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Clean numerical simulation:a new strategy to obtain reliable solutions of chaotic dynamic systems

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Applied Mathematics and Mechanics(English Edition) 2018年第11期39卷 1529-1546页

作者： Xiaoming LI Shijun LIAO School of Naval Architecture Ocean and Civil EngineeringShanghai Jiao Tong UniversityShanghai 200240China State Key Laboratory of Ocean Engineering Shanghai 200240China Collaborative Innovative Center for Advanced Ship and Deep-Sea Exploration Shanghai 200240China Ministry-of-Education Key Laboratory of Scientific and Engineering Computing Shanghai 200240China

It is well-known that chaotic dynamic systems, e.g., three-body system and turbulent flow, have sensitive dependence on the initial conditions(SDIC). Unfortunately,numerical noises, i.e., truncation error and round-off error, always exist in practice. Thus,due to the SDIC, the long-term accurate prediction of chaotic dynamic systems is practically impossible. In this paper, a new strategy for chaotic dynamic systems, i.e., the clean numerical simulation(CNS), is briefly described, and applied to a few Hamiltonian chaotic systems. With negligible numerical noises, the CNS can provide convergent(reliable) chaotic trajectories in a long enough interval of time. This is very important for Hamiltonian systems, and thus should have many applications in various fields. It is found that the traditional numerical methods in double precision cannot give not only reliable trajectories but also reliable Fourier power spectra and autocorrelation functions(ACFs). In addition, even the statistic properties of chaotic systems cannot be correctly obtained by means of traditional numerical algorithms in double precision, as long as these statistics are time-dependent. The CNS results strongly suggest that one had better be very careful on the direct numerical simulation(DNS) results of statistically unsteady turbulent flows, although DNS results often agree well with experimental data when the turbulent flow is in a statistical stationary state.

关键词： chaos numerical noise clean numerical simulation(CNS) reliability of computation

An improved gradient method with approximately optimal stepsize based on conic model for unconstrained optimization

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arXiv 2019年

作者： Liu, Zexian Liu, Hongwei State Key Laboratory of Scientific and Engineering Computing Institute of Computational Mathematics and Scientific/Engineering computing AMSS Chinese Academy of Sciences Beijing1000190 China School of Mathematics and Statistics Xidian University Xi’an710126 China

A new type of stepsize, which was recently introduced by Liu and Liu (Optimization, 67(3), 427-440, 2018), is called approximately optimal stepsize and is quit efficient for gradient method. Interestingly, all gradient methods can be regarded as gradient methods with approximately optimal stepsizes. In this paper, based on the work (Numer. Algorithms 78(1), 21-39, 2018), we present an improved gradient method with approximately optimal stepsize based on conic model for unconstrained optimization. If the objective function f is not close to a quadratic on the line segment between the current and latest iterates, we construct a conic model to generate approximately optimal stepsize for gradient method if the conic model can be used;otherwise, we construct some quadratic models to generate approximately optimal stepsizes for gradient method. The convergence of the proposed method is analyzed under suitable conditions. Numerical comparisons with some well-known conjugate gradient software packages such as CG_DESCENT (SIAM J. Optim. 16(1), 170-192, 2005) and CGOPT (SIAM J. Optim. 23(1), 296-320, 2013) indicate the proposed method is very promising. Copyright © 2019, The Authors. All rights reserved.

关键词： Gradient methods

Fully Conjugated Benzobisoxazole-Bridged Covalent Organic Frameworks for Boosting Photocatalytic Hydrogen Evolution

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Angewandte Chemie 2025年

作者： Dr. Si Ma Dr. Zhongping Li Yuxin Hou Jiali Li Zhenwei Zhang Prof. Dr. Tianqi Deng Prof. Dr. Gang Wu Prof. Dr. Rui Wang Prof. Dr. Shuo-wang Yang Prof. Dr. Xiaoming Liu College of Chemistry Jilin University Changchun 130012 P.R. China Department of Chemical and Biomolecular Engineering Yonsei University Seoul 03722 Republic of Korea State Key Laboratory of Silicon and Advanced Semiconductor Materials & School of Materials Science and Engineering Zhejiang University Hangzhou 310027 P.R. China Institute of Advanced Semiconductors & Zhejiang Provincial Key Laboratory of Power Semiconductor Materials and Devices ZJU-Hangzhou Global Scientific and Technological Innovation Center Zhejiang University Hangzhou 311215 P.R. China Institute of High Performance Computing Agency for Science Technology and Research 1 Fusionopolis Way #16-16 Connexis Singapore 138632 Singapore Key Laboratory of Aerospace Information Materials and Physics (NUAA) College of Physics Nanjing University of Aeronautics and Astronautics Nanjing 211106 P.R. China

2D covalent organic frameworks (2D-COFs) have attracted extensive interest in solar energy to hydrogen conversion. However, insufficient light harvesting and difficult exciton dissociation severely limit the improvement of photocatalytic activity for COFs, thereby impeding the progression of this advanced field. In this work, two benzobisoxazole-bridged and fully conjugated 2D-COFs with triazine (COF-JLU44) and pyrene (COF-JLU45) units were constructed for the first time via Knoevenagel polycondensation, and they hold long-range ordered structures, largely acceptable surface area, and fascinating photoelectric properties. Significantly, COF-JLU45 exhibits an impressive hydrogen evolution rate of 272.5 mmol g −1 h −1 and superior reusability in the presence of 1.0 wt% Pt under light irradiation, coupled with a remarkable apparent quantum yield of 12.9% at a long wavelength of 600 nm. Multiple spectroscopy and theoretical simulation demonstrate the ingenious design of COF-JLU45 widen its light absorption and effectively promote the exciton dissociation. This finding contributes valuable insights for constructing metal-free photocatalysts for solar energy conversion and utilization.

关键词： Benzobisoxazole Covalent organic frameworks Hydrogen evolution Photocatalysis sp2 Carbon linkage

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

GLOBAL-IN-TIME SOLVABILITY AND BLOW-UP FOR A NON-ISOSPECTRAL TWO-COMPONENT CUBIC CAMASSA-HOLM SYSTEM IN A CRITICAL BESOV SPACE

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arXiv 2020年

作者： Zhang, Lei Qiao, Zhijun SCHOOL OF MATHEMATICS AND STATISTICS HUBEI KEY LABORATORY OF ENGINEERING MODELING AND SCIENTIFIC COMPUTING HUAZHONG UNIVERSITY OF SCIENCE AND TECHNOLOGY WUHANHUBEI430074 China SCHOOL OF MATHEMATICAL AND STATISTICAL SCIENCES THE UNIVERSITY OF TEXAS RIO GRANDE VALLEY EDINBURGTX78539 United States

In this paper, we prove the global Hadamard well-posedness of strong solutions to anon-isospectral two-component cubic Camassa-Holm system in the critical Besov space B22,1(T). 1 Our results shows that in comparison with the well-known work for classic Camassa-Holm-type equations, the existence of global solution only relies on the L1-integrability of the variable coefficients α(t) and γ(t), but nothing to do with the shape or smoothness of the initial data. The key ingredient of the proof hinges on the careful analysis of the mutual effect among two component forms, the uniform bound of approximate solutions, and several crucial estimates of cubic nonlinearities in low-regularity Besov spaces via the Littlewood-Paley decomposition theory. A reduced case in our results yields the global existence of solutions in a Besov space for two kinds of well-known isospectral peakon system with weakly dissipative terms. Moreover, we derive two kinds of precise blow-up criteria for a strong solution in both critical and non-critical Besov spaces, as well as providing specific characterization for the lower bound of the blow-up time, which implies the global existence with additional conditions on the time-dependent parameters α(t) an γ(t). Copyright © 2020, The Authors. All rights reserved.

关键词： Banach spaces

Efficient Learning of Continuous-Variable Quantum States

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arXiv 2023年

作者： Wu, Ya-Dong Zhu, Yan Chiribella, Giulio Liu, Nana John Hopcroft Center for Computer Science Shanghai Jiao Tong University Shanghai200240 China QICI Quantum Information and Computation Initiative Department of Computer Science The University of Hong Kong Pokfulam Road Hong Kong Department of Computer Science Parks Road OxfordOX1 3QD United Kingdom Perimeter Institute for Theoretical Physics WaterlooONN2L 2Y5 Canada Institute of Natural Sciences Shanghai Jiao Tong University Shanghai200240 China Ministry of Education Key Laboratory in Scientific and Engineering Computing Shanghai Jiao Tong University Shanghai200240 China School of Mathematical Sciences Shanghai Jiao Tong University Shanghai200240 China University of Michigan-Shanghai Jiao Tong University Joint Institute Shanghai200240 China

The characterization of continuous-variable quantum states is crucial for applications in quantum communication, sensing, simulation and computing. However, a full characterization of multimode quantum states requires a number of experiments that grows exponentially with the number of modes. Here we propose an alternative approach where the goal is not to reconstruct the full quantum state, but rather to estimate its characteristic function at a given set of points. For multimode states with reflection symmetry, we show that the characteristic function at M points can be estimated using only O(log M) copies of the state, independently of the number of modes. When the characteristic function is known to be positive, as in the case of squeezed vacuum states, the estimation is achieved by an experimentally friendly setup using only beamsplitters and homodyne measurements. © 2023, CC BY-NC-SA.

关键词： Quantum communication

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