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Convergence analysis of PM-BDF2 method for quasiperiodic parabolic equations

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arXiv 2024年

作者： Jiang, Kai Li, Meng Zhang, Juan Zhang, Lei Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing Ministry of Education Department of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China School of Mathematical Sciences Institute of Natural Sciences MOE-LSC Shanghai Jiao Tong University Shanghai200240 China

Numerically solving parabolic equations with quasiperiodic coefficients is a significant challenge due to the potential formation of space-filling quasiperiodic structures that lack translational symmetry or decay. In this paper, we introduce a highly accurate numerical method for solving time-dependent quasiperiodic parabolic equations. We discretize the spatial variables using the projection method (PM) and the time variable with the second-order backward differentiation formula (BDF2). We provide a complexity analysis for the resulting PM-BDF2 method. Furthermore, we conduct a detailed convergence analysis, demonstrating that the proposed method exhibits spectral accuracy in space and second-order accuracy in time. Numerical results in both one and two dimensions validate these convergence results, highlighting the PMBDF2 method as a highly efficient algorithm for addressing quasiperiodic parabolic equations. Copyright © 2024, The Authors. All rights reserved.

关键词： Convergence of numerical methods

High-resolution boundary variation diminishing scheme for two-phase compressible flow with cavitation and evaporation

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Journal of computational Physics 2024年 513卷

作者： Wakimura, Hiro Li, Tatsuin Shyue, Keh-Ming Aoki, Takayuki Xiao, Feng Department of Mechanical Engineering Tokyo Institute of Technology 2-12-1 i6-29 Ookayama Meguro-ku Tokyo152-8550 Japan Department of Mathematics Institute of Applied Mathematical Sciences National Taiwan University Taipei10617 Taiwan Global Scientific Information and Computing Center Tokyo Institute of Technology 2-12-1 i7-3 Ookayama Meguro-ku Tokyo152-8550 Japan

We propose high-resolution numerical schemes for two-phase compressible flow simulations to reproduce dynamically created gas/vapor-liquid interfaces with phase change. In the Godunov-type finite volume framework, suppressing numerical dissipation errors in numerical schemes is crucial for capturing the discontinuous solutions. The MUSCL scheme has second-order accuracy for smooth solutions and non-oscillatory behavior near discontinuous solutions. However, the MUSCL scheme introduces excessive numerical dissipation and diffuses the discontinuities nonphysically, leading to difficulties in distinguishing gas/vapor and liquid phases, and thus causing a blur in the interfaces during the multi-phase flow simulations. The hybrid-type boundary variation diminishing (BVD) scheme in this paper combines the MUSCL scheme and the THINC scheme to reduce the numerical dissipation errors near the discontinuities. The MUSCL-THINC-BVD scheme applies the MUSCL scheme for smooth solutions and the THINC scheme for discontinuous solutions, resulting in the successful capture of the discontinuities including the dynamically created gas/vapor-liquid interfaces. The Adaptive THINC-BVD scheme, which switches two types of THINC schemes with different values of gradient parameter, also captures the discontinuities clearly. The numerical results of the benchmark tests show that the proposed BVD schemes can lucidly reproduce the vapor-liquid interfaces newly created during the dynamical process of phase change. © 2024 Elsevier Inc.

关键词： Finite volume method

Securing Neural Network in the Cloud: A Systematic Approach of Cloud Security Based on ANN-SVM Model

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Securing Neural Network in the Cloud: A Systematic Approach ...

Sustainable Expert Systems (ICSES), International Conference on

作者： Ar Arunarani V. Vijayagopal L. Kartheesan Vijay Kumar Dwivedi Rajesh Kumar A Neerav Nishant Department of Computational Intelligence School of Computing College of Engineering SRM Institute of Science and Technology Chennai Tamilnadu India Faculty of Computer Applications Dr. M.G.R. Educational and Research Institute Chennai Tamilnadu India Department of Computer Science and Engineering School of Computing Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology Chennai Tamilnadu India Department of Mathematics Vishwavidyalaya Engineering College Surguja Chhattisgarh India Department of Computer Science and Engineering N.S.N. College of Engineering and Technology Karur Tamilnadu India Department of Computer Science and Engineering School of Engineering Babu Banarasi Das University Lucknow India

ISBN: (数字)9798331540364

ISBN: (纸本)9798331540371

Many services have made the transition to the cloud because of the increased reliability and efficiency of cloud computing. Due to their capacity to ease service access while also securing communications in the public network, the three-factor Mutual Authentication and Key Agreement (MAKA) protocols are seeing increased use in multi-server designs. A lack of formal security proof or high communication and computation costs makes the limited three-factor MAKA protocols that are currently available susceptible to a wide range of attacks. The proposed approach consists of three phases, which are preprocessing, feature selection, and model training. Text scheduling and normalization make up preprocessing. In feature selection, the PSO is an optimization method that uses a population-based approach to find optimal solutions. An ANN-SVM was employed throughout the model's training phase. With an average accuracy of 93.30%, the proposed technique surpasses ANN and SVM.

关键词： Training Support vector machines Protocols Accuracy Systematics Cloud computing security Neural networks Feature extraction Vectors Reliability

Normalizing field flows: Solving forward and inverse stochastic differential equations using physics-informed flow models

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arXiv 2021年

作者： Guo, Ling Wu, Hao Zhou, Tao Department of Mathematics Shanghai Normal University Shanghai China School Odmathematical Sciences Tongji University Shanghai China LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China

We introduce in this work the normalizing field flows (NFF) for learning random fields from scattered measurements. More precisely, we construct a bijective transformation (a normalizing flow characterizing by neural networks) between a Gaussian random field with the Karhunen-Loève (KL) expansion structure and the target stochastic field, where the KL expansion coefficients and the invertible networks are trained by maximizing the sum of the log-likelihood on scattered measurements. This NFF model can be used to solve data-driven forward, inverse, and mixed forward/inverse stochastic partial differential equations in a unified framework. We demonstrate the capability of the proposed NFF model for learning Non Gaussian processes and different types of stochastic partial differential equations. Copyright © 2021, The Authors. All rights reserved.

关键词： Uncertainty analysis

Efficient CI-Based One-Bit Precoding for Multiuser Downlink Massive MIMO Systems with PSK Modulation

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arXiv 2021年

作者： Wu, Zheyu Jiang, Bo Liu, Ya-Feng Shao, Mingjie Dai, Yu-Hong The State Key Laboratory of Scientific and Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China The Ministry of Education Key Laboratory for NSLSCS School of Mathematical Sciences Nanjing Normal University Nanjing210023 China The School of Information Science and Engineering Shandong University Qingdao China

In this paper, we consider the one-bit precoding problem for the multiuser downlink massive multiple-input multiple-output (MIMO) system with phase shift keying (PSK) modulation. We focus on the celebrated constructive interference (CI)-based problem formulation. We first establish the NP-hardness of the problem (even in the single-user case), which reveals the intrinsic difficulty of globally solving the problem. Then, we propose a novel negative 1 penalty model for the considered problem, which penalizes the one-bit constraint into the objective by a negative 1-norm term, and show the equivalence between (global and local) solutions of the original problem and the penalty problem when the penalty parameter is sufficiently large. We further transform the penalty model into an equivalent min-max problem and propose an efficient alternating proximal/projection gradient descent ascent (APGDA) algorithm for solving it, which performs a proximal gradient decent over one block of variables and a projection gradient ascent over the other block of variables alternately. The APGDA algorithm enjoys a low per-iteration complexity and is guaranteed to converge to a stationary point of the min-max problem and a local minimizer of the penalty problem. To further reduce the computational cost, we also propose a low-complexity implementation of the APGDA algorithm, where the values of the variables will be fixed in later iterations once they satisfy the one-bit constraint. Numerical results show that, compared to the state-of-the-art CI-based algorithms, both of the proposed algorithms generally achieve better bit-error-rate (BER) performance with lower computational cost. Copyright © 2021, The Authors. All rights reserved.

关键词： Gradient methods

Foundation Models for Atomistic Simulation of Chemistry and Materials

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arXiv 2025年

作者： Yuan, Eric C.-Y. Liu, Yunsheng Chen, Junmin Zhong, Peichen Raja, Sanjeev Kreiman, Tobias Vargas, Santiago Xu, Wenbin Head-Gordon, Martin Yang, Chao Blau, Samuel M. Cheng, Bingqing Krishnapriyan, Aditi Head-Gordon, Teresa Kenneth S. Pitzer Theory Center Department of Chemistry United States Department of Chemical and Biomolecular Engineering United States Electrical Engineering and Computer Science United States Bakar Institute of Digital Materials for the Planet University of California BerkeleyCA94720 United States Chemical Sciences Division United States Applied Mathematics and Computational Research Division United States Energy Technologies Area United States National Energy Research Scientific Computing Center Lawrence Berkeley National Laboratory BerkeleyCA94720 United States

Given the power of large language and large vision models, it is of profound and fundamental interest to ask if a foundational model based on data and parameter scaling laws and pre-training strategies is possible for learned simulations of chemistry and materials. The scaling of large and diverse datasets and highly expressive architectures for chemical and materials sciences should result in a foundation model that is more efficient and broadly transferable, robust to out-of-distribution challenges, and easily fine-tuned to a variety of downstream observables, when compared to specific training from scratch on targeted applications in atomistic simulation. In this Perspective we aim to cover the rapidly advancing field of machine learned interatomic potentials (MLIP), and to illustrate a path to create chemistry and materials MLIP foundation models at larger scale. © 2025, CC BY-SA.

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An automatic approach to develop the fourth-order and L2-stable lattice Boltzmann model for diagonal-anisotropic diffusion equations

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arXiv 2025年

作者： Chen, Ying Chai, Zhenhua Shi, Baochang School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Institute of Interdisciplinary Research for Mathematics and Applied Science 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 The State Key Laboratory of Intelligent Manufacturing Equipment and Technology Huazhong University of Science and Technology Wuhan430074 China

This paper discusses how to develop a high-order multiple-relaxation-time lattice Boltzmann (MRT-LB) model for the general d(≥ 1)-dimensional diagonal-anisotropic diffusion equation. Such an MRT-LB model considers the transformation matrix constructed in a natural way and the DdQ(2d2 + 1) [(2d2 + 1) discrete velocities in d-dimensional space] lattice structure. A key step in developing the high-order MRT-LB model is to determine the additional adjustable relaxation parameters and weight coefficients, which are used to eliminate the truncation errors at some certain orders of the MRT-LB model, while ensuring the stability of the MRT-LB model. In this work, we first present a unified MRT-LB model for the d-dimensional diagonal-anisotropic diffusion equation. Then, through the direct Taylor expansion, we analyze the macroscopic modified equations of the MRT-LB model up to fourth-order at the diffusive scaling, and further derive the conditions that ensure the MRT-LB model to be fourth-order consistent with the diagonal-anisotropic diffusion equation. In particular, when the diagonal-anisotropic diffusion equation is reduced to the isotropic type, we propose another MRT-LB model with the DdQ(2d+ 1) lattice structure [fewer discrete velocities than the DdQ(2d2 + 1) lattice structure], and the fourth-order conditions are similarly derived. Additionally, we also construct the fourth-order initialization scheme for the present LB method. After that, the condition which guarantees that the MRT-LB model can satisfy the stability structure is explicitly given, and we would like to point out that from a numerical perspective, once the stability structure is satisfied, the MRT-LB model must be L2 stable. In combination with the fourth-order consistent and L2 stability conditions, the relaxation parameters and weight coefficients of the MRT-LB model can be automatically given by a simple computer code. Finally, we perform numerical simulations of several benchmark problems, and f

关键词： Anisotropy

A novel strong duality-based reformulation for trilevel infrastructure models in energy systems development

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arXiv 2023年

作者： Herrala, Olli Gabriel, Steven A. Oliveira, Fabricio Ekholm, Tommi Aalto University Department of Mathematics and Systems Analysis Espoo Finland University of Maryland Department of Mechanical Engineering/Applied Mathematics Statistics & Scientific Computing Program College ParkMD United States Norwegian University of Science and Technology Department of Industrial Economics and Technology Management Trondheim Norway Finnish Meteorological Institute Helsinki Finland

We explore the class of trilevel equilibrium problems with a focus on energy-environmental applications and present a novel single-level reformulation for such problems, based on strong duality. To the best of our knowledge, only one alternative single-level reformulation for trilevel problems exists. This reformulation uses a representation of the bottom-level solution set, whereas we propose a reformulation based on strong duality. Our novel reformulation is compared to this existing formulation, discussing both model sizes and computational performance. In particular, we apply this trilevel framework to a power market model, exploring the possibilities of an international policymaker in reducing emissions of the system. Using the proposed methods, we are able to obtain globally optimal solutions for a five-node case study representing the Nordic countries and assess the impact of a carbon tax on the electricity production portfolio. Copyright © 2023, The Authors. All rights reserved.

关键词： Energy resources

A high-order finite element methods for nonlinear convection-diffusion equation on time-varying domain

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arXiv 2021年

作者： Ma, Chuwen Zheng, Weiying LSEC NCMIS Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China School of Mathematical Science University of Chinese Academy of Sciences Beijing100049 China

A high-order finite element method is proposed to solve the nonlinear convection-diffusion equation on a time-varying domain whose boundary is implicitly driven by the solution of the equation. The method is semi-implicit in the sense that the boundary is traced explicitly with a high-order surface-tracking algorithm, while the convection-diffusion equation is solved implicitly with high-order backward differentiation formulas and fictitious-domain finite element methods. By numerical experiments on severely deforming domains, we show that optimal convergence orders are obtained in energy norm for both third-order and fourth-order methods. Copyright © 2021, The Authors. All rights reserved.

关键词： Finite element method