arXiv subjects

Jinqiao Duan

Publications and source records attributed to Jinqiao Duan.

At least 127 records · Page 7Linked to original sources

arXiv2020

Effective Wave Factorization for a Stochastic Schrödinger Equation

We study the homogenization of a stochastic Schrödinger equation with a large periodic potential in solid state physics. Denoting by $\varepsilon$ the period, the potential is scaled as $\varepsilon^{-2}$. Under a generic assumption on the spectral properties of the associated cell problem, we prove that the solution can be approximately factorized as the product of a fast oscillating cell eigenfunction and of a slowly varying solution of an effective equation. Our method is based on two-scale convergence and Bloch waves theory.

math.AP↗

arXiv2020

The tipping times in an Arctic sea ice system under influence of extreme events

In light of the rapid recent retreat of Arctic sea ice, the extreme weather events triggering the variability in Arctic ice cover has drawn increasing attention. A non-Gaussian $α$-stable Lévy process is thought to be an appropriate model to describe such extreme event. The maximal likely trajectory, based on the nonlocal Fokker-Planck equation, is applied to a nonautonomous Arctic sea ice system under $α$-stable Lévy noise. Two types of tipping times, the early-warning tipping time and the disaster-happening tipping time, are used to predict the critical time for the maximal likely transition from a perennially ice-covered state to a seasonally ice-free one, and from a seasonally ice-free state to a perennially ice-free one, respectively. We find that the increased intensity of extreme events results in shorter warning time for sea ice melting, and that an enhanced greenhouse effect will intensify this influence, making the arrival of warning time significantly earlier. Meanwhile, for the enhanced greenhouse effect, we discover that increased intensity and frequency of extreme events will advance the disaster-happening tipping time, in which an ice-free state is maintained throughout the year in the Arctic Ocean. Finally, we identify values of Lévy index $α$ and noise intensity $ε$ in $αε$-space that can trigger a transition between the Arctic sea ice state. These results provide an effective theoretical framework for studying Arctic sea ice variations under the influence of extreme events.

physics.ao-ph↗

arXiv2020

Detecting the maximum likelihood transition path from data of stochastic dynamic systems

In recent years, the discovery of complex dynamic systems in various fields through data-driven methods has attracted widespread attention. This method has played the role of data and has become an advantageous tool for us to study complex phenomena. In this work, we propose a framework for detecting the dynamic behavior, such as the maximum likelihood transition path, of stochastic dynamic systems from data. For the stochastic dynamic system, we need to use the Kramers-Moyal formula to convert it into a deterministic form for processing, then use the extended SINDy method to obtain the parameters of stochastic dynamic systems, and finally calculate the maximum likelihood transition path. We give two examples of stochastic dynamical systems driven by additive and multiplicative Gaussian noise, and demonstrate the validity of the method by reproducing the known dynamical system behavior.

math.DS↗

arXiv2020

An averaging principle for fractional stochastic differential equations with Lévy noise

This paper is devoted to the study of an averaging principle for fractional stochastic differential equations in Rnwith Lévy motion, using an integral transform method. We obtain a time-averaged equation under suitable assumptions. Furthermore, we show that the solutions of averaged equation approach the solutions of the original equation. Our results in this paper provide better understanding for effective approximation of fractional dynamical systems with non-Gaussian Lévy noise.

math.PR↗

arXiv2020

Numerical analysis and applications of Fokker-Planck equations for stochastic dynamical systems with multiplicative $α$-stable noises

The Fokker-Planck equations (FPEs) for stochastic systems driven by additive symmetric $α$-stable noises may not adequately describe the time evolution for the probability densities of solution paths in some practical applications, such as hydrodynamical systems, porous media, and composite materials. As a continuation of previous works on additive case, the FPEs for stochastic dynamical systems with multiplicative symmetric $α$-stable noises are derived by the adjoint operator method, which satisfy the nonlocal partial differential equations. A finite difference method for solving the nonlocal Fokker-Planck equation (FPE) is constructed, which is shown to satisfy the discrete maximum principle and to be convergent. Moreover, an example is given to illustrate this method. For asymmetric case, general finite difference schemes are proposed, and some analyses of the corresponding numerical schemes are given. Furthermore, the corresponding result is successfully applied to the nonlinear filtering problem.

math.DS↗

arXiv2020

Most Probable Dynamics of Stochastic Dynamical Systems with Exponentially Light Jump Fluctuations

The emergence of the exit events from a bounded domain containing a stable fixed point induced by non-Gaussian Lévy fluctuations plays a pivotal role in practical physical systems. In the limit of weak noise, we develop a Hamiltonian formalism under the Lévy fluctuations with exponentially light jumps for one- and two-dimensional stochastic dynamical systems. This formalism is based on a recently proved large deviation principle for dynamical systems under non-Gaussian Lévy perturbations. We demonstrate how to compute the most probable exit path and the quasi-potential by several examples. Meanwhile, we explore the impacts of the jump measure on the quasi-potential quantitatively and on the most probable exit path qualitatively. Results show that the quasi-potential can be well estimated by an approximate analytical expression. Moreover, we discover that although the most probable exit paths are analogous to the Gaussian case for the isotropic noise, the anisotropic noise leads to significant changes of the structure of the exit paths. These findings shed light on the underlying qualitative mechanism and quantitative feature of the exit phenomenon induced by non-Gaussian noise.

math.ST↗

arXiv2020

The role of slow manifolds in parameter estimation for a multiscale stochastic system with $α$-stable Lévy noise

This work is about parameter estimation for a fast-slow stochastic system with non-Gaussian $α$-stable Lévy noise. When the observations are only available for slow components, a system parameter is estimated and the accuracy for this estimation is quantified by $p$-moment with $p\in(1, α)$, with the help of a reduced system through random slow manifold approximation. This method provides an advantage in computational complexity and cost, due to the dimension reduction in stochastic systems. To numerically illustrate this method, and to corroborate that the parameter estimator based on the reduced slow system is a good approximation for the true parameter value of the original system, a prototypical example is present.

math.DS↗

arXiv2020

Recurrent Solutions of a Nonautonomous Modified Swift-Hohenberg Equation

We consider recurrent solutions of the nonautonomous modified Swift-Hohenberg equation $$u_t+Δ^2u+2Δu+au+b|\nabla u|^2+u^3=g(t,x).$$ We employ Conley index theory to show that, if the forcing $g:\mathbb{R}\rightarrow L^2(Ω)$ is a recurrent function, then there are at least two recurrent solutions in $H_0^2(Ω)$ under appropriate assumptions on the parameters $a$, $b$ and $g$.

math.AP↗

arXiv2020

Maximal Likely Phase Lines for a Reduced Ice Growth Model

We study the impact of Brownian noise on transitions between metastable equilibrium states in a stochastic ice sheet model. Two methods to accomplish different objectives are employed. The maximal likely trajectory by maximizing the probability density function and numerically solving the Fokker-Planck equation shows how the system will evolve over time. We have especially studied the maximal likely trajectories starting near the ice-free metastable state, and examined whether they evolve to or near the ice-covered metastable state for certain parameters, in order to gain insights into how the ice sheet formed. Furthermore, for the transition from ice-covered metastable state to the ice-free metastable state, we study the most probable path for various noise parameters via the Onsager-Machlup least action principle. This enables us to predict and visualize the melting process of the ice sheet if such a rare event ever does take place.

math.DS↗

arXiv2020

Maximum Likelihood Estimation of Stochastic Differential Equations with Random Effects Driven by Fractional Brownian Motion

Stochastic differential equations and stochastic dynamics are good models to describe stochastic phenomena in real world. In this paper, we study N independent stochastic processes Xi(t) with real entries and the processes are determined by the stochastic differential equations with drift term relying on some random effects. We obtain the Girsanov-type formula of the stochastic differential equation driven by Fractional Brownian Motion through kernel transformation. Under some assumptions of the random effect, we estimate the parameter estimators by the maximum likelihood estimation and give some numerical simulations for the discrete observations. Results show that for the different H, the parameter estimator is closer to the true value as the amount of data increases.

math.ST↗

arXiv2019

Global Solution and Blow-up of the Stochastic Nonlinear Schrödinger system

We study the stochastic Nonlinear Schrödinger system with multiplicative white noise in energy space $H^1$. Based on deterministic and stochastic Strichartz estimates, we prove the local well-posedness and uniqueness of mild solution. Then we prove the global well-posedness in the mass subcritical case and the defocusing case. For the mass subcritical case, we also investigate the global existence when the $L^2$ norm of initial value is small enough. In addition, we study the blow-up phenomenon and give a sharp criteria.

math.AP↗

arXiv2019

The persistence of synchronization under $α$-stable noise

This work is about the synchronization of nonlinear coupled dynamical systems driven by $α$-stable noise. Firstly, we provide a novel technique to construct the relationship between synchronized system and slow-fast system. Secondly, we show that the slow component of original systems converges to the mild solution of the averaging equation under $L^{p}(1<p<α)$ sense. Finally, using the results of averaging principle for stochastic dynamical system with two-time scales, we show that the synchronization effect is persisted provided equilibria are replaced by stationary random solutions.

math.PR↗

arXiv2019

Mean exit time for stochastic dynamical systems driven by tempered stable Lévy fluctuations

We use the mean exit time to quantify macroscopic dynamical behaviors of stochastic dynamical systems driven by tempered Lévy fluctuations, which are solutions of nonlocal elliptic equations. Firstly, we construct a new numerical scheme to compute and solve the mean exit time associated with the one dimensional stochastic system. Secondly, we extend the analytical and numerical results to two dimensional case: horizontal-vertical and isotropic case. Finally, we verify the effectiveness of the presented schemes with numerical experiments in several examples.

math.DS↗

arXiv2019

Learning and Meta-Learning of Stochastic Advection-Diffusion-Reaction Systems from Sparse Measurements

Physics-informed neural networks (PINNs) were recently proposed in [1] as an alternative way to solve partial differential equations (PDEs). A neural network (NN) represents the solution while a PDE-induced NN is coupled to the solution NN, and all differential operators are treated using automatic differentiation. Here, we first employ the standard PINN and a stochastic version, sPINN, to solve forward and inverse problems governed by a nonlinear advection-diffusion-reaction (ADR) equation, assuming we have some sparse measurements of the concentration field at random or pre-selected locations. Subsequently, we attempt to optimize the hyper-parameters of sPINN by using the Bayesian optimization method (meta-learning), and compare the results with the empirically selected hyper-parameters of sPINN.

physics.comp-ph↗

arXiv2019

The maximum likelihood climate change for global warming under the influence of greenhouse effect and Lévy noise

An abrupt climatic transition could be triggered by a single extreme event, an $α$-stable non-Gaussian Lévy noise is regarded as a type of noise to generate such extreme events. In contrast with the classic Gaussian noise, a comprehensive approach of the most probable transition path for systems under $α$-stable Lévy noise is still lacking. We develop here a probabilistic framework, based on the nonlocal Fokker-Planck equation, to investigate the maximum likelihood climate change for an energy balance system under the influence of greenhouse effect and Lévy fluctuations. We find that a period of the cold climate state can be interrupted by a sharp shift to the warmer one due to larger noise jumps, and the climate change for warming $1.5\rm ^oC$ under an enhanced greenhouse effect generates a step-like growth process. These results provide important insights into the underlying mechanisms of abrupt climate transitions triggered by a Lévy process.

cond-mat.stat-mech↗

arXiv2019

Discovering mean residence time and escape probability from data of stochastic dynamical systems

We present a method to learn mean residence time and escape probability from data modeled by stochastic differential equations. This method is a combination of machine learning from data (to extract stochastic differential equations as models) and stochastic dynamics (to quantify dynamical behaviors with deterministic tools). The goal is to learn and understand stochastic dynamics based on data. This method is applicable to sample path data collected from complex systems, as long as these systems can be modeled as stochastic differential equations.

math.DS↗

arXiv2019

Action functionals for stochastic differential equations with Lévy noise

By using large deviation theory that deals with the decay of probabilities of rare events on an exponential scale, we study the longtime behaviors and establish action functionals for scaled Brownian motion and Lévy processes with existing finite exponential moments. Based on extended contraction principle, Legendre transform and Lévy symbols, we derive the action functionals for stochastic differential equations driven by Lévy processes.

math.DS↗

arXiv2019

Compactly Generated Shape Index Theory and its Application to a Retarded Nonautonomous Parabolic Equation

We establish the compactly generated shape (H-shape) index theory for local semiflows on complete metric spaces via more general shape index pairs, and define the H-shape cohomology index to develop the Morse equations. The main advantages are that the quotient space $N/E$ is not necessarily metrizable for the shape index pair $(N,E)$ and $N\sm E$ need not to be a neighborhood of the compact invariant set. Moreover, in this new theory, the phase space is not required to be separable. We apply H-shape index theory to an abstract retarded nonautonomous parabolic equation to obtain the existence of bounded full solutions.

math.DS↗