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A wong-zakai approximation for random slow manifolds with application to parameter estimation

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arXiv 2018年

作者： He, Ziying Zhang, Xinyong Jiang, Tao Liu, Xianming Center for Mathematical Sciences Huazhong University of Sciences and Technology Wuhan430074 China School of Mathematics and Statistics Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Sciences and Technology Wuhan430074 China Department of Mathematical Sciences Tsinghua University Beijing100084 China Collaborative Innovation Center of China Pilot Reform Exploration and Assessment Hubei Sub-Center Hubei University of Economics Wuhan430205 China

We study a Wong-Zakai approximation for the random slow manifold of a slow-fast stochastic dynamical system. We first deduce the existence of the random slow manifold about an approximation system driven by an integrated Ornstein-Uhlenbeck (O-U) process. Then we compute the slow manifold of the approximation system, in order to gain insights of the long time dynamics of the original stochastic system. By restricting this approximation system to its slow manifold, we thus get a reduced slow random system. This reduced slow random system is used to accurately estimate a system parameter of the original system. An example is presented to illustrate this *** Codes Primary: 37L55, 35R60, Secondary: 60H15, 58J65 Copyright © 2018, The Authors. All rights reserved.

关键词： Parameter estimation

A stochastic pitchfork bifurcation in most probable phase portraits

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arXiv 2018年

作者： Wang, Hui Chen, Xiaoli Duan, Jinqiao Center for Mathematical Sciences School of Mathematics and Statistics Hubei Key Laboratory for Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China Department of Applied Mathematics Illinois Institute of Technology ChicagoIL60616 United States Center for Mathematical Sciences Huazhong University of Science and Technology Wuhan430074 China

We study stochastic bifurcation for a system under multiplicative stable Lévy noise (an important class of non-Gaussian noise), by examining the qualitative changes of equilibrium states in its most probable phase portraits. We have found some peculiar bifurcation phenomena in contrast to the deterministic counterpart: (i) When the non-Gaussianity parameter in Lévy noise varies, there is either one, two or none backward pitchfork type bifurcations;(ii) When a parameter in the vector field varies, there are two or three forward pitchfork bifurcations;(iii) The non-Gaussian Lévy noise clearly leads to fundamentally more complex bifurcation scenarios, since in the special case of Gaussian noise, there is only one pitchfork bifurcation which is reminiscent of the deterministic situation. Copyright © 2018, The Authors. All rights reserved.

关键词： Gaussian noise (electronic)

Maxwell-Stefan theory based lattice Boltzmann model for diffusion in multicomponent mixtures

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arXiv 2018年

作者： Chai, Zhenhua Guo, Xiuya Wang, Lei Shi, Baochang School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China School of Mathematics and Physics China University of Geosciences Wuhan430074 China School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China

The phenomena of diffusion in multicomponent (more than two components) mixtures are very universal in both science and engineering, and from mathematical point of view, they are usually described by the Maxwell-Stefan (MS) based continuum equations. In this paper, we propose a multiple-relaxation-time lattice Boltzmann (LB) model for the mass diffusion in multicomponent mixtures, and also perform a Chapman-Enskog analysis to show that the MS based continuum equations can be correctly recovered from the developed LB model. In addition, considering the fact that the MS based continuum equations are just a diffusion type of partial differential equations, we can also adopt much simpler lattice structures to reduce the computational cost of present LB model. We then conduct some simulations to test this model, and find that the results are in good agreement with some available works. Besides, the reverse diffusion, osmotic diffusion and diffusion barrier phenomena are also captured. Finally, compared to the kinetic theory based LB models for multicomponent gas diffusion, the present model does not include any complicated interpolations, and its collision process can be still implemented locally. Therefore, the advantages of single-component LB method can also be preserved in present LB model. Copyright © 2018, The Authors. All rights reserved.

关键词： Mixtures

A role of random slow manifolds in detecting stochastic bifurcation

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arXiv 2018年

作者： He, Ziying Cai, Rui Duan, Jinqiao Liu, Xianming Center for Mathematical Sciences Huazhong University of Sciences and Technology Wuhan430074 China School of Mathematics and Statistics Huazhong University of Sciences and Technology Wuhan430074 China Hubei Key Laboratory for Engineering Modeling and Scientific Computing Huazhong University of Sciences and Technology Wuhan430074 China Department of Applied Mathematics Illinois Institute of Technology ChicagoIL60616 United States

We consider the relation for the stochastic equilibrium states between the reduced system on a random slow manifold and the original system. This provides a theoretical basis for the reduction about sophisticated detailed models by the random slow manifold without significant damage to the overall qualitative properties. Based on this result, we reveal a role of random slow manifolds in detecting stochastic bifurcation by an example. The example exhibits a stochastic bifurcation phenomenon and possesses a random slow manifold that carries the stochastic bifurcation information of the system. Specifically, the lower dimensional reduced system on the random slow manifold retains the stochastic bifurcation phenomenon from the original *** Codes 60H10, 58J65 Copyright © 2018, The Authors. All rights reserved.

关键词： Perturbation techniques

Asymptotic behavior of the principal eigenvalue of a linear second order elliptic operator with small/large diffusion coefficient and its application

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arXiv 2018年

作者： Peng, Rui Zhang, Guanghui Zhou, Maolin School of Mathematics and Statistics Jiangsu Normal University Xuzhou Jiangsu Province221116 China School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China School of Science and Technology University of New England ArmidaleNSW2351 Australia

In this article, we are concerned with the following eigenvalue problem of a linear second order elliptic operator: − D∆φ − 2α∇m(x) · ∇φ + V (x)φ = λφ in Ω, complemented by a general boundary condition including Dirichlet boundary condition and Robin boundary condition: ∂n∂φ + β(x)φ = 0 on ∂Ω, where β ∈ C(∂Ω) allows to be positive, sign-changing or negative, and n(x) is the unit exterior normal to ∂Ω at x. The domain Ω ⊂ RN is bounded and smooth, the constants D > 0 and α > 0 are, respectively, the diffusive and advection coefficients, and m ∈ C2(Ω) ¯ , V ∈ C(Ω) ¯ are given functions. We aim to investigate the asymptotic behavior of the principal eigenvalue of the above eigenvalue problem as the diffusive coefficient D → 0 or D → ∞. Our results, together with those of [4, 5, 10] where the Nuemann boundary case (i.e., β = 0 on ∂Ω) and Dirichlet boundary case were studied, reveal the important effect of advection and boundary conditions on the asymptotic behavior of the principal eigenvalue. We also apply our results to a reaction-diffusion-advection equation which is used to describe the evolution of a single species living in a heterogeneous stream environment and show some interesting behaviors of the species persistence and extinction caused by the buffer zone and small/large diffusion rate. Copyright © 2018, The Authors. All rights reserved.

关键词： Boundary conditions

A lattice Boltzmann model for two-phase flow in porous media

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arXiv 2018年

作者： Chai, Zhenhua Liang, Hong Du, Rui Shi, Baochang School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China State Key Laboratory of Coal Combustion Huazhong University of Science and Technology Wuhan430074 China Department of Physics Hangzhou Dianzi University Hangzhou310018 China School of Mathematics Southeast University Nanjing210096 China

In this paper, a lattice Boltzmann (LB) model with double distribution functions is proposed for two-phase flow in porous media where one distribution function is used for pressure governed by the Poisson equation, and the other is applied for saturation evolution described by the convection-diffusion equation with a source term. We first performed a Chapman-Enskog analysis, and show that the macroscopic nonlinear equations for pressure and saturation can be recovered correctly from present LB model. Then in the framework of LB method, we develop a local scheme for pressure gradient or equivalently velocity, which may be more efficient than the nonlocal second-order finite-difference schemes. We also perform some numerical simulations, and the results show that the developed LB model and local scheme for velocity are accurate and also have a second-order convergence rate in space. Finally, compared to the available pore-scale LB models for two-phase flow in porous media, the present LB model has more potential in the study of the large-scale problems. Copyright © 2018, The Authors. All rights reserved.

关键词： Two phase flow

Most Probable Transition Pathways and Maximal Likely Trajectories in a Genetic Regulatory System

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arXiv 2018年

作者： Cheng, Xiujun Wang, Hui Wang, Xiao Duan, Jinqiao Li, Xiaofan Center for Mathematical Sciences School of Mathematics and Statistics Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China School of Mathematics and Statistics Henan University Kaifeng475000 China Department of Applied Mathematics Illinois Institute of Technology Chicago60616 United States School of Mathematics and Statistics Zhengzhou University Zhengzhou450001 China

We study the most probable transition pathways and maximal likely trajectories in a genetic regulation model of the transcription factor activator’s concentration evolution, with Gaussian noise and non-Gaussian stable Lévy noise in the synthesis reaction rate taking into account, respectively. We compute the most probable transition pathways by the Onsager-Machlup least action principle, and calculate the maximal likely trajectories by spatially maximizing the probability density of the system path, i.e., the solution of the associated nonlocal Fokker-Planck equation. We have observed the rare most probable transition pathways in the case of Gaussian noise, for certain noise intensity, evolution time scale and system parameters. We have especially studied the maximal likely trajectories starting at the low concentration metastable state, and examined whether they evolve to or near the high concentration metastable state (i.e., the likely transcription regime) for certain parameters, in order to gain insights into the transcription processes and the tipping time for the transcription likely to occur. This enables us: (i) to visualize the progress of concentration evolution (i.e., observe whether the system enters the transcription regime within a given time period);(ii) to predict or avoid certain transcriptions via selecting specific noise parameters in particular regions in the parameter space. Moreover, we have found some peculiar or counter-intuitive phenomena in this gene model system, including: (a) A smaller noise intensity may trigger the transcription process, while a larger noise intensity can not, under the same asymmetric Lévy noise. This phenomenon does not occur in the case of symmetric Lévy noise;(b) The symmetric Lévy motion always induces transition to high concentration, but certain asymmetric Lévy motions do not trigger the switch to transcription. These findings provide insights for further experimental research, in order to achieve or to avoid sp

关键词： Trajectories

Anomalous time-scaling of extreme events in infinite systems and birkhoff sums of infinite observables

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arXiv 2018年

作者： Galatolo, Stefano Holland, Mark Persson, Tomas Zhang, Yiwei Dipartimento di Matematica Via Buonattoti 1 Pisa56127 Italy North Park Road EXETEREX4 4QF United Kingdom Centre for Mathematical Sciences Lund University Box 118 Lund221 00 Sweden Scool of Mathematics and Statistics Center for Mathematical Sciences Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Sciences and Technology Wuhan430074 China

We establish quantitative results for the statistical behaviour of infinite systems. We consider two kinds of infinite system: i) a conservative dynamical system (f, X, µ) preserving a σ-finite measure µ such that µ(X) = ∞;ii) the case where µ is a probability measure but we consider the statistical behaviour of an observable φ: X → [0, ∞) which is non-integrable: R φ dµ = ∞. In the first part of this work we study the behaviour of Birkhoff sums of systems of the kind ii). For certain weakly chaotic systems, we show that these sums can be strongly oscillating. However, if the system has superpolynomial decay of correlations or has a Markov structure, then we show this oscillation cannot happen. In this case we prove a general relation between the behavior of φ, the local dimension of µ, and the scaling rate of the growth of Birkhoff sums of φ as time tends to infinity. We then establish several important consequences which apply to infinite systems of the kind i). This includes showing anomalous scalings in extreme event limit laws, or entrance time statistics. We apply our findings to non-uniformly hyperbolic systems preserving an infinite measure, establishing anomalous scalings in the case of logarithm laws of entrance times, dynamical Borel-Cantelli lemmas, almost sure growth rates of extremes, and dynamical run length functions.37A40, 37A50, 37A25, 60G75, 37D25 Copyright © 2018, The Authors. All rights reserved.

关键词： Chaotic systems

A fully discrete energy stable scheme for a phase-field moving contact line model with variabledensities and viscosities

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arXiv 2018年

作者： Zhu, Guangpu Chen, Huangxin Sun, Shuyu Yao, Jun Qingdao266580 China Sch. of Math. Sci. and Fujian Prov. Key Lab. on Math. Modeling and High Perf. Scientific Computing Xiamen University Xiamen361005 China Computational Transport Phenomena Laboratory Division of Physical Science and Engineering King Abdullah University of Science and Technology Thuwal23955-6900 Saudi Arabia

In this work, we propose a fully discrete energy stable scheme for the phase-field moving contact line model with variable densities and viscosities. The mathematical model consists of a Cahn-Hilliard equation, a Navier-Stokes equation and the generalized Navier boundary condition. A scalar auxiliary variable is introduced to transform the nonlinear potential into an equivalent form, allowing the nonlinear potential to be treated effectively and semi-explicitly. A pressure stabilization method is used to decouple the computation of velocity and pressure. Some subtle implicit-explicit treatments are adopted to deal with convention and stress terms. An artificial term is added to balance the explicit nonlinear term originating from the surface energy at fluid-solid interface. We establisharigorous proof of energy stability for the proposed time-marching *** a finite difference methodon staggered grids isusedto spatially discretize the constructed time-marchingscheme. We prove that the fully discrete scheme satisfiesthe discrete energy dissipationlaw. Numerical results demonstrate accuracy and energy stability of the proposed scheme. Using our numerical scheme, we analyze theevolution of kinetic energy, mixing energy and surface energy at fluid-solid interfaceduring the bubble rising process. Three-dimensional droplet spreading is also investigated onachemically patterned surface. Our numerical simulation accurately predicts the expected energy evolutions and it successfully reproduces expected phenomenathat an oil droplet contractsinwardson a hydrophobiczone and spreadsoutwards quickly on a hydrophiliczone. Copyright © 2018, The Authors. All rights reserved.

关键词： Finite difference method