Discovery of rare event testing for stochastic simulations of diffusion processes

AbstractOptimal control of diffusion processes is intimately connected to the problem of solving certain Hamilton–Jacobi–Bellman equations. Building on recent machine learning inspired approaches towards high-dimensional PDEs, we investigate the potential of iterative diffusion optimisation techniques, in particular considering applications in importance sampling and rare event simulation, and focusing on problems without diffusion control, with linearly controlled drift and running costs that depend quadratically on the control. More generally, our methods apply to nonlinear parabolic PDEs with a certain shift invariance. The choice of an appropriate loss function being a central element in the algorithmic design, we develop a principled framework based on divergences between path measures, encompassing various existing methods. Motivated by connections to forward-backward SDEs, we propose and study the novel log-variance divergence, showing favourable properties of corresponding Monte Carlo estimators. The promise of the developed approach is exemplified by a range of high-dimensional and metastable numerical examples.

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Reactive boundary conditions for stochastic simulations of reaction–diffusion processes

Physical Biology ◽

10.1088/1478-3975/4/1/003 ◽

2007 ◽

Vol 4 (1) ◽

pp. 16-28 ◽

Cited By ~ 111

Author(s):

Radek Erban ◽

S Jonathan Chapman

Keyword(s):

Boundary Conditions ◽

Diffusion Processes ◽

Reaction Diffusion ◽

Stochastic Simulations

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Time-averaging and emerging nonergodicity upon resetting of fractional Brownian motion and heterogeneous diffusion processes

10.1101/2021.04.28.441681 ◽

2021 ◽

Author(s):

Wei Wang ◽

Andrey G. Cherstvy ◽

Holger Kantz ◽

Ralf Metzler ◽

Igor M. Sokolov

Keyword(s):

Brownian Motion ◽

Fractional Brownian Motion ◽

Diffusion Processes ◽

Stochastic Simulations ◽

Time Averaging ◽

Weak Ergodicity ◽

Experimental Systems ◽

Long Time ◽

Breaking Parameter ◽

Ergodicity Breaking

How different are the results of constant-rate resetting of anomalous-diffusion processes in terms of their ensemble-averaged versus time-averaged mean-squared displacements (MSDs versus TAMSDs) and how does the process of stochastic resetting impact nonergodicity? These are the main questions addressed in this study. Specifically, we examine, both analytically and by stochastic simulations, the implications of resetting on the MSD-and TAMSD-based spreading dynamics of fractional Brownian motion (FBM) with a long-time memory, of heterogeneous diffusion processes (HDPs) with a power-law-like space-dependent diffusivity D(x) = D0 |x| γ, and of their “combined” process of HDP-FBM. We find, i.a., that the resetting dynamics of originally ergodic FBM for superdiffusive choices of the Hurst exponent develops distinct disparities in the scaling behavior and magnitudes of the MSDs and mean TAMSDs, indicating so-called weak ergodicity breaking (WEB). For subdiffusive HDPs we also quantify the nonequivalence of the MSD and TAMSD, and additionally observe a new trimodal form of the probability density function (PDF) of particle’ displacements. For all three reset processes (FBM, HDPs, and HDP-FBM) we compute analytically and verify by stochastic computer simulations the short-time (normal and anomalous) MSD and TAMSD asymptotes (making conclusions about WEB) as well as the long-time MSD and TAMSD plateaus, reminiscent of those for “confined” processes. We show that certain characteristics of the reset processes studied are functionally similar, despite the very different stochastic nature of their nonreset variants. Importantly, we discover nonmonotonicity of the ergodicity breaking parameter EB as a function of the resetting rate r. For all the reset processes studied, we unveil a pronounced resetting-induced nonergodicity with a maximum of EB at intermediate r and EB ∼ (1/r)-decay at large r values. Together with the emerging MSD-versus-TAMSD disparity, this pronounced r-dependence of the EB parameter can be an experimentally testable prediction. We conclude via discussing some implications of our results to experimental systems featuring resetting dynamics.

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Rare event simulation for diffusion processes via two-stage importance sampling

Monte Carlo Methods and Applications ◽

10.1515/mcma-2013-0019 ◽

2014 ◽

Vol 20 (2) ◽

Author(s):

Adam Metzler ◽

Alexandre Scott

Keyword(s):

Brownian Motion ◽

Importance Sampling ◽

Diffusion Processes ◽

Practical Importance ◽

Rare Event ◽

Region Of Interest ◽

Sampling Procedure ◽

Importance Measure ◽

Standard Brownian Motion ◽

Two Stage

Abstract.We consider the problem of estimating expected values of functionals of real-valued diffusions over regions in path space that have very small probability. We propose a two-stage importance sampling procedure that first converts the problem into one involving standard Brownian motion and then addresses the rare event problem in this simpler setting. In order to identify an effective yet practical importance measure we propose using a time-dependent deterministic drift that minimizes the relative entropy between the corresponding importance measure and the conditional law of the standard Brownian motion, given that its trajectory lies in the region of interest. We provide numerical evidence that (i) our entropy-based criteria performs favourably with an alternative, but less general and less practical, criteria based on large deviations and (ii) our two-stage procedure performs admirably in cases where the region of interest is so rare that crude estimators fail completely.

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Multiresolution stochastic simulations of reaction–diffusion processes

Physical Chemistry Chemical Physics ◽

10.1039/b810795e ◽

2008 ◽

Vol 10 (39) ◽

pp. 5963 ◽

Cited By ~ 3

Author(s):

Basil Bayati ◽

Philippe Chatelain ◽

Petros Koumoutsakos

Keyword(s):

Diffusion Processes ◽

Reaction Diffusion ◽

Stochastic Simulations

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Analytical approximations for spatial stochastic gene expression in single cells and tissues

Journal of The Royal Society Interface ◽

10.1098/rsif.2015.1051 ◽

2016 ◽

Vol 13 (118) ◽

pp. 20151051 ◽

Cited By ~ 16

Author(s):

Stephen Smith ◽

Claudia Cianci ◽

Ramon Grima

Keyword(s):

Gene Expression ◽

Diffusion Coefficients ◽

Brownian Dynamics ◽

Diffusion Processes ◽

Single Cells ◽

Reaction Diffusion ◽

Intrinsic Noise ◽

Stochastic Simulations ◽

Analytical Approximations ◽

Diffusive Effects

Gene expression occurs in an environment in which both stochastic and diffusive effects are significant. Spatial stochastic simulations are computationally expensive compared with their deterministic counterparts, and hence little is currently known of the significance of intrinsic noise in a spatial setting. Starting from the reaction–diffusion master equation (RDME) describing stochastic reaction–diffusion processes, we here derive expressions for the approximate steady-state mean concentrations which are explicit functions of the dimensionality of space, rate constants and diffusion coefficients. The expressions have a simple closed form when the system consists of one effective species. These formulae show that, even for spatially homogeneous systems, mean concentrations can depend on diffusion coefficients: this contradicts the predictions of deterministic reaction–diffusion processes, thus highlighting the importance of intrinsic noise. We confirm our theory by comparison with stochastic simulations, using the RDME and Brownian dynamics, of two models of stochastic and spatial gene expression in single cells and tissues.

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Stochastic Simulations of Diffusion Processes

Diffusion in Random Fields - Geosystems Mathematics ◽

10.1007/978-3-030-15081-5_3 ◽

2019 ◽

pp. 47-89

Author(s):

Nicolae Suciu

Keyword(s):

Diffusion Processes ◽

Stochastic Simulations

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Meridional Circulation, Turbulence, and Diffusion Processes in Stars

International Astronomical Union Colloquium ◽

10.1017/s0252921100080477 ◽

1976 ◽

Vol 32 ◽

pp. 109-116 ◽

Cited By ~ 1

Author(s):

S. Vauclair

Keyword(s):

Convection Zone ◽

Diffusion Processes ◽

Meridional Circulation ◽

Diffusion Time ◽

Work In Progress ◽

First Results ◽

Stellar Envelopes ◽

And Diffusion ◽

Turbulence And Diffusion ◽

Hr Diagram

This paper gives the first results of a work in progress, in collaboration with G. Michaud and G. Vauclair. It is a first attempt to compute the effects of meridional circulation and turbulence on diffusion processes in stellar envelopes. Computations have been made for a 2 Mʘstar, which lies in the Am - δ Scuti region of the HR diagram.Let us recall that in Am stars diffusion cannot occur between the two outer convection zones, contrary to what was assumed by Watson (1970, 1971) and Smith (1971), since they are linked by overshooting (Latour, 1972; Toomre et al., 1975). But diffusion may occur at the bottom of the second convection zone. According to Vauclair et al. (1974), the second convection zone, due to He II ionization, disappears after a time equal to the helium diffusion time, and then diffusion may happen at the bottom of the first convection zone, so that the arguments by Watson and Smith are preserved.

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