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CPAFT: A Consistent Parallel Advancing Front Technique for Unstructured Triangular/Tetrahedral Mesh Generation

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arXiv 2024年

作者： Ma, Chengdi Huang, Jizu Luo, Hao Yang, Chao School of Mathematical Sciences Peking University Beijing100871 China Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China PKU-Changsha Institute for Computing and Digital Economy Peking University Hunan410006 China

Compared with the remarkable progress made in parallel numerical solvers of partial differential equations, the development of algorithms for generating unstructured triangular/tetrahedral meshes has been relatively sluggish. In this paper, we propose a novel, consistent parallel advancing front technique (CPAFT) by combining the advancing front technique, the domain decomposition method based on space-filling curves, the distributed forest-of-overlapping-trees approach, and the consistent parallel maximal independent set algorithm. The newly proposed CPAFT algorithm can mathematically ensure that the generated unstructured triangular/tetrahedral meshes are independent of the number of processors and the implementation of domain decomposition. Several numerical tests are conducted to validate the parallel consistency and outstanding parallel efficiency of the proposed algorithm, which scales effectively up to two thousand processors. This is, as far as we know, the first parallel unstructured triangular/tetrahedral mesh generator with scalability to O(1,000) CPU *** Codes 65M50, 65M55, 68W10 Copyright © 2024, The Authors. All rights reserved.

关键词： Domain decomposition methods

Deep Learning-Based Image Analysis to Track Vehicles 2

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Deep Learning-Based Image Analysis to Track Vehicles

2nd International Conference on Disruptive Technologies, ICDT 2024

作者： Prabakaran, P. Jayasudha, R. Sandhu, Mukta Gupta, Sandeep School of Computing Sciences Vels Institute of Science Technology and Advanced Studies Department of Computer Applications Chennai India Dr.N.G.P.Institute of Technology Department of Mathematics Coimbatore India Shri Vishwakarma Skill University Skill Department of Computer Science and Engineering Haryana Palwal India Department of Electrical Engineering Dehradun India

ISBN: (纸本)9798350371055

There are many ways to track the traffic conditions on roads. With the rise of AI-based image processing technology, there has been a surge in interest in developing traffic monitoring systems that rely on camera vision data. An improved road monitoring system may be derived from the data collected in this research by placing a camera at a junction. The technique detects and classifies vehicles using a deep-learning-based methodology (YOLOv4). By comparing the positions of the vehicle with the map HD, lane-by-lane trajectories of the vehicle may be approximated. The predicted vehicle trajectories are used to calculate each lane's wait lengths and traffic volumes, broken down by travel direction. This technique was evaluated using five distinct metrics: vehicle detection rates, trajectory prediction, type of vehicle categorization, traffic volume estimate, and queue length assessment. Thousands of samples were used to assess its performance. Based on the findings, it is acceptable to assume that the system can identify vehicles with a 98% success rate and an error rate of less than 25% regarding lane-by-lane trip volume estimation and vehicle type classification. Therefore, this study's approach reveals the viability of collecting comprehensive traffic information from a camera set up at a junction. Current traffic monitoring schemes have a reasonable likelihood of being developed by this study's significant contribution, which is a strategy that combines AI with HD map approaches. © 2024 IEEE.

关键词： Artificial Intelligence HD Map Traffic Monitoring System Vehicle Type Classification YOLOv4

Low-rank alternating direction doubling algorithm for solving large-scale continuous time algebraic Riccati equations

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arXiv 2024年

作者： Zhang, Juan Xun, Wenlu Key Laboratory of Intelligent Computing and Information Processing Ministry of Education Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan China School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan China

This paper proposes an effective low-rank alternating direction doubling algorithm (R-ADDA) for computing numerical low-rank solutions to large-scale sparse continuous-time algebraic Riccati matrix equations. The method is based on the alternating direction doubling algorithm (ADDA), utilizing the low-rank property of matrices and employing Cholesky factorization for solving. The advantage of the new algorithm lies in computing only the 2k-th approximation during the iterative process, instead of every approximation. Its efficient low-rank formula saves storage space and is highly effective from a computational perspective. Finally, the effectiveness of the new algorithm is demonstrated through theoretical analysis and numerical experiments. Copyright © 2024, The Authors. All rights reserved.

关键词： Matrix algebra

Optimization methods for solving matrix equations

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arXiv 2024年

作者： Zhang, Juan Luo, Xiao Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan China School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan China

In this paper, we focus on using optimization methods to solve matrix equations by transforming the problem of solving the Sylvester matrix equation or continuous algebraic Riccati equation into an optimization problem. Initially, we use a constrained convex optimization method (CCOM) to solve the Sylvester matrix equation with 2,1-norm, where we provide a convergence analysis and numerical examples of CCOM;however, the results show that the algorithm is not efficient. To address this issue, we employ classical quasi-Newton methods such as DFP and BFGS algorithms to solve the Sylvester matrix equation and present the convergence and numerical results of the algorithm. Additionally, we compare these algorithms with the CG algorithm and AR algorithm, and our results demonstrate that the presented algorithms are effective. Furthermore, we propose a unified framework of the alternating direction multiplier method (ADMM) for directly solving the continuous algebraic Riccati equation (CARE), and we provide the convergence and numerical results of ADMM. Our experimental results indicate that ADMM is an effective optimization algorithm for solving CARE. Finally, to improve the effectiveness of the optimization method for solving Riccati equation, we propose the Newton-ADMM algorithm framework, where the outer iteration of this method is the classical Newton method, and the inner iteration involves using ADMM to solve Lyapunov matrix equations inexactly. We also provide the convergence and numerical results of this algorithm, which our results demonstrate are more efficient than ADMM for solving CARE. Copyright © 2024, The Authors. All rights reserved.

关键词： Newton-Raphson method

Cryptography: Recent research trends of encrypting mathematics

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Materials Today: Proceedings 2022年 56卷 3247-3253页

作者： Kikani, Ria Jatin Verma, Kirti Navalakhe, Rachna Shrivastava, Gourav Shrivastava, Vineeta Department of Information Technology Silver Oak College of Engineering and Technology GTU Gujarat India Department of Engineering Mathematics Lakshmi Narain College of Technology MP Bhopal India Department of Applied Mathematics and Computational Science Shri G.S. Institute of Technology and Science MP Indore India Department of Computer Science and Engineering SAGE University MP Bhopal India Department of Computer Science and Engineering Lakshmi Narain College of Technology MP Bhopal India

In today's world where technology and mathematics are progressing hand in hand there are so many things to be considered and thought of when it comes to network security. Cryptography plays a prominent and an important role when it comes to securing the system from unwanted hazardous attacks. As the Internet has made great strides in different domains it has impacted the human life in several ways and hence data security has become the question of the hour. In such a scenario cryptography plays a big role as it ensures that the data transmitted is received as it is without any alteration or modification. However, we have discussed the algorithms and various methods required in order to maintain the security. In the subsequent sections we shall be discussing about theoretical computer science, cryptography attacks, algorithms and its application in everyday life. © 2021

关键词： Cryptography

When are dynamical systems learned from time series data statistically accurate? 38

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When are dynamical systems learned from time series data sta...

38th Conference on Neural Information Processing Systems, NeurIPS 2024

作者： Park, Jeongjin Yang, Nicole Tianjiao Chandramoorthy, Nisha School of Computational Science and Engineering Georgia Institute of Technology AtlantaGA30332 United States Department of Mathematics Emory University AtlantaGA30322 United States Department of Statistics The University of Chicago ChicagoIL60637 United States

Conventional notions of generalization often fail to describe the ability of learned models to capture meaningful information from dynamical data. A neural network that learns complex dynamics with a small test error may still fail to reproduce its physical behavior, including associated statistical moments and Lyapunov exponents. To address this gap, we propose an ergodic theoretic approach to generalization of complex dynamical models learned from time series data. Our main contribution is to define and analyze generalization of a broad suite of neural representations of classes of ergodic systems, including chaotic systems, in a way that captures emulating underlying invariant, physical measures. Our results provide theoretical justification for why regression methods for generators of dynamical systems (Neural ODEs) fail to generalize, and why their statistical accuracy improves upon adding Jacobian information during training. We verify our results on a number of ergodic chaotic systems and neural network parameterizations, including MLPs, ResNets, Fourier Neural layers, and RNNs. © 2024 Neural information processing systems foundation. All rights reserved.

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Low-rank generalized alternating direction implicit iteration method for solving matrix equations

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arXiv 2024年

作者： Zhang, Juan Xun, Wenlu Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan China School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan China

This paper presents an effective low-rank generalized alternating direction implicit iteration (R-GADI) method for solving large-scale sparse and stable Lyapunov matrix equations and continuous-time algebraic Riccati matrix equations. The method is based on generalized alternating direction implicit iteration (GADI), which exploits the low-rank property of matrices and utilizes the Cholesky factorization approach for solving. The advantage of the new algorithm lies in its direct and efficient low-rank formulation, which is a variant of the Cholesky decomposition in the Lyapunov GADI method, saving storage space and making it computationally effective. When solving the continuous-time algebraic Riccati matrix equation, the Riccati equation is first simplified to a Lyapunov equation using the Newton method, and then the R-GADI method is employed for computation. Additionally, we analyze the convergence of the R-GADI method and prove its consistency with the convergence of the GADI method. Finally, the effectiveness of the new algorithm is demonstrated through corresponding numerical experiments. Copyright © 2024, The Authors. All rights reserved.

关键词： Riccati equations

Generalization Error Estimates of Machine Learning Methods for Solving High Dimensional Schrödinger Eigenvalue Problems

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arXiv 2024年

作者： Yu, Hao Guo, Yixiao Ming, Pingbing LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China

We propose a machine learning method for computing eigenvalues and eigenfunctions of the Schrödinger operator on a d-dimensional hypercube with Dirichlet boundary conditions. The cut-off function technique is employed to construct trial functions that precisely satisfy the homogeneous boundary conditions. This approach eliminates the error caused by the standard boundary penalty method, improves the overall accuracy of the method, as demonstrated by the typical numerical examples. Under the assumption that the eigenfunctions belong to a spectral Barron space, we derive an explicit convergence rate of the generalization error of the proposed method, which does not suffer from the curse of dimensionality. We verify the assumption by proving a new regularity shift result for the eigenfunctions when the potential function belongs to an appropriate spectral Barron space. Moreover, we extend the generalization error bound to the normalized penalty method, which is widely used in practice. Copyright © 2024, The Authors. All rights reserved.

关键词： Boundary conditions

A Diffuse-Interface Lattice Boltzmann Method for the Dendritic Growth with Thermosolutal Convection

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Communications in computational Physics 2023年第4期33卷 1164-1188页

作者： Chengjie Zhan Zhenhua Chai Baochang Shi Ping Jiang Shaoning Geng Dongke Sun School of Mathematics and Statistics Huazhong University of Science and TechnologyWuhan430074P.R.China Institute of Interdisciplinary Research for Mathematics and Applied Science Huazhong University of Science and TechnologyWuhan 430074P.R.China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and TechnologyWuhan430074P.R.China The State Key Laboratory of Digital Manufacturing Equipment and Technology School of Mechanical Science and EngineeringHuazhong University of Science and TechnologyWuhan 430074P.R.China School of Mechanical Engineering Southeast UniversityNanjing 211189P.R.China

In this work,we proposed a diffuse-interface model for the dendritic growth with thermosolutal *** this model,the sharp boundary between the fluid and solid dendrite is firstly replaced by a thin but nonzero thickness diffuse interface,which is described by the order parameter,and the diffuse-interface based governing equations for the dendritic growth are *** solve the model for the dendritic growth with thermosolutal convection,we also developed a diffuse-interface multirelaxation-time lattice Boltzmann(LB)*** this method,the order parameter in the phase-field equation is combined into the force caused by the fluid-solid interaction,and the treatment on the complex fluid-solid interface can be *** addition,four LB models are designed for the phase-field equation,concentration equation,temperature equation and the Navier-Stokes equations in a unified ***,we performed some simulations of the dendritic growth to test the present diffuse-interface LB method,and found that the numerical results are in good agreements with some previous works.

关键词： Dendritic growth diffuse interface lattice Boltzmann method phase-field method.