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

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arXiv 2020年

作者： Chao, Ying Wei, Pingyuan Duan, Jinqiao School of Mathematics and Statistics Center for Mathematical Sciences 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

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 ∈ (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. Copyright © 2020, The Authors. All rights reserved.

关键词： Parameter estimation

The non-cutoff Vlasov-Maxwell-Boltzmann system with weak angular singularity

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Science China Mathematics 2018年第1期61卷 111-136页

作者： Yingzhe Fan Yuanjie Lei Shuangqian Liu Huijiang Zhao School of Mathematics and Statistics Wuhan University School of Mathematics and Statistics Huazhong University of Science and Technology Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Department of Mathematics Jinan University Computational Science Hubei Key Laboratory Wuhan University

We establish the global existence of small-amplitude solutions near a global Maxwellian to the Cauchy problem of the Vlasov-Maxwell-Boltzmann system for non-cutoff soft potentials with weak angular singularity. This extends the work of Duan et al.(2013), in which the case of strong angular singularity is considered, to the case of weak angular singularity.

关键词： non-cutoff Vlasov-Maxwell-Boltzmann system global solutions near Maxwellians weak angular singularity time-velocity weighted energy method

Cole-Hopf Transformation Based Lattice Boltzmann Model for One-dimensional Burgers' Equation

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Communications in Theoretical Physics 2018年第3期69卷 329-335页

作者： Xiao-Tong Qi Bao-Chang Shi Zhen-Hua Chai School of Mathematics and Statistics Huazhong University of Science and TechnologyWuhan 430074China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and TechnologyWuhan 430074China

In this paper,we present a Cole-Hopf transformation based lattice Boltzmann(LB)model for solving one-dimensional Burgers'equation,and compared to available LB models,the effect of nonlinear convection term can be *** Chapman-Enskog analysis,it can be found that the converted diffusion equation based on the Cole-Hopf transformation can be recovered correctly from present LB *** numerical tests are also performed to validate the present LB model,and the numerical results show that,similar to previous LB models,the present model also has a second-order convergence rate in space,but it is more accurate than the previous ones.

关键词： Burgers'equation Cole-Hopf transformation lattice Boltzmann model

A multiple-relaxation-time lattice boltzmann model based four-level finite-difference scheme for one-dimensional diffusion equation

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arXiv 2020年

作者： Lin, Yuxin Hong, Ning Shi, Baochang Chai, Zhenhua School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China School of General Education Wuchang University of Technology Wuhan430223 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China

In this paper, we first present a multiple-relaxation-time lattice Boltzmann (MRT-LB) model for one-dimensional diffusion equation where the D1Q3 (three discrete velocities in one-dimensional space) lattice structure is considered. Then through the theoretical analysis, we derive an explicit four-level finite-difference scheme from this MRT-LB model. The results show that the four-level finite-difference scheme is unconditionally stable, and through adjusting the weight coefficient ω0 and the relaxation parameters s1 and s2 corresponding to the first and second moments, it can also have a sixth-order accuracy in space. Finally, we also test the four-level finite-difference scheme through some numerical simulations, and find that the numerical results are consistent with our theoretical analysis. © 2020, CC BY.

关键词： Finite difference method

OBSERVABLE SETS, POTENTIALS AND SCHRÖDINGER EQUATIONS

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arXiv 2020年

作者： Huang, Shanlin Wang, Gengsheng Wang, Ming School of Mathematics and Statistics Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China Center for Applied Mathematics Tianjin University Tianjin300072 China School of Mathematics and Physics China University of Geosciences Wuhan430074 China

We characterize observable sets for 1-dim Schrödinger equations in R: i∂tu = (−∂2x + x2m)u (with m ∈ N:= {0, 1, . . . }). More precisely, we obtain what follows: First, when m = 0, E ⊂ R is an observable set at some time if and only if it is thick, namely, there is γ > 0 and L > 0 so that E \ [x, x + L] ≥ γL for each x ∈ R;Second, when m = 1 (m ≥ 2 resp.), E is an observable set at some time (at any time resp.) if and only if it is weakly thick, namely lim |E T[−x, x]| > 0. x x→+∞ From these, we see how potentials x2m affect the observability (including the geometric structures of observable sets and the minimal observable time). Besides, we obtain several supplemental theorems for the above results, in particular, we find that a half line is an observable set at time T > 0 for the above equation with m = 1 if and only if T > π2 .MSC Codes 93B07, 35J10 Copyright © 2020, The Authors. All rights reserved.

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Characterizations of stabilizable sets for some parabolic equations in Rn

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arXiv 2020年

We consider the parabolic type equation in Rn: (∂t + H)y(t, x) = 0, (t, x) ∈ (0, ∞) × Rn;y(0, x) ∈ L2(Rn), (0.1) where H can be one of the following operators: (i) a shifted fractional Laplacian;(ii) a shifted Hermite operator;(iii) the Schrödinger operator with some general potentials. We call a subset E ⊂ Rn as a stabilizable set for (0.1), if there is a linear bounded operator K on L2(Rn) so that the semigroup {e−t(H−χEK)}t≥0 is exponentially stable. (Here, χE denotes the characteristic function of E, which is treated as a linear operator on L2(Rn).) This paper presents different geometric characterizations of the stabilizable sets for (0.1) with different H. In particular, when H is a shifted fractional Laplacian, E ⊂ Rn is a stabilizable set for (0.1) if and only if E ⊂ Rn is a thick set, while when H is a shifted Hermite operator, E ⊂ Rn is a stabilizable set for (0.1) if and only if E ⊂ Rn is a set of positive measure. Our results, together with the results on the observable sets for (0.1) obtained in [1, 18, 24, 32], reveal such phenomena: for some H, the class of stabilizable sets contains strictly the class of observable sets, while for some other H, the classes of stabilizable sets and observable sets coincide. Besides, this paper gives some sufficient conditions on the stabilizable sets for (0.1) where H is the Schrödinger operator with some general potentials. Copyright © 2020, The Authors. All rights reserved.

关键词： Partial differential equations

CONVERGENCE ANALYSIS FOR MINIMUM ACTION METHODS COUPLED WITH A FINITE DIFFERENCE METHOD

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arXiv 2021年

作者： Hong, Jialin Jin, Diancong Sheng, Derui Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 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 Department of Applied Mathematics The Hong Kong Polytechnic University Hung Hom Kowloon Hong Kong

The minimum action method (MAM) is an effective approach to numerically solving minimums and minimizers of Freidlin–Wentzell (F-W) action functionals, which is used to study the most probable transition path and probability of the occurrence of transitions for stochastic differential equations (SDEs) with small noise. In this paper, we focus on MAMs based on a finite difference method, and present the convergence analysis of minimums and minimizers of the discrete F-W action functional. The main result shows that the convergence orders of the minimum of the discrete F-W action functional in the cases of multiplicative noises and additive noises are 1/2 and 1, respectively. Our main result also reveals the convergence of the stochastic θ-method for SDEs with small noise in terms of large deviations. Copyright © 2021, The Authors. All rights reserved.

关键词： Finite difference method

Lattice Boltzmann simulations of melting in a rectangular cavity heated locally from below at high Rayleigh number

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arXiv 2019年

作者： Zhao, Yong Wu, Yao Chai, Zhenhua Shi, Baochang Hubei Key Laboratory of Engineering Modeling and Scientific Computing School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China

This work presented a block triple-relaxation-time (B-TriRT) lattice Boltzmann model for simulating melting in a rectangular cavity heated from below at high Rayleigh (Ra) number (Ra = 108). The test of benchmark problem show that present B-TriRT can dramatically reduce the numerical diffusion across the phase interface. In addition, the influences of location of the heated region are investigated. The results indicate that the location of heated region plays an essential role in melting rate and the full melting occur earliest when the heated region is located at the middle region. Copyright © 2019, The Authors. All rights reserved.

关键词： Melting

Precise asymptotics on the Birkhoff sums for dynamical systems

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arXiv 2020年

作者： Fang, Lulu Takahasi, Hiroki Zhang, Yiwei School of Science Nanjing University of Science and Technology Nanjing210094 China Department of Mathematics Keio University Yokohama223-8522 Japan School 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 two precise asymptotic results on the Birkhoff sums for dynamical systems. These results are parallel to that on the arithmetic sums of independent and identically distributed random variables previously obtained by Hsu and Robbins, Erdős, Heyde. We apply our results to the Gauss map and obtain new precise asymptotics in the theorem of Lévy on the regular continued fraction expansion of irrational numbers in (0, 1).MSC Codes 37A44, 37A50, 60F10 © 2020, CC BY.

关键词： Dynamical systems