scholarly journals Spectral Properties of Effective Dynamics from Conditional Expectations

Entropy ◽  
2021 ◽  
Vol 23 (2) ◽  
pp. 134
Author(s):  
Feliks Nüske ◽  
Péter Koltai ◽  
Lorenzo Boninsegna ◽  
Cecilia Clementi
Keyword(s):  
Stochastic Dynamics ◽  
Diffusion Processes ◽  
Time Window ◽  
General Procedure ◽  
Approximation Error ◽  
High Dimensional ◽  
Effective Dynamics ◽  
Conditional Expectations ◽  
Conditioning Approach ◽  
Effective Models

The reduction of high-dimensional systems to effective models on a smaller set of variables is an essential task in many areas of science. For stochastic dynamics governed by diffusion processes, a general procedure to find effective equations is the conditioning approach. In this paper, we are interested in the spectrum of the generator of the resulting effective dynamics, and how it compares to the spectrum of the full generator. We prove a new relative error bound in terms of the eigenfunction approximation error for reversible systems. We also present numerical examples indicating that, if Kramers–Moyal (KM) type approximations are used to compute the spectrum of the reduced generator, it seems largely insensitive to the time window used for the KM estimators. We analyze the implications of these observations for systems driven by underdamped Langevin dynamics, and show how meaningful effective dynamics can be defined in this setting.

Jump Diffusion Inference in Complex Scenes

2006 ◽  
Author(s):  
Ulf Grenander ◽  
Michael I. Miller
Keyword(s):  
Markov Process ◽  
Parameter Space ◽  
Diffusion Processes ◽  
Jump Process ◽  
Jump Diffusion ◽  
Jump Diffusion Processes ◽  
Transition Distribution ◽  
Conditional Expectations ◽  
Sampling Algorithms ◽  
Lie Manifolds

This chapter explores random sampling algorithms introduced in for generating conditional expectations in hypothesis spaces in which there is a mixture of discrete, disconnected subsets. Random samples are generated via the direct simulation of a Markov process whose state moves through the hypothesis space with the ergodic property that the transition distribution of the Markov process converges to the posterior distribution. This allows for the empirical generation of conditional expectations under the posterior. To accommodate the connected and disconnected nature of the state spaces, the Markov process is forced to satisfy jump–diffusion dynamics. Through the connected parts of the parameter space (Lie manifolds) the algorithm searches continuously, with sample paths corresponding to solutions of standard diffusion equations. Across the disconnected parts of parameter space the jump process determines the dynamics. The infinitesimal properties of these jump–diffusion processes are selected so that various sample statistics converge to their expectation under the posterior.

scholarly journals Publisher Correction: Potential landscape of high dimensional nonlinear stochastic dynamics with large noise

2019 ◽  
Vol 9 (1) ◽  
Author(s):  
Ying Tang ◽  
Ruoshi Yuan ◽  
Gaowei Wang ◽  
Xiaomei Zhu ◽  
Ping Ao
Keyword(s):  
Stochastic Dynamics ◽  
High Dimensional ◽  
Nonlinear Stochastic Dynamics ◽  
Potential Landscape ◽  
Correction Potential

scholarly journals Dynamical phase diagram of an auto-regulating gene in fast switching conditions

2020 ◽  
Author(s):  
Chen Jia ◽  
Ramon Grima
Keyword(s):  
Steady State ◽  
Stochastic Dynamics ◽  
Deterministic Model ◽  
Time Window ◽  
Burst Size ◽  
Regulatory Gene ◽  
Initial Distribution ◽  
Observation Time ◽  
Time Dependent ◽  
Protein Distribution

AbstractWhile the steady-state behavior of stochastic gene expression with auto-regulation has been extensively studied, its time-dependent behavior has received much less attention. Here, under the assumption of fast promoter switching, we derive and solve a reduced chemical master equation for an auto-regulatory gene circuit with translational bursting and cooperative protein-gene interactions. The analytical expression for the time-dependent probability distribution of protein numbers enables a fast exploration of large swaths of parameter space. For a unimodal initial distribution, we identify three distinct types of stochastic dynamics: (i) the protein distribution remains unimodal at all times; (ii) the protein distribution becomes bimodal at intermediate times and then reverts back to being unimodal at long times (transient bimodality) and (iii) the protein distribution switches to being bimodal at long times. For each of these, the deterministic model predicts either monostable or bistable behaviour and hence there exist six dynamical phases in total. We investigate the relationship of the six phases to the transcription rates, the protein binding and unbinding rates, the mean protein burst size, the degree of cooperativity, the relaxation time to the steady state, the protein mean and the type of feedback loop (positive or negative). We show that transient bimodality is a noise-induced phenomenon that occurs when protein expression is sufficiently bursty and we use theory to estimate the observation time window when it is manifest.

scholarly journals Solving high-dimensional Hamilton–Jacobi–Bellman PDEs using neural networks: perspectives from the theory of controlled diffusions and measures on path space

2021 ◽  
Vol 2 (4) ◽  
Author(s):  
Nikolas Nüsken ◽  
Lorenz Richter
Keyword(s):  
Diffusion Processes ◽  
Rare Event ◽  
Central Element ◽  
High Dimensional ◽  
Diffusion Control ◽  
Parabolic Pdes ◽  
Bellman Equations ◽  
Nonlinear Parabolic Pdes ◽  
Controlled Diffusions ◽  
Hamilton Jacobi Bellman

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.

scholarly journals Perturbation-based inference for diffusion processes: Obtaining effective models from multiscale data

2018 ◽  
Vol 28 (08) ◽  
pp. 1565-1597
Author(s):  
Sebastian Krumscheid
Keyword(s):  
Diffusion Processes ◽  
Point Of View ◽  
Coarse Grained ◽  
Stability And Convergence ◽  
Estimation Of Parameters ◽  
Inference Procedure ◽  
Inference Problem ◽  
Discrete Time Observations ◽  
Effective Models ◽  
Analysis Point

We consider the inference problem for parameters in stochastic differential equation (SDE) models from discrete time observations (e.g. experimental or simulation data). Specifically, we study the case where one does not have access to observations of the model itself, but only to a perturbed version that converges weakly to the solution of the model. Motivated by this perturbation argument, we study the convergence of estimation procedures from a numerical analysis point of view. More precisely, we introduce appropriate consistency, stability, and convergence concepts and study their connection. It turns out that standard statistical techniques, such as the maximum likelihood estimator, are not convergent methodologies in this setting, since they fail to be stable. Due to this shortcoming, we introduce and analyse a novel inference procedure for parameters in SDE models which turns out to be convergent. As such, the method is particularly suited for the estimation of parameters in effective (i.e. coarse-grained) models from observations of the corresponding multiscale process. We illustrate these theoretical findings via several numerical examples.

scholarly journals Large deviations for metastable states of Markov processes with absorbing states with applications to population models in stable or randomly switching environment

2022 ◽  
Vol 2022 (1) ◽  
pp. 013206
Author(s):  
Cécile Monthus
Keyword(s):  
Large Deviations ◽  
Markov Processes ◽  
Large Time ◽  
Diffusion Processes ◽  
Large Deviation ◽  
Time Window ◽  
Extinction Rate ◽  
Empirical Density ◽  
Full Optimization ◽  
Absorbing States

Abstract The large deviations at level 2.5 are applied to Markov processes with absorbing states in order to obtain the explicit extinction rate of metastable quasi-stationary states in terms of their empirical time-averaged density and of their time-averaged empirical flows over a large time-window T. The standard spectral problem for the slowest relaxation mode can be recovered from the full optimization of the extinction rate over all these empirical observables and the equivalence can be understood via the Doob generator of the process conditioned to survive up to time T. The large deviation properties of any time-additive observable of the Markov trajectory before extinction can be derived from the level 2.5 via the decomposition of the time-additive observable in terms of the empirical density and the empirical flows. This general formalism is described for continuous-time Markov chains, with applications to population birth–death model in a stable or in a switching environment, and for diffusion processes in dimension d.

scholarly journals Effective dynamics using conditional expectations

Nonlinearity ◽  
2010 ◽  
Vol 23 (9) ◽  
pp. 2131-2163 ◽  
Author(s):  
Frédéric Legoll ◽  
Tony Lelièvre
Keyword(s):  
Effective Dynamics ◽  
Conditional Expectations

Accelerating Stochastic Dynamics Simulation With Continuous-Time Quantum Walks

2016 ◽  
Author(s):  
Yan Wang
Keyword(s):  
Path Integral ◽  
Continuous Time ◽  
Stochastic Dynamics ◽  
Diffusion Processes ◽  
Dimensional Space ◽  
Planck Equation ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Dynamics Simulation ◽  
Time Quantum

Stochastic diffusion is a general phenomenon observed in various national and engineering systems. It is typically modeled by either stochastic differential equation (SDE) or Fokker-Planck equation (FPE), which are equivalent approaches. Path integral is an accurate and effective method to solve FPEs. Yet, computational efficiency is the common challenge for path integral and other numerical methods, include time and space complexities. Previously, one-dimensional continuous-time quantum walk was used to simulate diffusion. By combining quantum diffusion and random diffusion, the new approach can accelerate the simulation with longer time steps than those in path integral. It was demonstrated that simulation can be dozens or even hundreds of times faster. In this paper, a new generic quantum operator is proposed to simulate drift-diffusion processes in high-dimensional space, which combines quantum walks on graphs with traditional path integral approaches. Probability amplitudes are computed efficiently by spectral analysis. The efficiency of the new method is demonstrated with stochastic resonance problems.

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