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An Energy-Stable Finite Element Method for Nonlinear Maxwell's Equations

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SSRN

SSRN 2022年

作者： Li, Lingxiao Lyu, Maohui Zheng, Weiying School of Mathematics and Statistics Henan University Kaifeng475004 China Center for Applied Mathematics of Henan Province Henan University Zhengzhou450046 China 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

In this paper, we consider the three-dimensional (3D) nonlinear Maxwell’s equations in an optical medium characterized by the linear Lorentz dispersion, the nonlinear Kerr effect, and the delayed Raman scattering. By introducing auxiliary variables, we propose a fully-discrete and energy-stable finite element method for the nonlinear problem. The discrete scheme is unconditionally stable and admits second-order convergence. To solve the nonlinear discrete problem, we propose a Newton-Krylov iterative method, where the linearized algebraic problems are solved with a preconditioned GMRES method based on the auxiliary space preconditioning for Maxwell’s equations. Numerical experiment for a spatial soliton shows agreements with the two-dimensional numerical simulations in the literature. Moreover, we also demonstrate the efficiency and parallel scalability of the finite element method numerically on 1–512 computer processors. © 2022, The Authors. All rights reserved.

关键词： Finite element method

Disconnection Flow-Mediated Grain Rotation

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arXiv 2023年

作者： Qiu, Caihao Salvalaglio, Marco Srolovitz, David J. Han, Jian Department of Materials Science and Engineering City University of Hong Kong Hong Kong Institute of Scientific Computing TU Dresden Dresden01062 Germany Dresden Center for Computational Materials Science TU Dresden Dresden01062 Germany Department of Mechanical Engineering The University of Hong Kong Pokfulam Road Hong Kong

Interface migration in microstructures is mediated by the motion of line defects with step and dislocation character, i.e., disconnections. We propose a continuum model for arbitrarily-curved grain boundaries or heterophase interfaces accounting for disconnections’ role in grain rotation. Numerical simulations show that their densities evolve as grain size and shape change, generating stresses and increasing or decreasing misorientation;the predictions agree with molecular dynamics simulations for pure capillarity-driven evolution, and are interpreted through an extended Cahn-Taylor model. Copyright © 2023, The Authors. All rights reserved.

关键词： Molecular dynamics

Stress-constrained topology optimization of lattice-like structures using component-wise reduced order models

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arXiv 2022年

作者： McBane, Sean Choi, Youngsoo Willcox, Karen Oden Institute for Computational Engineering & Sciences University of Texas AustinTX78712 United States Center for Applied Scientific Computing Lawrence Livermore National Laboratory LivermoreCA94550 United States

Lattice-like structures can provide a combination of high stiffness with light weight that is useful in many applications, but a resolved finite element mesh of such structures results in a computationally expensive discretization. This computational expense may be particularly burdensome in many-query applications, such as optimization. We develop a stress-constrained topology optimization method for lattice-like structures that uses component-wise reduced order models as a cheap surrogate, providing accurate computation of stress fields while greatly reducing run time relative to a full order model. We demonstrate the ability of our method to produce large reductions in mass while respecting a constraint on the maximum stress in a pair of test problems. The ROM methodology provides a speedup of about 150x in forward solves compared to full order static condensation and provides a relative error of less than 5% in the relaxed stress. Copyright © 2022, The Authors. All rights reserved.

关键词： Constrained optimization

Correction to: Students’ Development of a Logarithm Function in Python Using Taylor Expansions: a Teaching Design Case Study

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Digital Experiences in mathematics Education 2022年第2期8卷 256-257页

作者： Sand, Odd Petter Lockwood, Elise Caballero, Marcos D. Mørken, Knut Centre for Computing in Science Education (CCSE) University of Oslo Oslo Norway Department of Mathematics Oregon State University Corvallis USA Department of Physics and Astronomy Department of Computational Mathematics Science and Engineering and CREATE for STEM Institute Michigan State University East Lansing USA

An Efficient Novel Approach for Iris Recognition and Segmentation Based on the Utilization of Deep Learning

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An Efficient Novel Approach for Iris Recognition and Segment...

Evolutionary Algorithms and Soft computing Techniques (EASCT), International Conference on

作者： Syed Nawaz Pasha Anitha D R Anbunathan Gobinath C Madhura G K School of Computer Science & Artificial Intelligence SR University Warangal India Department of Computational Intelligence School of Computing SRM Institute of Science and Technology Kattankulathur India Department of Mathematics SA Engineering College Chennai India School of Computing and Information Technology Reva University Bangalore India Department of AI and ML Nitte Meenakshi Institute of Technology Bengaluru India

Iris recognition stands out as one of the most stable and accurate biometric identification systems such as facial, voice, fingerprint, handwriting, and iris recognition itself. Deep Learning (DL) plays a pivotal role in enhancing security by leveraging Convolutional Neural Networks (CNNs) for iris recognition. The acquisition of iris images and real-time processing have become focal points demanding heightened attention in the realm of DL. Iris recognition is prized for its high level of protection, making it challenging to reconstruct iris data easily from the database. The CASIA-Iris Dataset serves as a benchmark to verify the uniqueness of iris patterns and evaluate the performance of iris recognition. However, due to limitations in iris image access, the application of CNNs in the segmentation approach remains critical for iris localization and pupil boundary identification, ensuring both efficiency and specificity in iris images. The results obtained from this study reveal that the proposed CNN, utilizing DL models, achieves an impressive accuracy of 99.89% on the CASIA-Iris dataset, better performance of other existing models, such as U-Net and VGG-Net, and ensuring accurate classification.

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Persistent homology and topological statistics of hyperuniform point clouds

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Physical Review Research 2024年第2期6卷 023107-023107页

作者： Marco Salvalaglio Dominic J. Skinner Jörn Dunkel Axel Voigt Institute of Scientific Computing Technische Universität Dresden 01062 Dresden Germany Dresden Center for Computational Materials Science (DCMS) TU Dresden 01062 Dresden Germany NSF-Simons Center for Quantitative Biology Northwestern University 2205 Tech Drive Evanston Illinois 60208 USA Department of Mathematics Massachusetts Institute of Technology 77 Massachusetts Avenue Cambridge Massachusetts 01239 USA Cluster of Excellence Physics of Life Technische Universität Dresden 01062 Dresden Germany

Hyperuniformity, the suppression of density fluctuations at large length scales, is observed across a wide variety of domains, from cosmology to condensed matter and biological systems. Although the standard definition of hyperuniformity only utilizes information at the largest scales, hyperuniform configurations have distinctive local characteristics. However, the influence of global hyperuniformity on local structure has remained largely unexplored; establishing this connection can help uncover long-range interaction mechanisms and detect hyperuniform traits in finite-size systems. Here, we study the topological properties of hyperuniform point clouds by characterizing their persistent homology and the statistics of local graph neighborhoods. We find that varying the structure factor results in configurations with systematically different topological properties. Moreover, these topological properties are conserved for subsets of hyperuniform point clouds, establishing a connection between finite-sized systems and idealized reference arrangements. Comparing distributions of local topological neighborhoods reveals that the hyperuniform arrangements lie along a primarily one-dimensional manifold reflecting an order-to-disorder transition via hyperuniform configurations. The results presented here complement existing characterizations of hyperuniform phases of matter, and they show how local topological features can be used to detect hyperuniformity in size-limited simulations and experiments.

关键词： Exotic phases of matter Patterns in complex systems Disordered systems Random & disordered media Strongly correlated systems Data analysis Topological tools Topology

On the convergence of stochastic gradient descent with bandwidth-based step size

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arXiv 2021年

作者： Wang, Xiaoyu Yuan, Ya-Xiang Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China University of Chinese Academy of Sciences No.19A Yuquan Road Beijing100049 China State Key Laboratory of Scientific/Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China

We investigate the stochastic gradient descent (SGD) method where the step size lies within a banded region instead of being given by a fixed formula. The optimal convergence rate under mild conditions and large initial step size is proved. Our analysis provides comparable theoretical error bounds for SGD associated with a variety of step sizes. In addition, the convergence rates for some existing step size strategies, e.g., triangular policy and cosine-wave, can be revealed by our analytical framework under the boundary constraints. The bandwidth-based step size provides efficient and flexible step size selection in optimization. We also propose a 1/t up-down policy and give several non-monotonic step sizes. Numerical experiments demonstrate the efficiency and significant potential of the bandwidth-based step-size in many *** Codes 65K05, 65K10, 90C15 Copyright © 2021, The Authors. All rights reserved.

关键词： Bandwidth

CONVERGENCE OF THE PLANEWAVE APPROXIMATIONS FOR QUANTUM INCOMMENSURATE SYSTEMS

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arXiv 2022年

作者： Wang, Ting Chen, Huajie Zhou, Aihui Zhou, Yuzhi Massatt, Daniel School of Mathematical Sciences Beijing Normal University Beijing100875 China 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 CAEP Software Center for High Performance Numerical Simulation Beijing100088 China Institute of Applied Physics and Computational Mathematics Beijing100094 China Mathematics Department Louisiana State University LA70802 United States

Incommensurate structures come from stacking the single layers of low-dimensional materials on top of one another with misalignment such as a twist in orientation. While these structures are of significant physical interest, they pose many theoretical challenges due to the loss of periodicity. This paper studies the physical observables of continuum Schrödinger operators for incommensurate systems. We characterize the density of states and local density of states in real and reciprocal spaces, and develop novel numerical methods to approximate them. In particular, we (i) justify the thermodynamic limit of the density of states and local density of states via the operator kernel characterization;and (ii) propose efficient numerical schemes based on planewave approximation and trapezoidal rule in reciprocal space. We present both rigorous analysis and numerical simulations to support the reliability and efficiency of our numerical *** Codes 65Z05, 81-08, 47G30 Copyright © 2022, The Authors. All rights reserved.

关键词： Numerical methods

Saturated feedback stabilizability to trajectories for the Schlögl parabolic equation

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arXiv 2021年

作者： Azmi, Behzad Kunisch, Karl Rodrigues, Sérgio S. Johann Radon Institute for Computational and Applied Mathematics ÖAW Altenbergerstr. 69 Linz4040 Austria Institute for Mathematics and Scientific Computing 621 Heinrichstrasse 36 Graz8010 Austria

It is shown that there exist a finite number of indicator functions, which allow us to track an arbitrary given trajectory of the Schlögl model, by means of an explicit saturated feedback control input whose magnitude is bounded by a constant independent of the given targeted trajectory. Simulations are presented showing the stabilizing performance of the explicit feedback constrained control. Further, such performance is compared to that of a receding horizon constrained control minimizing the classical energy cost *** Codes 93C20, 93D15, 93B45, 35K58 Copyright © 2021, The Authors. All rights reserved.

关键词： Feedback control