Browsing by Author "Vanden-Eijnden, E"

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Open Access
A Mathematical Theory of Optimal Milestoning (with a Detour via Exact Milestoning)
(2017-04-23) Lin, L; Lu, J; Vanden-Eijnden, E
Milestoning is a computational procedure that reduces the dynamics of complex systems to memoryless jumps between intermediates, or milestones, and only retains some information about the probability of these jumps and the time lags between them. Here we analyze a variant of this procedure, termed optimal milestoning, which relies on a specific choice of milestones to capture exactly some kinetic features of the original dynamical system. In particular, we prove that optimal milestoning permits the exact calculation of the mean first passage times (MFPT) between any two milestones. In so doing, we also analyze another variant of the method, called exact milestoning, which also permits the exact calculation of certain MFPTs, but at the price of retaining more information about the original system's dynamics. Finally, we discuss importance sampling strategies based on optimal and exact milestoning that can be used to bypass the simulation of the original system when estimating the statistical quantities used in these methods.
Open Access
Anomalous dissipation in a stochastically forced infinite-dimensional system of coupled oscillators
(Journal of Statistical Physics, 2007-09-01) Mattingly, JC; Suidan, TM; Vanden-Eijnden, E
We study a system of stochastically forced infinite-dimensional coupled harmonic oscillators. Although this system formally conserves energy and is not explicitly dissipative, we show that it has a nontrivial invariant probability measure. This phenomenon, which has no finite dimensional equivalent, is due to the appearance of some anomalous dissipation mechanism which transports energy to infinity. This prevents the energy from building up locally and allows the system to converge to the invariant measure. The invariant measure is constructed explicitly and some of its properties are analyzed. © 2007 Springer Science+Business Media, LLC.
Open Access
Methodological and computational aspects of parallel tempering methods in the infinite swapping limit
(2018-02-14) Lu, J; Vanden-Eijnden, E
A variant of the parallel tempering method is proposed in terms of a stochastic switching process for the coupled dynamics of replica configuration and temperature permutation. This formulation is shown to facilitate the analysis of the convergence properties of parallel tempering by large deviation theory, which indicates that the method should be operated in the infinite swapping limit to maximize sampling efficiency. The effective equation for the replica alone that arises in this infinite swapping limit simply involves replacing the original potential by a mixture potential. The analysis of the geometric properties of this potential offers a new perspective on the issues of how to choose of temperature ladder, and why many temperatures should typically be introduced to boost the sampling efficiency. It is also shown how to simulate the effective equation in this many temperature regime using multiscale integrators. Finally, similar ideas are also used to discuss extensions of the infinite swapping limits to the technique of simulated tempering.
Open Access
Simple systems with anomalous dissipation and energy cascade
(Communications in Mathematical Physics, 2007-11-01) Mattingly, JC; Suidan, T; Vanden-Eijnden, E
We analyze a class of dynamical systems of the type ȧn(t) = cn-1 an-1(t) - cn an+1(t) + f n(t), n ∈ ℕ, a 0=0, where f n (t) is a forcing term with fn(t) ≠ = 0 only for ≤n n* < ∞ and the coupling coefficients c n satisfy a condition ensuring the formal conservation of energy 1/2 Σn |a n(t)|2. Despite being formally conservative, we show that these dynamical systems support dissipative solutions (suitably defined) and, as a result, may admit unique (statistical) steady states when the forcing term f n (t) is nonzero. This claim is demonstrated via the complete characterization of the solutions of the system above for specific choices of the coupling coefficients c n . The mechanism of anomalous dissipations is shown to arise via a cascade of the energy towards the modes with higher n; this is responsible for solutions with interesting energy spectra, namely E |an|2 scales as n-α as n→∞. Here the exponents α depend on the coupling coefficients c n and E denotes expectation with respect to the equilibrium measure. This is reminiscent of the conjectured properties of the solutions of the Navier-Stokes equations in the inviscid limit and their accepted relationship with fully developed turbulence. Hence, these simple models illustrate some of the heuristic ideas that have been advanced to characterize turbulence, similar in that respect to the random passive scalar or random Burgers equation, but even simpler and fully solvable. © 2007 Springer-Verlag.