhood has very small probability. The Wentzell-Freidlin theory gives an estimate of the probability distribution of the random process in the path space. We state the estimate by first introducing an action functional in the path space. We denote by C [0,T ] the set of continuous functions on the interval [0,T]withvaluesin R n. In this space, we define the metric ρ T (ϕ, ψ) = sup ϕ(t) ψ(t). 0 t T

4 4 W. E, W. REN, E. VANDEN-EIJNDEN For ϕ C [0,T ], we define an action functional associated with (.) as (.3) S T [ϕ] = T 0 σ ( ϕ b(ϕ)) dt if the integral is finite. Otherwise, we set S T [ϕ] tobe+. The Wentzell-Freidlin estimates are:. For any δ>0,γ >0, and K>0 there exists an ε 0 > 0 such that (.4) P{ρ T (X ε,ϕ) <δ} exp{ ε (S T [ϕ]+γ)} for ε<ε 0,whereT>0andϕ C [0,T ] are such that ϕ(0) = x 0 and T + S T [ϕ] K.. For any δ>0,γ > 0, and s 0 > 0 there exists an ε 0 > 0 such that for 0 <ε ε 0 and s<s 0 we have (.5) P{ρ T (X ε, Φ T (s)) δ} exp{ (s γ)} ε where Φ T (s) ={ϕ C [0,T ],ϕ(0) = x 0,S T [ϕ] s} for s>0 and ρ T (X ε, Φ T (s)) = sup ρ T (X ε,ϕ). ϕ Φ T (s) For details of the proofs of these estimates, we refer to []. Roughly speaking, the estimates (.4) and (.5) tell us that the probability that X ε stays in a δ-neighborhood of a path ϕ is (.6) P{ρ T (X ε,ϕ) <δ} exp{ ε S T [ϕ]}. The above estimates can be used to calculate the probability of various events associated with (.) by constrained minimization of the action functional. For instance, let a and b be two states of the system and fix atimet. Then the probability P T that the system moves from a to a δ-neighborhood of b within time T can be estimated using the Wentzell- Freidlin theory: (.7) lim ε ln P T = min S T [ϕ], ε 0 ϕ

5 MINIMUM ACTION METHOD 5 where the minimization in (.7) is constrained by (.8) ϕ(0) = a, ϕ(t ) = b. The minimizer of (.7), or the MAP ϕ gives the most probable path for the transition from a to b, in the sense that the probability that the system moves by another path is exponentially smaller in ε. So far we have only discussed finite dimensional systems. In the following, we will be interested in the application of Wentzell-Freidlin theory to infinite dimensional systems described by stochastic PDEs. There are several subtleties in extending the Wentzell-Freidlin theory to PDEs, one of which is associated with the fact that the noise we are most interested in, the spatio-temporal white noise, is too singular for the stochastic PDE to be well-defined in dimensions higher than one []. We will not discuss these issues here. Instead we refer to [, 6, 9, 0] for existing results on large deviation theory for PDEs. We will assume that the formal extension of the action functional from finite to infinite dimension is valid. 3 Minimum Action Method: Application to the One-dimensional Ginzburg-Landau Model We will discuss the minimum action method by way of an example: the thermally activated switching of a bistable system modeled by the Ginzburg-Landau equation (3.) u t = δu xx δ V (u), x [0, ] with Dirichlet boundary conditions (3.) u(0,t)=0, u(,t)=0. We take V (u) to be the standard double-well potential: (3.3) V (u) = 4 ( u ). In the equation (3.), δ is a small parameter which indicates that the dynamics corresponding to the reaction term, δ V (u), is fast while the diffusion is slow. The system can be considered as the gradient flow associated with the energy (3.4) E[u] = 0 (δu x +δ V (u))dx.

6 6 W. E, W. REN, E. VANDEN-EIJNDEN u + (x) u (x) 0 x 0 x Figure 3.. Two equilibrium states of (3.) for δ = The dynamics in (3.) has two stable equilibrium states, u + (x)andu (x), which minimize the energy (3.4). When δ is small, u ± (x) =± except in two thin layers of width δ at x =0andx =, as shown in figure 3.. Now we add to (3.) a small noise term modeling thermal effects: (3.5) u t = δu xx δ V (u)+ εη, where ε is proportional to the temperature of the system and η is a spatiotemporal white-noise with covariance (3.6) E (η(x, t)η(y,s)) = δ(x y) δ(t s). It is shown in [6] (see also [5]) that (3.5) makes sense at least in dimension one. The presence of the noise in (3.5) destroys the long-time stability of the equilibria u ±. For instance, if the initial state is u +, there is a finite probability P T that the system switches to u in any time interval [0,T]. From large deviation theory [6], the probability P T satisfies (3.7) lim ε ln P T = min S T [u], ε 0 u where the action functional is given by T (3.8) S T [u] = (u t δu xx + δ V (u)) dxdt. 0 0 The minimization problem in (3.7) is constrained by the conditions (3.9) u(0,t)=u(,t)=0, u(x, 0) = u + (x), u(x, T )=u (x). The minimizer of (3.8) constrained by (3.9) defines the optimal switching path between u + and u during the time interval [0,T].

7 MINIMUM ACTION METHOD 7 3. Minimization of the Action Functional To find the optimal switching path, we minimize the action (3.8) in a simple two-step numerical procedure. In the first step we discretize the action functional using finite difference. In the second step we minimize the discretized action functional using the limited memory BFGS (L- BFGS) method (see the appendix). Other methods can also be used in either steps, for example, finite element can be used in the first step. We made these choices for their simplicity. We discretize the space-time domain [0, ] [0,T]withameshwith sizes x =/I and t = t/j, and we define the grid point (x i,t j )by x i = i x, t j = j t, i =0,,...,I, j =0,,...,J. We also define x i+ t j+ =(i + ) x, =(j + ) t. The numerical approximation to u(x i,t j ) is denoted by U i,j. In order to simplify the expression, we introduce the force (3.0) p(x, t) =u t δu xx + δ V (u), so the action can be written as T (3.) S T [u] = p (x, t) dxdt. 0 0 We use the trapezoidal rule to discretize the spatial integral, and the midpoint rule for the temporal integral and we obtain (3.) S T (U) = I x t where P i,j+ finite difference: (3.3) P i,j+ J i= j=0 P i,j+, is the numerical approximation to p(x i,t j+ ) by central = U i,j+ U i,j t + δ V ( U i,j+ + U i,j ) δ (U i+,j+ U i,j+ + U i,j+ x + U i+,j U i,j + U i,j x ).

8 8 W. E, W. REN, E. VANDEN-EIJNDEN The discretized boundary conditions corresponding to (3.9) are (3.4) U 0,j = U I,j =0, j =0,,...,J, U i,0 = u + (x i ), U i,j = u (x i ), i =0,,...,I. BFGS method requires the gradient of (3.); this is given by: S T U i,j = ( x + x t V (U i,j + U i,j ) δ t + δ x ( x x t V (U i,j + U i,j+ ) δ t δ x δ t ( P x i,j + P i,j+ + P i+,j ) P i,j ) P i,j+ + P i+,j+ for i =,,...,I, j =,,...,J. In the BFGS method, the initial approximation to the inverse of the Hessian plays the role of preconditioners. A good choice of the preconditioner is vital for the efficiency of the algorithm. A general choice is to use the diagonal matrix, (3.5) H 0 k = γ ki, where γ k is defined in the appendix. For our problem, we found that the linear part of the Hessian of the action (3.8), which contains the highest order derivatives, is a better preconditioner: (3.6) H 0 k = B =( t + δ 4 x). The operator B can be inverted efficiently by fast Fourier transform (FFT). In the following we will compare the behavior for different choices of H 0 k. 3. Numerical Results In Table 3. we illustrate the behavior of the L-BFGS method for different choices of Hk 0 and the memory parameter m. In this problem, the parameter δ is 0.05, T is, and we used points in the discretization. For a complex system, the evaluation of the objective function and its gradient dominates the cost of the computation. Therefore, in the table we present the number of function and gradient evaluations, n fg,requiredfor the L -norm of the gradient to reach a certain tolerance. We also present the number of iterations, denoted by n it, in the parenthesis. Note that ),

9 MINIMUM ACTION METHOD 9 Hk 0 = I H0 k = γ ki Hk 0 = B n fg (n it ) n fg (n it ) n fg (n it ) m= (898) 8697 (7838) 744 (709) m=5 749 (874) 330 (74) 658 (638) m=6 89 (909) 0070 (977) 573 (547) m=7 690 (84) 9745 (9500) 570 (553) m=8 798 (8935) 000 (9938) 56 (544) m= (8656) 9446 (908) 564 (550) m=0 864 (965) 8765 (8547) 540 (50) Table 3.. Performance of L-BFGS for various choices of the storage parameter m and initial approximations to the Hessian. n fg is the total number of function and gradient evaluations, and n it is the total number of iterations required to decrease the L -norm of the gradient to 0 0. n fg is always bigger than n it, since at each iteration step, several function and gradient evaluations might be needed in the line search if the search direction p k is not well scaled (see the appendix for the definition of p k ). The first conclusion that can be drawn from this table is that the number of function and gradient evaluations is insensitive to the memory parameter m. But since the cost of each iteration increases with the amount of storage, the method is most efficient for moderate m, for example, m = 6 or 7. Secondly, the table shows that for fixed m, the Hessian approximation by (3.6) improves the convergence significantly, which indicates that (3.6) is a good preconditioner for this problem. Furthermore, for Hk 0 = γ ki and Hk 0 = B, the two values n fg and n it are of the same order, which indicates that the search direction p k is well scaled, and, as a result, the initial step length α k = is accepted in most iterations. However, for Hk 0 = I, thenumbern fg is about twice the size of n it, which means that p k is not well scaled, and the line search is required to obtain a suitable step length. Figure 3. shows the sequence of profiles of the optimal path u at different times in [0,T] for various values of T at a fixed δ =0.03. The switching proceeds by nucleation followed by propagation of domain walls. For large T, the switching proceeds by propagating one domain wall from one boundary to another, as shown in figure 3.(a). As T becomes smaller, however, the number of nucleation events increases. In figure 3.3, we display the values of the action against T for the various local minimizers shown in figure 3.. Figure 3.4 shows the space-time plot of the square

10 0 W. E, W. REN, E. VANDEN-EIJNDEN of the force p(x, t) corresponding to the switching event in figure 3.(f). This can be interpreted as the minimal noise necessary to induce the switching. The peaks correspond to the nucleations. These results are explained next. 3.3 Nucleation versus Propagation in One Dimension The results in the last section can be understood as follows. (Our discussion here is rather qualitative; more rigorous results will be presented in []. For a complementary discussion of nucleation versus propagation in one dimension, see [8].) Consider the critical points of the energy (3.4), i.e., the solutions of (3.7) δu xx δ V (u) =0 with u x=0 = u x= = 0. Besides u + and u, corresponding to the minimizers of the energy, there are also saddle point configurations with an increasing number of domain walls. Our result shows that, for large T,the switching path crosses the saddle point configuration with minimum energy, i.e. the configuration with a single domain wall shown in figure 3.(a) (this result is standard, see [6] and also [4]). As T is decreased, the optimal switching path crosses (the vicinity of) saddle point configurations with increasing energy, i.e. it involves more nucleations and therefore more domain walls, giving rise to a cascade of nucleation events. The reason is that both the nucleation and domain wall motion are noise induced. As T decreases, at fixed number of nucleations the speed of propagation of the domain wall must increase in order to complete the switching during the allowed time T. This is energy consuming and, at certain critical values of T it becomes more favorable to make an additional nucleation. The same type of cascade of nucleations is also observed if T is fixed but δ is decreased. To quantify these observations, we estimate the cost of nucleation and propagation of domain walls. We assume there are n nucleations and correspondingly n domain walls participating in the switching. We also assume that the nucleation and propagation costs can be estimated independently (which isn t immediately obvious since a nucleation in this system is not a saddle point of the energy). In this picture, each nucleation corresponds to the creation of two domain walls separated by a distance λ satisfying δ λ ; propagation corresponds to the subsequent motion of each pair of interfaces by an O() distance across the domain to achieve switching.

11 MINIMUM ACTION METHOD (a) (b) 0 0 (c) (d) 0 0 (e) (f) 0 0 Figure 3.. Snapshots of profiles of the minimizer u during a switching from u + (top curve) to u (bottom curve) at equally spaced times in [0,T]fordifferentT at fixed δ =0.03. (a)t =7;(b)T =;(c)t =; (d) T =0.8; (e) T =0.6; (f) T =0.4.

12 W. E, W. REN, E. VANDEN-EIJNDEN 35 5 (f) S T [u] (e) 5 (d) (c) (b) 5 (a) T Figure 3.3. The action as a function of the switching time T for the six minimizers shown in figure p (x,t) 0 0 x 0 t Figure 3.4. figure 3.. The space-time value of p (x, t) for the minimizer (f) in

13 MINIMUM ACTION METHOD 3 Thus, for nucleation events, we look for the minimizer of (3.8) localized in a region of size O(δ) bothinspaceandtime.sowetake ( x x u(x, t) =v, t ) δ δ and obtain: (3.8) S T [v] = (v τ v ξξ + V (v)) dξdτ. 0 R The minimum cost of a nucleation event is given by the minimum of the above action constrained by v(ξ,0) =, v(ξ, ) =+ if ξ [ λδ, λδ ]. The minimum is achieved by following backward in time the orbit associated with v τ = v ξξ V (v), which connects the state where v =+ifξ [ λδ, λδ ]tov =. Indeed, the action (3.8) can be written as S T [v] = (v τ + v ξξ V (v)) dξdτ + v τ (v ξξ V (v))dξdτ; 0 R 0 R along the orbit, the first term vanishes as can be seen by reversing time in the integration, whereas the second term reduces to twice the difference of energy between the state where v =+ifξ [ λδ, λδ ]andthe minimum energy state v = (which has zero energy). In the limit as λδ, the energy of the state where v =+ifξ [ λδ, λδ ] can be estimated as twice the energy of the tangent hyperbolic profile, v(ξ) = tanh(ξ/ ), which solves v ξξ = V (v) with boundary conditions lim v(ξ) =±. ξ ± We denote twice this energy by C and therefore obtain the following estimate for the cost of one nucleation: (3.9) A nucl = C. Notice that it is because we can take the double limit when λ 0, δ 0 with λδ and obtain a limiting cost that the nucleation events can be well-defined in the system.

14 4 W. E, W. REN, E. VANDEN-EIJNDEN For propagation of domain walls, we look for a minimizer of (3.8) with a sharp transition between + and at the interface location specified by x ct =0,wherec =/(nt ) is the speed of the interface. So we take ( x ct ) u(x, t) =w δ and obtain (3.0) S T [u] = T (cw η + w ηη V (w)) dη. δ R The minimum cost of propagating one domain wall is thus given by (3.) A prop = δ TM(c), where M(c) is the minimum of the integral in (3.0) over w subject to suitable constraints, e.g., w( ) = and w(+ ) = +. For small c, the minimum is achieved when w ηη V (w)b = O(c), i.e. when w(η) = tanh(η/ ) + c w(η). Therefore to leading order we obtain M(c) =C c, where ( C =inf sech (η/ )/ + w ηη V ( tanh(η/ ) ) w R subject to lim η ± w = 0. Therefore, (3.) A prop = C c δ T. w) dη, Using (3.9) and (3.), we obtain the total cost of the nucleation and propagation of n domain walls: (3.3) A(n, δ, T )=C n + C nδt. For fixed δ and T, the optimal number of nucleations is given by ( ) / (3.4) n C =mina(n, δ, T )=, n C δt and from (3.7), one has ( ) C C / (3.5) lim ε ln P T =, ε 0 δt which gives the envelop of the curves in figure 3.3.

15 MINIMUM ACTION METHOD 5 4 Application to the Two-dimensional Ginzburg-Landau Model 4. Numerical results As an example in two dimensions, we consider (4.) u t = δ u δ V (u) intheunitsquareω=[0, ] [0, ]. We will consider two different Dirichlet boundary conditions: (4.) u x=0 = u x= =, u y=0 = u y= =, or (4.3) u x=0 = u x= = u y=0 = u y= =0. In both cases, (4.) has two stable equilibrium states. One of them, u +, which is the top figure in each column in figure 4., is close to u = except at the boundary layers at y =0andy = in case (4.), or at the edge of the square in case (4.3). Another stable state, u,whichisthe bottom figure in each column in figure 4., is close to u = except at the boundary layers at x =0andx = in case (4.), or at the edge of the square in case (4.3). Similar to the one-dimensional problem, assuming the system switches from u + to u before a given time T, we minimize the action (4.4) S T [u] = T (u t δ u + δ V (u)) dxdydt 0 Ω to find the optimal switching path between the two states. The numerical algorithm we use is a direct extension of the method for the onedimensional model. 4. Nucleation versus Propagation in Higher Dimensions In figure 4., we show the time sequences of the switching process for different values of δ at fixed T =. The paths (a), (b), (c) and (d) correspond to the first case with boundary condition (4.), and (e) corresponds to (4.3). The overall trend is consistent with what was found in the one dimensional example, namely there are more and more nucleation events as δ 0.

16 6 W. E, W. REN, E. VANDEN-EIJNDEN (a) (b) (c) (d) (e) Figure 4.. Snapshots of profiles of the minimizer u during a switching from u+ (top figure in each column) to u (bottom figure in each column) at five equally spaced times in [0, T ] for T = and different δ and different boundary conditions: (a)-(d) correspond to (4.), (e) corresponds to (4.3). In (a) δ = 0.04; (b) δ = 0.03; (c) δ = 0.0; (d) δ = 0.0; (e) δ = 0.0. The gray scale is from white for u = to black for u =.

17 MINIMUM ACTION METHOD 7 As in one-dimensional case, we can estimate the optimal number of nucleations for fixed T and δ in two dimensions using test functions. To estimate the cost of one nucleation, we proceed with the same type of test functions as in one-dimensional case and obtain (4.5) A nucl = C δ. where C is a constant. (Note that this supposes that one can clearly distinguish between nucleation and propagation events, which is less clear in dimension two than in dimension one since the energy of the nucleus now depends on its radius; this means C is given by a variational problem which is different from the one in dimension one and optimizes on the radius of the nucleus. We shall not dwell on this issue here; see [] for details.) To describe the propagation of the interface, we take ( ϕ(x, y) t ) u(x, y, t) =w, δ where w(ξ) is a smooth function and the level set ϕ(x, y) =t specifies the location of the interface at time t. The normal velocity of the interface is given by c = ϕ. Substituting u into (4.4) an using η = cξ instead of t as time integration variable we obtain to leading order in δ: (4.6) S T [u] = cw η w ηη + V (w) dxdydη, δ R D where D is the domain of the motion of the interface (in the domain of definition of ϕ(x, y)). The minimum cost of the propagation of one interface is given by the minimum of (4.6) over ϕ and w: (4.7) A prop =min S T [u] =C δ min ϕ ϕ,w ϕ D dxdy, where C is the same constant as given in the one-dimensional case. The minimum in (4.7) is achieved by ( (4.8) ϕ(x, y) =B (x x ) +(y y ) ) 3/4, where (x,y ) is the location of the nucleation and B is a constant. The minimizer (4.8) corresponds to a circular interface whose radius is zero at t = 0 and grows as ( ) t /3 (4.9) ρ(t) =. B

18 8 W. E, W. REN, E. VANDEN-EIJNDEN We assume there are n nucleations in a domain with unit area, then the radius of each interface has to grow to O(n / )attimet in order to accomplish the switching. This specifies B: (4.0) B = C 3 Tn 3/4, where C 3 is a constant. Substituting (4.8) and (4.0) into (4.7), we obtain the cost of propagating each of the n interfaces: (4.) A prop = C 4 (δtn 3/), where C 4 is a constant. It follows from (4.5) and (4.) that the minimum of S T [u] withn nucleations satisfies: (4.) A(n, δ, T )=C nδ + C 4 (δtn /). For fixed δ and T, the optimal number of nucleations is given by ( ) d/(d+) (4.3) n C4 =mina(n, δ, T )= n C dδ d, T and the corresponding action is (4.4) A(n,δ,T)=(+d ) ( dc C d 4 δt d ) /(d+), where d = is the dimensionality of the domain. (4.3) and (4.4) are valid in higher dimensions. 5 Non-Gradient Systems The methodology can be applied equally well to non-gradient systems. In this section we illustrate this on the example of the Brusselator: u t = ( ) u xx ++u v ( + A)u (5.) α v t = v xx + Au u v, where A and α are two parameters. We consider these equations on x [0, ] and impose Neumann boundary conditions (5.) u x (0,t)=u x (,t)=0, v x (0,t)=v x (,t)=0.

19 MINIMUM ACTION METHOD 9 The Brusselator was introduced as a simple model of nonlinear chemical system in which the relative concentration of products can oscillate in time as in, e.g., the Belousov-Zhabotinski reaction [7, 0]. The Brusselator has a stable fixed point at (,A). We will be interested in finding the optimal path from the stable fixed point to another point in the phase space, under the influence of a small noise: u t = ( ) u xx ++u v ( + A)u + εη (x, t) (5.3) α v t = v xx + Au u v + εη (x, t), where η, η are two independent spatio-temporal white-noises and we impose reflecting boundary condition at {u =0} and {v =0}. The action functional associated with (5.3) is S T [u, v] = T ( u t ( dxdt u xx ++u v ( + A)u)) 0 0 α (5.4) + T ( v t v xx Au + u v) dxdt. 0 0 We minimize this action functional using a straightforward extension of the method described earlier. In figure 5., we plot the spatial means ū = 0 udx and v = 0 vdx of the optimal pathways for various final states and the contour lines of the corresponding actions. One can see from figure 5. that the optimal paths contain large excursions. This is one of the special features of non-gradient systems. More thorough discussion of this example as well as other examples of infinite dimensional non-gradient systems will be presented elsewhere [6]. 6 Discussion In summary, the least action principle provided by the Wentzell-Freidlin theory of large deviations is exploited as a numerical tool for finding the optimal dynamical path for spatially extended systems driven by a small noise. We presented the numerical results for the Ginzburg-Landau system in one and two dimensions, as well applications to non-gradient systems. A quasi-newton method, the limited memory BFGS method, is used to minimize the Wentzell-Freidlin action. Other numerical issues such as the preconditioners are also discussed. We also present analytical results on the nucleation and propagation of domain walls for the Ginzburg-Landau models. The theoretical estimates agree very well with the numerical results.

20 0 W. E, W. REN, E. VANDEN-EIJNDEN v 0.4 v u u Figure 5.. Left panel: The spatial means ū = udx and v = vdx of 0 0 the optimal switching pathways from the stable position (,A) to various other states. Note that there are more than one local minimizer of the action in the vicinity of (0.8, 0.). Also shown are the two nullclines (dashed lines). Right panel: The contour lines of the corresponding action as a function of the final states. In the calculation, the parameters are α =0.,A=0.5 andt =. Although it requires the calculation of the Hessian of the energy functional and it is in general more expensive than methods based on computing the minimal energy path (such as the string method or NEB), the minimum action method has the advantage of being applicable to non-gradient systems and finite-time events. While the latter have exponentially small probability in small samples, they are relevant in large samples because their probability increases with system size and eventually tends to one. Appendix: Limited Memory BFGS Method The BFGS method is one of the most popular and efficient quasi-newton method for large-scale problems[8]. It only requires the gradient of the objective function to be supplied at each iterate, while achieves superlinear convergence. Suppose f(x) is the objective function that we are minimizing, and x k is the current iterate. The BFGS method defines the next one as (A.) x k+ = x k + α k p k. The step length α k is chosen to satisfy the Wolfe conditions: (A.) { fk+ f k + c α k f T k p k f T k+ p k c f T k p k

21 MINIMUM ACTION METHOD with 0 <c <c <. The search direction p k is given by (A.3) p k = H k f k where H k is an approximation of the inverse of the Hessian of f(x) atx k. In BFGS method, H k is defined recursively by (A.4) where H k+ =(I ρ k s k y T k ) H k (I ρ k y k s T k )+ρ k s k s T k (A.5) s k = x k+ x k, y k = f k+ f k, and ρ k = yk T s. k The initial Hessian approximation H 0 plays the role of preconditioners for the problem. In fact, it can be easily shown that the BFGS method is identical to the preconditioned conjugate gradient method with the preconditioner B 0 = H0 when applied to strongly convex quadratic functions. One general strategy for the choice of H 0 that has proved to be effective in practice is to use H 0 = γ I, where the scaling factor γ is defined by (A.6) γ k = yt k s k yk T y. k When applied to a smooth convex function with an arbitrary starting point x 0 and a symmetric positive definite matrix H 0, the BFGS method can be proved to be globally convergent. Furthermore, The rate of convergence is superlinear, which is adequate for practical problems. In the BFGS method, H k defined by (A.4), is usually a dense matrix, so the cost of storing and manipulating it is prohibitively large for large-scale systems. The limited memory BFGS method, abbreviated as L-BFGS, can be used to circumvent this problem. The idea is to store and use the m most recent vector pairs {s i,y i } to construct the approximation of the Hessian. Once the new iterate x k+ is computed, the oldest vector pair in the set {s i,y i }, which is less likely to be relevant to the behavior of the objective function at the current iteration, is discarded in order to save the storage and the computational cost. Acknowledgment. We are grateful to Bob Kohn, Cyrill Muratov, Felix Otto, and Maria Reznikoff for stimulating discussions. The work of E is partially supported by NSF via grant DMS Ren is partially supported by NSF via grant DMS Vanden-Eijnden is partially supported by NSF via grant DMS , and by AMIAS (Association of Members of the Institute for Advanced Study).

22 W. E, W. REN, E. VANDEN-EIJNDEN Bibliography [] Berkov, D. V. Numerical calculation of the energy barrier distribution in disordered many-particle systems: the path integral method. J. Magn. Mag. Mat. 86 (998), [] Da Prato, G.; Zabczyk, J. Stochastic equations in infinite dimensions. Encyclopedia of Mathematics and its Applications, 44. Cambridge University Press, Cambridge, 99. [3] E, W.; Ren, W.; Vanden-Eijnden, E.. String Method for the Study of Rare Events. Phys. Rev. B. 66 (00), [4] E, W.; Ren, W.; Vanden-Eijnden, E. Probing Multi-Scale Energy Landscapes Using the String Method. unpublished [5] E, W.; Ren, W.; Vanden-Eijnden, E. Energy Landscape and Thermally Activated Switching of Submicron-sized Ferromagnetic Elements. J. Appl. Phys. 93 (003), [6] Faris, W. G.; Jona-Lasinio, G. Large fluctuations for a nonlinear heat equation with noise. J. Phys. A 5 (98), no. 0, [7] Feynman, R. P.; Hibbs, A. R. Quantum mechanics and path integrals. New York, McGraw-Hill, 965. [8] Fogedby, H. C.; Hertz, J.; and Svane, A. Soliton-dynamical approach to a noisy Ginzburg-Landau model. preprint. [9] Freidlin, M. I. Limit theorems for large deviations and reaction-diffusion equations. Ann. Probab. 3 (985), no. 3, [0] Freidlin, M. I. Semi-linear PDEs and limit theorems for large deviations. Ecole d Eté de Probabilités de Saint-Flour XX 990, 09, Lecture Notes in Math., 57, Springer, Berlin, 99. [] Freidlin, M. I.; Wentzell, A. D. Random Perturbations of Dynamical Systems. nd ed. Springer, New York, 998 [] Holden, H.; Øksendal, B.; Ubøe, J.; Zhang, T. Stochastic partial differential equations. Birkhauser Boston, Inc., Boston, 996. [3] Jónsson, H.; Mills, G.; Jacobsen, K. W. Nudged elastic band Method for Finding Minimum Energy Paths of Transitions. in: Classical and Quantum Dynamics in Condensed Phase Simulations. ed. Berne, B. J.; Ciccotti, G.; Coker, D. F. World Scientific, 998 [4] Maier, R. S.; Stein, D. L. Droplet Nucleation and Domain Wall Motion in a Bounded Interval. Phys. Rev. Lett. 87 (00), [5] Marcus, R. Parabolic Itô equations with monotone nonlinearities. J. Funct. Anal. 9 (978), no. 3, [6] Muratov, C; Ren, W.; Vanden-Eijnden, E.; E, W. in preparation. [7] Nicolis, G.; Prigogine, I. Self-Organization in Nonequilibrium Systems. Wiley- Interscience, New York, 977. [8] Nocedal, J. Numerical Optimization. Springer Series in Operations Research, Springer-Verlag, New York, 999

23 MINIMUM ACTION METHOD 3 [9] Olender, R.; Elber, R. Calculation of classical trajectories with a very large time step: Formalism and numerical examples. J. Chem. Phys. 05 (996), 999. [0] Reichl,L.E.A Modern Course in Statistical Physics. University of Texas Press, Austin, 980. [] Ren, W. Numerical Methods for the Study of Energy Landscapes and Rare Events. PhD thesis, New York University, 00. [] Reznikoff, M.; Ren, W.; E, W.; Kohn, R. V.; Otto, F.; Vanden-Eijnden, E. in preparation. Weinan E Department of Mathematics and PACM Princeton University Washington Road Princeton, NJ weinan@princeton.edu Weiqing Ren School of Mathematics Institute for Advanced Study Einstein Drive Princeton, NJ weiqing@ias.edu Eric Vanden-Eijnden Courant Institute 5 Mercer Street New York, NY 00 eve@cims.nyu.edu