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Efficient numerical methods for computing stationary states of spherical Landau-Brazovskii model

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arXiv 2025年

作者： Qiu, Qun Si, Wei Ji, Guanghua Jiang, Kai School of Mathematical Sciences Beijing Normal University Beijing100875 China School of Mathematics Sciences Peking University Beijing100871 China School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education Xiangtan University Hunan Xiangtan411105 China

In this paper, we develop a set of efficient methods to compute stationary states of the spherical Landau-Brazovskii (LB) model in a discretization-then-optimization way. First, we discretize the spherical LB energy functional into a finite-dimensional energy function by the spherical harmonic expansion. Then five optimization methods are developed to compute stationary states of the discretized energy function, including the accelerated adaptive Bregman proximal gradient, Nesterov, adaptive Nesterov, adaptive nonlinear conjugate gradient and adaptive gradient descent methods. To speed up the convergence, we propose a principal mode analysis (PMA) method to estimate good initial configurations and sphere radius. The PMA method also reveals the relationship between the optimal sphere radius and the dominant degree of spherical harmonics. Numerical experiments show that our approaches significantly reduce the number of iterations and the computational time. Copyright © 2025, The Authors. All rights reserved.

关键词： Conjugate gradient method

Higher order log-concavity of the overpartition function and its consequences

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arXiv 2022年

作者： Mukherjee, Gargi Zhang, Helen W.J. Zhong, Ying Institute for Algebra Johannes Kepler University Altenberger Straße 69 LinzA-4040 Austria School of Mathematics Hunan University Changsha410082 China Hunan Provincial Key Laboratory of Intelligent Information Processing and Applied Mathematics Changsha410082 China

Let p(n) denote the overpartition function. In this paper, we study the asymptotic higher order log-concavity property of the overpatition function in a similar framework done by Hou and Zhang for the partition function. This will enable us to move on further in order to prove log-concavity of overpartitions, explicitly by studying the asymptotic expansion of the quotient (Equation presented) upto a certain order so that one can finally ends up with the phenomena of 2-log-concavity and higher order Turán property of p(n) by following a sort of unified *** Codes 05A20, 11N37, 65G99 © 2022, CC BY.

关键词：

Irrational-window-filter projection method and application to quasiperiodic Schrödinger eigenproblems

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arXiv 2024年

作者： Jiang, Kai Li, Xueyang Ma, Yao Zhang, Juan Zhang, Pingwen Zhou, Qi Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing Ministry of Education School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China School of Mathematics and Statistics Wuhan University Wuhan430072 China School of Mathematical Sciences Peking University Beijing100871 China

In this paper, we propose a new algorithm, the irrational-window-filter projection method (IWFPM), for quasiperiodic systems with concentrated spectral point distribution. Based on the projection method (PM), IWFPM filters out dominant spectral points by defining an irrational window and uses a corresponding index-shift transform to make the FFT available. The error analysis on the function approximation level is also given. We apply IWFPM to 1D, 2D, and 3D quasiperiodic Schrödinger eigenproblems (QSEs) to demonstrate its accuracy and efficiency. IWFPM exhibits a significant computational advantage over PM for both extended and localized quantum states. More importantly, by using IWFPM, the existence of Anderson localization in 2D and 3D QSEs is numerically *** Codes 35P05, 35J10, 65D15, 65T50, 81-08 © 2024, CC BY-NC-SA.

关键词： Wiener filtering

The interior penalty virtual element method for fourth-order singular perturbation problems

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arXiv 2023年

作者： Feng, Fang Yu, Yue School of Mathematics and Statistics Nanjing University of Science and Technology Nanjing China School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing Ministry of Education Xiangtan University Hunan Xiangtan411105 China

This paper is dedicated to the numerical solution of a fourth-order singular perturbation problem using the interior penalty virtual element method (IPVEM) proposed in [42]. The study introduces modifications to the jumps and averages in the penalty term, as well as presents an automated mesh-dependent selection of the penalty parameter. Drawing inspiration from the modified Morley finite element methods, we leverage the conforming interpolation technique to handle the lower part of the bilinear form. Through our analysis, we establish optimal convergence in the energy norm and provide a rigorous proof of uniform convergence concerning the perturbation parameter in the lowest-order case. © 2023, CC BY-NC-ND.

关键词： Numerical methods

ALCencryption:A Secure and Efficient Algorithm for Medical Image Encryption

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Computer Modeling in Engineering & Sciences 2020年第12期125卷 1083-1100页

作者： Jiao Ge College of Computer Science and Technology Hengyang Normal UniversityHengyang421002China Hunan Provincial Key Laboratory of Intelligent Information Processing and Application Hengyang421002China

With the rapid development of medical informatization and the popularization of digital imaging equipment,DICOM images contain the personal privacy of patients,and there are security risks in the process of storage and transmission,so it needs to be *** order to solve the security problem of medical images on mobile devices,a safe and efficient medical image encryption algorithm called ALCencryption is *** algorithm first analyzes the medical image and distinguishes the color image from the gray *** gray images,the improved Arnold map is used to scramble them according to the optimal number of iterations,and then the diffusion is realized by the Logistic and Chebyshev map cross-diffusion *** color image is encrypted by cross-diffusion algorithm of double chaotic *** and efficiency analysis show that the ALCencryption algorithm has the characteristics of small neighboring pixels,large key space,strong key sensitivity,high safety and short encryption *** is suitable for medical image encryption of mobile devices with high real-time requirements.

关键词： Patient privacy DICOM medical image encryption scrambling degree cross-diffusion

PRISTA-Net: Deep Iterative Shrinkage Thresholding Network for Coded Diffraction Patterns Phase Retrieval

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arXiv 2023年

作者： Liu, Aoxu Fan, Xiaohong Yang, Yin Zhang, Jianping The School of Mathematics and Computational Science Xiangtan University Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan411105 China The School of Mathematics and Computational Science Xiangtan University the Key Laboratory of Intelligent Computing & Information Processing the Ministry of Education Xiangtan411105 China The School of Mathematics and Computational Science Xiangtan University the National Center for Applied Mathematics in Hunan Xiangtan411105 China

The problem of phase retrieval (PR) involves recovering an unknown image from limited amplitude measurement data and is a challenge nonlinear inverse problem in computational imaging and image processing. However, many of the PR methods are based on black-box network models that lack interpretability and plug-and-play (PnP) frameworks that are computationally complex and require careful parameter tuning. To address this, we have developed PRISTA-Net, a deep unfolding network (DUN) based on the first-order iterative shrinkage thresholding algorithm (ISTA). This network utilizes a learnable nonlinear transformation to address the proximal-point mapping sub-problem associated with the sparse priors, and an attention mechanism to focus on phase information containing image edges, textures, and structures. Additionally, the fast Fourier transform (FFT) is used to learn global features to enhance local information, and the designed logarithmic-based loss function leads to significant improvements when the noise level is low. All parameters in the proposed PRISTA-Net framework, including the nonlinear transformation, threshold parameters, and step size, are learned end-to-end instead of being manually set. This method combines the interpretability of traditional methods with the fast inference ability of deep learning and is able to handle noise at each iteration during the unfolding stage, thus improving recovery quality. Experiments on Coded Diffraction Patterns (CDPs) measurements demonstrate that our approach outperforms the existing state-of-the-art methods in terms of qualitative and quantitative evaluations. Our source codes are available at https://***/liuaxou/PRISTA-Net. Copyright © 2023, The Authors. All rights reserved.

关键词： Mapping

Explainable Semantic Federated Learning Enabled Industrial Edge Network for Fire Surveillance

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arXiv 2024年

作者： Dong, Li Peng, Yubo Jiang, Feibo Wang, Kezhi Yang, Kun School of Computer Science Hunan University of Technology and Business Changsha410205 China Xiangjiang Laboratory Changsha410205 China School of Intelligent Software and Engineering Nanjing University Suzhou China Hunan Provincial Key Laboratory of Intelligent Computing and Language Information Processing Hunan Normal University Changsha410081 China Department of Computer Science Brunel University London United Kingdom

In fire surveillance, Industrial Internet of Things (IIoT) devices require transmitting large monitoring data frequently, which leads to huge consumption of spectrum resources. Hence, we propose an Industrial Edge Semantic Network (IESN) to allow IIoT devices to send warnings through Semantic communication (SC). Thus, we should consider (1) Data privacy and security. (2) SC model adaptation for heterogeneous devices. (3) Explainability of semantics. Therefore, first, we present an eXplainable Semantic Federated Learning (XSFL) to train the SC model, thus ensuring data privacy and security. Then, we present an Adaptive Client Training (ACT) strategy to provide a specific SC model for each device according to its Fisher information matrix, thus overcoming the heterogeneity. Next, an Explainable SC (ESC) mechanism is designed, which introduces a leakyReLU-based activation mapping to explain the relationship between the extracted semantics and monitoring data. Finally, simulation results demonstrate the effectiveness of XSFL. Copyright © 2024, The Authors. All rights reserved.

关键词： Federated learning

Solving linear Bayesian inverse problems using a fractional total variation-Gaussian (FTG) prior and transport map

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arXiv 2021年

作者： Sun, Zejun Zheng, Guang-Hui College of Mathematics Hunan University Changsha410082 China College of Mathematics Hunan Provincial Key Laboratory of Intelligent Information Processing and Applied Mathematics Hunan University Changsha410082 China

The Bayesian inference is widely used in many scientific and engineering problems, especially in the linear inverse problems in infinite-dimensional setting where the unknowns are functions. In such problems, choosing an appropriate prior distribution is an important task. In particular, when the function to infer has much detail information, such as many sharp jumps, corners, and the discontinuous and nonsmooth oscillation, the so-called total variation-Gaussian (TG) prior is proposed in function space to address it. However, the TG prior is easy to lead the blocky (staircase) effect in numerical results. In this work, we present a fractional order-TG (FTG) hybrid prior to deal with such problems, where the fractional order total variation (FTV) term is used to capture the detail information of the unknowns and simultaneously uses the Gaussian measure to ensure that it results in a well-defined posterior measure. For the numerical implementations of linear inverse problems in function spaces, we also propose an efficient independence sampler based on a transport map, which uses a proposal distribution derived from a diagonal map, and the acceptance probability associated to the proposal is independent of discretization dimensionality. And in order to take full advantage of the transport map, the hierarchical Bayesian framework is applied to flexibly determine the regularization parameter. Finally we provide some numerical examples to demonstrate the performance of the FTG prior and the efficiency and robustness of the proposed independence sampler method. Copyright © 2021, The Authors. All rights reserved.

关键词： Differential equations

Error Analysis of Virtual Element Method for the Poisson-Boltzmann Equation

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arXiv 2023年

作者： Huang, Linghan Shu, Shi Yang, Ying Guangxi Guilin541004 China School of Mathematics and Computational Science Xiangtan University Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing Ministry of Education Hunan Xiangtan411105 China Guangxi541004 China

The Poisson-Boltzmann equation is a nonlinear elliptic equation with Dirac distribution sources, which has been widely applied to the prediction of electrostatics potential of biological biomolecular systems in solution. In this paper, we discuss and analysis the virtual element method for the Poisson-Boltzmann equation on general polyhedral meshes. Under the low regularity of the solution of the whole domain, nearly optimal error estimates in both L2-norm and H1-norm for the virtual element approximation are obtained. The numerical experiment on different polyhedral meshes shows the efficiency of the virtual element method and verifies the proposed theoretical prediction. Copyright © 2023, The Authors. All rights reserved.

关键词： Poisson equation