Jarzynski’s Equality, Fluctuation Theorems, and Variance Reduction: Mathematical Analysis and Numerical Algorithms

Hartmann, Carsten; Schütte, Christof; Zhang, Wei

doi:10.1007/s10955-019-02286-4

Cited by 12 publications

(20 citation statements)

References 68 publications

(232 reference statements)

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“…Furthermore, applying inverse function theorem, we can verify that Π is smooth on Ω. We have the following result which connects the derivatives of Π to the projection map P in (21). Its proof is given in Appendix C.…”

Section: Numerical Schemes Sampling On σmentioning

confidence: 80%

“…For this purpose, it is necessary to study the properties of the limiting flow map Θ, since it is involved in the numerical scheme (72). In fact, we have the following important result, which characterizes the derivatives of Θ by the projection map P in (21). Proposition 4.…”

Section: Numerical Schemes Sampling On σmentioning

confidence: 95%

“…Proposition 1. Let Σ be the submanifold of M defined in (20), P be the projection matrix in (21), and ∆ Σ be the Laplace-Beltrami operator on Σ. We have…”

Section: Notations and Preparationsmentioning

confidence: 99%

“…Proposition 4. Let Θ be the limiting flow map in (70) and P be the projection map in (21). At each x ∈ Σ, we have…”

Section: Numerical Schemes Sampling On σmentioning

confidence: 99%

“…For instance, in Section 4, the Poisson equation on Σ related to (9) will be used to study the long time sampling efficiency of numerical schemes. Furthermore, the infinitesimal generator (9) plays a role in the subsequent work [31] in analyzing the approximation quality of the effective dynamics for the diffusion process (6), while the dynamics (8) has been used in [21] to study the fluctuation relations and Jarzynski's equality for nonequilibrium systems in the reaction coordinate case. Secondly, for Question (Q2), in Subsection 3.2 we study the constrained Langevin dynamics in [33], where we introduce a parameter ǫ > 0 and we modify the potential function U properly such that its marginal invariant measure is µ 1 .…”

Section: Introductionmentioning

confidence: 99%

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Ergodic SDEs on submanifolds and related numerical sampling schemes

Zhang

2020

ESAIM: M2AN

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In many applications, it is often necessary to sample the mean value of certain quantity with respect to a probability measure µ on the level set of a smooth function ξ :A specially interesting case is the so-called conditional probability measure, which is useful in the study of free energy calculation and model reduction of diffusion processes. By Birkhoff's ergodic theorem, one approach to estimate the mean value is to compute the time average along an infinitely long trajectory of an ergodic diffusion process on the level set whose invariant measure is µ. Motivated by the previous work of Ciccotti, Lelièvre, and Vanden-Eijnden [11], as well as the work of Lelièvre, Rousset, and Stoltz [33], in this paper we construct a family of ergodic diffusion processes on the level set of ξ whose invariant measures coincide with the given one. For the conditional measure, in particular, we show that the corresponding SDEs of the constructed ergodic processes have relatively simple forms, and, moreover, we propose a consistent numerical scheme which samples the conditional measure asymptotically. The numerical scheme doesn't require computing the second derivatives of ξ and the error estimates of its long time sampling efficiency are obtained.

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Section: Numerical Schemes Sampling On σmentioning

confidence: 80%

Section: Numerical Schemes Sampling On σmentioning

confidence: 95%

“…Proposition 1. Let Σ be the submanifold of M defined in (20), P be the projection matrix in (21), and ∆ Σ be the Laplace-Beltrami operator on Σ. We have…”

Section: Notations and Preparationsmentioning

confidence: 99%

“…Proposition 4. Let Θ be the limiting flow map in (70) and P be the projection map in (21). At each x ∈ Σ, we have…”

Section: Numerical Schemes Sampling On σmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Ergodic SDEs on submanifolds and related numerical sampling schemes

Zhang

2020

ESAIM: M2AN

Self Cite

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Some new results on relative entropy production, time reversal, and optimal control of time-inhomogeneous diffusion processes

Zhang

2021

Journal of Mathematical Physics

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This paper studies time-inhomogeneous diffusion processes, including both Brownian dynamics and Langevin dynamics. We derive upper bounds of the relative entropy production for a time-inhomogeneous process with respect to the transient invariant probability measures. We also study the time reversal of the reverse process in Crooks’s fluctuation theorem. We show that the time reversal of the reverse process coincides with the optimally controlled forward process that leads to a zero variance importance sampling estimator for free energy calculations based on Jarzynski’s equality.

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Stochastic gradient descent and fast relaxation to thermodynamic equilibrium: A stochastic control approach

Breiten

Hartmann

Neureither

et al. 2021

Journal of Mathematical Physics

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We study the convergence to equilibrium of an underdamped Langevin equation that is controlled by a linear feedback force. Specifically, we are interested in sampling the possibly multimodal invariant probability distribution of a Langevin system at small noise (or low temperature), for which the dynamics can easily get trapped inside metastable subsets of the phase space. We follow Chen et al. [J. Math. Phys. 56, 113302 (2015)] and consider a Langevin equation that is simulated at a high temperature, with the control playing the role of a friction that balances the additional noise so as to restore the original invariant measure at a lower temperature. We discuss different limits as the temperature ratio goes to infinity and prove convergence to a limit dynamics. It turns out that, depending on whether the lower (“target”) or the higher (“simulation”) temperature is fixed, the controlled dynamics converges either to the overdamped Langevin equation or to a deterministic gradient flow. This implies that (a) the ergodic limit and the large temperature separation limit do not commute in general and that (b) it is not possible to accelerate the speed of convergence to the ergodic limit by making the temperature separation larger and larger. We discuss the implications of these observations from the perspective of stochastic optimization algorithms and enhanced sampling schemes in molecular dynamics.

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Jarzynski’s Equality, Fluctuation Theorems, and Variance Reduction: Mathematical Analysis and Numerical Algorithms

Cited by 12 publications

References 68 publications

Ergodic SDEs on submanifolds and related numerical sampling schemes

Ergodic SDEs on submanifolds and related numerical sampling schemes

Some new results on relative entropy production, time reversal, and optimal control of time-inhomogeneous diffusion processes

Stochastic gradient descent and fast relaxation to thermodynamic equilibrium: A stochastic control approach

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