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FAST TIME-EXTRAPOLATION ALGORITHM(FTEA)OF CRANK-NICOLSON SCHEME FOR PARABOLIC PROBLEM

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FAST TIME-EXTRAPOLATION ALGORITHM(FTEA)OF CRANK-NICOLSON SCH...

2012China-Norway-Sweden Workshop on Computational mathematics(2012中国-挪威-瑞典计算数学研讨会)

作者： Chuanmiao Chen Key Laboratory of High Performance Computing and Stochastic Information Processing(Ministry of Education of China) ChangshaHunan 410081PRCCollege of Mathematics and Computer ScienceHunan Normal University

A fast time-extrapolation algorithm(FTEA)of Crank-Nicolson scheme on quasi-uniform mesh for parabolic problem is first *** decrease regularity requirements of solutions,new error estimates and time-extrapolation are p...

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Numerical approximation based on deep convolutional neural network for high-dimensional fully nonlinear merged PDEs and 2BSDEs

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arXiv 2022年

作者： Xiao, Xu Qiu, Wenlin Nikan, Omid Key Laboratory of Computing and Stochastic Mathematics Ministry of Education School of Mathematics and Statistics Hunan Normal University Hunan Changsha410081 China School of Mathematics Iran University of Science and Technology Narmak Tehran16846-13114 Iran

This paper proposes two efficient approximation methods to solve high-dimensional fully nonlinear partial differential equations (NPDEs) and second-order backward stochastic differential equations (2BSDEs), where such high-dimensional fully NPDEs are extremely difficult to solve because the computational cost of standard approximation methods grows exponentially with the number of dimensions. Therefore, we consider the following methods to overcome this difficulty. For the merged fully NPDEs and 2BSDEs system, combined with the time forward discretization and ReLU function, we use multi-scale deep learning fusion and convolutional neural network (CNN) techniques to obtain two numerical approximation schemes, respectively. Finally, three practical high-dimensional test problems involving Allen-Cahn, Black-Scholes-Barentblatt, and Hamiltonian-Jacobi-Bellman equations are given so that the first proposed method exhibits higher efficiency and accuracy than the existing method, while the second proposed method can extend the dimensionality of the completely NPDEs-2BSDEs system over 400 dimensions, from which the numerical results highlight the effectiveness of proposed *** Codes 65M22, 60H15, 65C30, 68T07 Copyright © 2022, The Authors. All rights reserved.

关键词： Convolution

Quasi-periodic solutions for differential equations with an elliptic-type degenerate equilibrium point under small perturbations

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arXiv 2017年

作者： Li, Xuemei Shang, Zaijiu Key Laboratory of High Performance Computing and Stochastic Information Processing Department of Mathematics Hunan Normal University Changsha Hunan410081 Institute of Mathematics Amss Chinese Academy of Science Beijing100080 China

This work focuses on the existence of quasi-periodic solutions for ordinary and delay differential equations (ODEs and DDEs for short) with an elliptic-type degenerate equilibrium point under quasi-periodic perturbations. We prove that under appropriate hypotheses there exist quasi-periodic solutions for perturbed ODEs and DDEs near the equilibrium point for most parameter values, then apply these results to the delayed van der Pol's oscillator with zero-Hopf singularity. Copyright © 2017, The Authors. All rights reserved.

关键词： Ordinary differential equations

SPECTRALITY OF A CLASS OF MORAN MEASURES ON 2

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arXiv 2024年

作者： Liu, Jing-Cheng Liu, Qiao-Qin Luo, Jun Jason Wang, Jia-Jie Key Laboratory of Computing and Stochastic Mathematics Ministry of Education School of Mathematics and Statistics Hunan Normal University Hunan Changsha410081 China College of Mathematics and Statistics Key Laboratory of Nonlinear Analysis and its Applications Chongqing University Ministry of Education Chongqing University Chongqing401331 China

Let µ{Mn},{Dn} be a Moran measure on 2 generated by a sequence of expanding matrices {Mn} ⊂ GL(2, ) and a sequence of integer digit sets {Dn} where (formula presented) with (formula presented). If | det(Mn)| > 4 for n ≥ 1, sup (formula presented) and #{Dn : n ≥ 1} n ∈ GL(2, 2) for n ≥ 2. If | det(Mn)| = 4 for n ≥ 1, we also establish a necessary and sufficient condition for a class of special Moran measures to be spectral *** Codes Primary 28A80, Secondary 42B10, 42C05 © 2024, CC BY-NC-ND.

关键词： Fourier transforms

Multifractal analysis of the growth rate of digits in Schneider's p-adic continued fraction dynamical system

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arXiv 2024年

作者： Song, Kunkun Wu, Wanlou Yu, Yueli Zeng, Sainan Key Laboratory of Computing and Stochastic Mathematics Ministry of Education School of Mathematics and Statistics Hunan Normal University Changsha410081 China School of Mathematics and Statistics Jiangsu Normal University Xuzhou221116 China School of Mathematics and Statistics Wuhan University Wuhan430072 China

Let p be the ring of p-adic integers and an(x) be the n-th digit of Schneider’s p-adic continued fraction of x ∊ pp. We study the growth rate of the digits {an(x)}n≥1 from the viewpoint of multifractal analysis. The Hausdorff dimension of the set (Formula presented) is completely determined for any ψ: → + satisfying ψ(n) → ∞ as n → ∞. As an application, we also calculate the Hausdorff dimension of the intersection sets (Formula presented) for the above function ψ and 0 ≤ α1 2 ≤ ∞. Copyright © 2024, The Authors. All rights reserved.

关键词： Dynamical systems

Superconvergence of differential structure for finite element methods on perturbed surface meshes

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arXiv 2019年

作者： Dong, Guozhi Guo, Hailong Guo, Ting School of Mathematics and Statistics HNP-LAMA Central South University Changsha410083 China School of Mathematics and Statistics The University of Melbourne ParkvilleVIC3010 Australia Key Laboratory of Computing and Stochastic Mathematics Ministry of Education School of Mathematics and Statistics Hunan Normal University Changsha410081 China

Superconvergence of differential structure on discretized surfaces is studied in this paper. The newly introduced geometric supercloseness provides us with a fundamental tool to prove the superconvergence of gradient recovery on deviated surfaces. An algorithmic framework for gradient recovery without exact geometric information is introduced. Several numerical examples are documented to validate the theoretical *** Codes 41A25, 65N15, 65N30 Copyright © 2019, The Authors. All rights reserved.

关键词： Finite element method

RIESZ TYPE THEOREMS FOR κ-PLURIHARMONIC MAPPINGS, INVARIANT HARMONIC QUASIREGULAR MAPPINGS AND HARMONIC QUASIREGULAR MAPPINGS

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arXiv 2023年

作者： Chen, Shaolin Huang, Manzi College of Mathematics and Statistics Hengyang Normal University Hunan Hengyang421002 China Hunan Provincial Key Laboratory of Intelligent Information Processing and Application 421002 China Key Laboratory of High Performance Computing and Stochastic Information Processing College of Mathematics and Statistics Hunan Normal University Hunan Changsha410081 China

The main purpose of this paper is to develop some methods to improve and generalize the main results in a recent paper by Liu and Zhu (Adv. Math., 2023, i.e., [19]). The paper consists of two parts. In the first part, we discuss the Riesz type theorem in the setting of n-dimensional complex spaces for all n ≥ 1. In this part, we first introduce the family of κ-pluriharmonic mappings of the n-dimensional complex unit ball. Then we establish two Riesz type theorems for these mappings, which are the n-dimensional versions of Theorems 1.1 and 1.2 in [19], respectively. Furthermore, even when n = 1, our first result shows that the assumption of the real parts of the mappings not being negative (or being negative) in [19, Theorem 1.1] is redundant;and our second result illustrates that the assumption of "quasiconformality" on the mappings in [19, Theorem 1.2] can be replaced by the weaker one of "quasiregularity". In the second part, we investigate the Riesz type theorem in the setting of n-dimensional real spaces for all n ≥ 2. In this part, first, we prove a Riesz type theorem for invariant harmonic quasiregular mappings of the unit n-dimensional real ball. Our result indicates that (i) the range of the parameter p discussed in [19, Theorem 1.3] can be changed from (1, 2) to (1, ∞);(ii) the assumption of the first coordinate functions of the mappings being non-zero in [19, Theorem 1.3] is redundant. In this way, we complete the discussions carried out in [19, Theorems 1.3 and 1.4]. Second, we obtain a Riesz type theorem for harmonic K-quasiregular mappings of the unit n-dimensional real ball. Our result demonstrates that the range of the parameter p discussed in [15, Theorem 2.1] can be changed from (1, 2) to (1, ∞).MSC Codes 30H10, 30C62 Copyright © 2023, The Authors. All rights reserved.

关键词： Harmonic analysis

An Enhanced Flower Pollination Algorithm for Solving Optimal Power Flow Problem

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SSRN

SSRN 2022年

作者： Zhong, keyu Xiao, Fen Gao, Xieping Key Laboratory of Computing and Stochastic Mathematics of Ministry of Education Hunan Normal University Changsha410081 China School of Mathematics and Statistics Hunan Normal University Changsha410081 China School of Computer Science Xiangtan University Hunan Xiangtan411105 China Hunan Provincial Key Laboratory of Intelligent Computing and Language Information Processing Hunan Normal University Changsha410081 China

The OPF system for renewable energy power has presented characteristics such as extremely high security constraints and more obvious multimodal properties, its solution is a challenging task and requires more efficient methods. In this paper, an enhanced flower pollination algorithm, EFPA-SF is proposed to solve the OPF problem. In EFPA-SF, there are four improvements: firstly, a random step size in the process of cross-pollination is proposed to improve the global search ability of EFPA-SF. Secondly, in the stage of self-pollination, the global optimal pollen is introduced to improve the ability of exploitation–exploration balance of the algorithm. In addition, if pollen individual crosses boundaries, boundary constraint is carried out by the previous position and boundary position of pollen, which is instrumental to escape from local optimum. Finally, the superiority of feasible solutions (SF) constraints handling method is integrated into the algorithm to satisfy the feasibility of the solutions. To verify the effectiveness of the proposed algorithm, the optimization capability of EFPA is first investigated and analyzed by using the most recent IEEE CEC2021 benchmark functions. Then, EFPA-SF is used to solve three engineering design problems and the performance is compared with the state-of-the-art algorithms, the validation results reveal that the proposed algorithm outperforms state-of-the-art algorithms, and demonstrate that it can be used for constrained optimization as well as for unconstrained optimization problems with good generality. Finally, EFA-SF is applied to the OPF problem on a modified IEEE-30 bus system that incorporates stochastic wind, solar, and small hydro power in addition to conventional thermal power generation. The comprehensive investigation reveals that EFA-SF can provide competitive solutions to the OPF problem compared to popular algorithms. © 2022, The Authors. All rights reserved.

关键词： Constraint handling

TWISTED QUANTUM AFFINIZATIONS AND QUANTIZATION OF EXTENDED AFFINE LIE ALGEBRAS

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arXiv 2020年

作者： Chen, Fulin Jing, Naihuan Kong, Fei Tan, Shaobin School of Mathematical Sciences Xiamen University Xiamen361005 China Department of Mathematics North Carolina State University RaleighNC27695 United States Key Laboratory of Computing and Stochastic Mathematics Ministry of Education School of Mathematics and Statistics Hunan Normal University Changsha410081 China

In this paper, for an arbitrary Kac-Moody Lie algebra g and a diagram automorphism µ of g satisfying certain natural linking conditions, we introduce and study a µ-twisted quantum affinization algebra (Equation presented) of g. When g is of finite type, (Equation presented) is Drinfeld's current algebra realization of the twisted quantum affine algebra. When µ = id and g in affine type, (Equation presented) is the quantum toroidal algebra introduced by Ginzburg, Kapranov and Vasserot. As the main results of this paper, we first prove a triangular decomposition for (Equation presented). Second, we give a simple characterization of the affine quantum Serre relations on restricted (Equation presented)-modules in terms of "normal order products". Third, we prove that the category of restricted (Equation presented)-modules is a monoidal category and hence obtain a topological Hopf algebra structure on the "restricted completion" of (Equation presented). Last, we study the classical limit of (Equation presented) and abridge it to the quantization theory of extended affine Lie algebras. In particular, based on a classification result of Allison-Berman-Pianzola, we obtain the ~-deformation of all nullity 2 extended affine Lie algebras. Copyright © 2020, The Authors. All rights reserved.

关键词： Algebra