arXiv subjects

Kenneth Uda

Publications and source records attributed to Kenneth Uda.

6 recordsLinked to original sources

arXiv2020

Time-periodic measures, random periodic orbits, and the linear response for dissipative non-autonomous stochastic differential equations

We consider a class of dissipative stochastic differential equations (SDE's) with time-periodic coefficients in finite dimension, and the response of time-asymptotic probability measures induced by such SDE's to sufficiently regular, small perturbations of the underlying dynamics. Understanding such a response provides a systematic way to study changes of statistical observables in response to perturbations, and it is often very useful for sensitivity analysis, uncertainty quantification, and for improving probabilistic predictions of nonlinear dynamical systems, especially in high dimensions. Here, we are concerned with the linear response to small perturbations in the case when the time-asymptotic probability measures are time-periodic. First, we establish sufficient conditions for the existence of stable random time-periodic orbits generated by the underlying SDE. Ergodicity of time-periodic probability measures supported on these random periodic orbits is subsequently discussed. Then, we derive the so-called fluctuation-dissipation relations which allow to describe the linear response of statistical observables to small perturbations away from the time-periodic ergodic regime in a manner which only exploits the unperturbed dynamics. The results are formulated in an abstract setting but they apply to problems ranging from aspects of climate modelling, to molecular dynamics, to the study of approximation capacity of neural~networks and robustness of their estimates.

math.PR↗

arXiv2019

Lagrangian uncertainty quantification and information inequalities for stochastic flows

We develop a systematic information-theoretic framework for quantification and mitigation of error in probabilistic Lagrangian (i.e., path-based) predictions which are obtained from dynamical systems generated by uncertain (Eulerian) vector fields. This work is motivated by the desire to improve Lagrangian predictions in complex dynamical systems based either on analytically simplified or data-driven models. We derive a hierarchy of general information bounds on uncertainty in estimates of statistical observables $\mathbb{E}^{\nu}[f]$, evaluated on trajectories of the approximating dynamical system, relative to the "true'' observables $\mathbb{E}^{\mu}[f]$ in terms of certain $\varphi$-divergences, $\mathcal{D}_\varphi(\mu\|\nu)$, which quantify discrepancies between probability measures $\mu$ associated with the original dynamics and their approximations $\nu$. We then derive two distinct bounds on $\mathcal{D}_\varphi(\mu\|\nu)$ itself in terms of the Eulerian fields. This new framework provides a rigorous way for quantifying and mitigating uncertainty in Lagrangian predictions due to Eulerian model error.

math.PR↗

arXiv2018

Averaging principle for stochastic differential equations in the random periodic regime

We present the validity of stochastic averaging principle for non-autonomous slow-fast stochastic differential equations (SDEs) whose fast motions admit random periodic solutions. Our investigation is motivated by some problems arising from multi-scale stochastic dynamical systems, where configurations are time dependent due to nonlinearity of the underlying vector fields and the onset of time dependent random invariant sets. Averaging principle with respect to uniform ergodicity of the fast motion is no longer available in this scenario. Lyapunov second method together with synchronous coupling and strong Feller property of Markovian flows of SDEs are used to prove the ergodicity of time periodic measures of the fast motion on certain minimal Poincare section and consequently identify the averaging limit.

math.PR↗

arXiv2018

Random periodic solutions and ergodicity for stochastic differential equations

In this paper, we establish some sufficient conditions for the existence of stable random periodic solutions of stochastic differential equations and ergodicity in the random periodic regime. The techniques involve the existence of Lyapunov type function, using two-point generator of the stochastic flow map, strong Feller argument and weak convergence.

math.PR↗

arXiv2018

Measures of path-based nonlinear expansion rates and Lagrangian uncertainty in stochastic flows

We develop a probabilistic characterisation of trajectorial expansion rates in non-autonomous stochastic dynamical systems that can be defined over a finite time interval and used for the subsequent uncertainty quantification in Lagrangian (trajectory-based) predictions. These expansion rates are quantified via certain divergences (pre-metrics) between probability measures induced by the laws of the stochastic flow associated with the underlying dynamics. We construct scalar fields of finite-time divergence/expansion rates, show their existence and space-time continuity for general stochastic flows. Combining these divergence rate fields with our 'information inequalities' derived in allows for quantification and mitigation of the uncertainty in path-based observables estimated from simplified models in a way that is amenable to algorithmic implementations, and it can be utilised in information-geometric analysis of statistical estimation and inference, as well as in a data-driven machine/deep learning of coarse-grained models. We also derive a link between the divergence rates and finite-time Lyapunov exponents for probability measures and for path-based observables.

math.DS↗

arXiv2015

Existence of random invariant periodic curves via random semiuniform ergodic theorem

We employ an extension of ergodic theory to the random setting to investigate the existence of random periodic solutions of random dynamical systems. Given that a random dynamical system has a dissipative structure, we proved that a random invariant compact set can be expressed as a union of finite of number of random periodic curves. The idea in this paper is closely related to the work recently considered by Zhao and Zheng [46].

math.PR↗