Noise-induced transitions in rugged energy landscapes

Duncan, Anthony; Kalliadasis, Serafim; Pavliotis, Grigorios A.; Pradas, Marc

doi:10.1103/physreve.94.032107

Cited by 10 publications

(16 citation statements)

References 61 publications

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“…This homogenization problem is (slightly) different from the standard one for the Fokker-Planck equation in a two-scale potential that was studied in [12,13] due to the self-consistency equation (3.12). In particular, in addition to the standard two-scale expansion for the solution of the Fokker-Planck equation (3.11), we also need to expand the solution of (3.12) into a power series in ǫ:…”

Section: Multiscale Analysis For the Mckean-vlasov Equation In A Two-mentioning

confidence: 87%

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Mean Field Limits for Interacting Diffusions in a Two-Scale Potential

Gomes

Pavliotis

2017

J Nonlinear Sci

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In this paper, we study the combined mean field and homogenization limits for a system of weakly interacting diffusions moving in a two-scale, locally periodic confining potential, of the form considered in Duncan et al. (Brownian motion in an N-scale periodic potential, arXiv:1605.05854, 2016b). We show that, although the mean field and homogenization limits commute for finite times, they do not, in general, commute in the long time limit. In particular, the bifurcation diagrams for the stationary states can be different depending on the order with which we take the two limits. Furthermore, we construct the bifurcation diagram for the stationary McKean–Vlasov equation in a two-scale potential, before passing to the homogenization limit, and we analyze the effect of the multiple local minima in the confining potential on the number and the stability of stationary solutions.

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Section: Multiscale Analysis For the Mckean-vlasov Equation In A Two-mentioning

confidence: 87%

“…5.4]. Such phenomena, including multiscaleinduced hysteresis effects, for a one-dimensional Brownian particle moving in a multiscale potential, were studied in detail in [12].…”

Section: Introductionmentioning

confidence: 99%

Mean Field Limits for Interacting Diffusions in a Two-Scale Potential

Gomes

Pavliotis

2017

J Nonlinear Sci

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“…Furthermore, the effective diffusion coefficient of the c 2017 SIAM and ASA. Published by SIAM and ASA under the terms of the Creative Commons 4.0 license simplified equation is not uniquely defined; see (18) and the fact that D(x) = A(x)A(x) T . Consequently, it is not possible to prove the strong convergence of the solution to the approximate SDE (32) to the solution to the homogenized SDE (20) without constraining the choice of the factorization.…”

Section: Resultsmentioning

confidence: 99%

“…SDEs of this form appear in several applications, e.g., in molecular dynamics [18,40]. Furthermore, several interesting semilinear singularly perturbed SPDEs can be written in this form; see section 6.…”

Section: (4b)mentioning

confidence: 99%

Spectral Methods for Multiscale Stochastic Differential Equations

Abdulle¹,

Pavliotis²,

Vaes³

2017

SIAM/ASA J. Uncertainty Quantification

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Abstract. This paper presents a new method for the solution of multiscale stochastic differential equations at the diffusive time scale. In contrast to averaging-based methods, e.g., the heterogeneous multiscale method (HMM) or the equation-free method, which rely on Monte Carlo simulations, in this paper we introduce a new numerical methodology that is based on a spectral method. In particular, we use an expansion in Hermite functions to approximate the solution of an appropriate Poisson equation, which is used in order to calculate the coefficients of the homogenized equation. Spectral convergence is proved under suitable assumptions. Numerical experiments corroborate the theory and illustrate the performance of the method. A comparison with the HMM and an application to singularly perturbed stochastic PDEs are also presented.

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“…Finally, we deal with CLT requirement (iii). First note that inequality (6) implies E[ρ 2 n,j ] ≤ κ 2 n,j + 2 E[ρ 2 n,j ; ρ n,j < −κ n,j ] for every n, j . Hence lim n→∞ sup j ≤m n κ n,j = 0 and the uniform integrability together imply that (9) lim…”

Section: Generalities Concerning Metropolis-hastings Log Acceptance Ratesmentioning

confidence: 99%

Counterexamples for optimal scaling of Metropolis–Hastings chains with rough target densities

Vogrinc

Kendall

2021

Ann. Appl. Probab.

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For sufficiently smooth targets of product form it is known that the variance of a single coordinate of the proposal in RWM (random walk Metropolis) and MALA (Metropolis adjusted Langevin algorithm) should optimally scale as n −1 and as n − 1 3 with dimension n, and that the acceptance rates should be tuned to 0.234 and 0.574. We establish counterexamples to demonstrate that smoothness assumptions of the order of C 1 (R) for RWM and C 3 (R) for MALA are indeed required if these scaling rates are to hold. The counterexamples identify classes of marginal targets for which these guidelines are violated, obtained by perturbing a standard normal density (at the level of the potential for RWM and the second derivative of the potential for MALA) using roughness generated by a path of fractional Brownian motion with Hurst exponent H . For such targets there is strong evidence that RWM and MALA proposal variances should optimally be scaled as n − 1 H and as n − 1 2+H and will then obey anomalous acceptance rate guidelines. Useful heuristics resulting from this theory are discussed. The paper develops a framework capable of tackling optimal scaling results for quite general Metropolis-Hastings algorithms (possibly depending on a random environment).

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Noise-induced transitions in rugged energy landscapes

Cited by 10 publications

References 61 publications

Mean Field Limits for Interacting Diffusions in a Two-Scale Potential

Mean Field Limits for Interacting Diffusions in a Two-Scale Potential

Spectral Methods for Multiscale Stochastic Differential Equations

Counterexamples for optimal scaling of Metropolis–Hastings chains with rough target densities

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