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Solving a Class of Infinite-Dimensional Tensor Eigenvalue Problems by Translational Invariant Tensor Ring Approximations

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arXiv 2021年

作者： van Beeumen, Roel Periša, Lana Kressner, Daniel Yang, Chao Applied Mathematics and Computational Research Division Lawrence Berkeley National Laboratory BerkeleyCA94720 United States Visage Technologies Diskettgatan 11A LinköpingSE-583 35 Sweden Department of Mathematics École Polytechnique Fédérale de Lausanne Station 8 Lausanne1015 Switzerland

We examine a method for solving an infinite-dimensional tensor eigenvalue problem Hx = λx, where the infinite-dimensional symmetric matrix H exhibits a translational invariant structure. We provide a formulation of this type of problem from a numerical linear algebra point of view and describe how a power method applied to e−Ht is used to obtain an approximation to the desired eigenvector. This infinite-dimensional eigenvector is represented in a compact way by a translational invariant infinite Tensor Ring (iTR). Low rank approximation is used to keep the cost of subsequent power iterations bounded while preserving the iTR structure of the approximate eigenvector. We show how the averaged Rayleigh quotient of an iTR eigenvector approximation can be efficiently computed and introduce a projected residual to monitor its convergence. In the numerical examples, we illustrate that the norm of this projected iTR residual can also be used to automatically modify the time step t to ensure accurate and rapid convergence of the power *** Codes 15A18, 15A21, 15A69, 65F15 Copyright © 2021, The Authors. All rights reserved.

关键词： Tensors

Geometric derivation and structure-preserving simulation of quasi-geostrophy on the sphere

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arXiv 2024年

作者： Luesink, Erwin Franken, Arnout D. Ephrati, Sagy R. Geurts, Bernard J. Department of Applied Mathematics Faculty EEMCS University of Twente PO Box 217 Enschede7500 AE Netherlands Department of Mathematical Sciences Chalmers University of Technology University of Gothenburg Gothenburg412 96 Sweden Center for Computational Energy Research Department of Applied Physics Eindhoven University of Technology Eindhoven Netherlands

We present a geometric derivation of the quasi-geostrophic equations on the sphere, starting from the rotating shallow water equations. We utilise perturbation series methods in vorticity and divergence variables. The derivation employs asymptotic analysis techniques, leading to a global quasi-geostrophic potential vorticity model on the sphere without approximation of the Coriolis parameter. The resulting model forms a closed system for the evolution of potential vorticity with a rich mathematical structure, including Lagrangian and Hamiltonian descriptions. Formulated using the Lie-Poisson bracket reveals the geometric invariants of the quasi-geostrophic model. Motivated by these geometric results, simulations of quasi-geostrophic flow on the sphere are presented based on structure-preserving Lie-Poisson time-integration. We explicitly demonstrate the preservation of Casimir invariants and show that the hyperbolic quasi-geostrophic equations can be simulated in a stable manner over long time. We show the emergence of longitudonal jets, wrapped around the circumference of the sphere in a general direction that is perpendicular to the axis of rotation. Copyright © 2024, The Authors. All rights reserved.

关键词： Spheres

Rydberg state excitation of hydrogen atoms in the intense 800-nm laser field

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Rydberg state excitation of hydrogen atoms in the intense 80...

第七届全国计算原子与分子物理学术会议

作者： Mingqing Liu Shilin Hu Xiaolei Hao Wilhelm Becker Jing Chen Institute of Applied Physics and Computational Mathematics Laboratory of Quantum Engineering and Quantum Metrology School of Physics and Astronomy Sun Yat-Sen University (Zhuhai Campus) Institute of Theoretical Physics and Department of Physics Shanxi University HEDPS Center for Applied Physics and Technology Peking University

Neutral excited states are formed after exposing ground state atoms to external laser *** studiedtheprocessesof Rydberg-State-Excitation(RSE)of atomic hydrogen driven by an intense 800-nm pulse laser *** calculating t... 详细信息

关键词： Rydberg state excitation of hydrogen atoms in the intense 800-nm laser field

Planar chiral meta-materials: controlling the polarization state of light in the far- and near-field

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Planar chiral meta-materials: controlling the polarization s...

Conference on Lasers and Electro-Optics Europe (CLEO EUROPE)

作者： D.M. Bagnall A. Papakostas A. Potts S.L. Prosvirnin S. Takahashi A.V. Zayats N.I. Zheludev Department of Electronics and Computer Science University of Southampton Southampton UK Department of Computational Mathematics Institute of Radio Astronomy Kharkiv Ukraine Department of Pure and Applied Physics The Queen's University of Belfast Belfast UK Department of Physics and Astronomy University of Southampton Southampton UK

We have extended our research to the study of near-field polarization effects arising from the planar chirality. The interaction between the neighboring elements in a metallic structure can be visualized with near-field measurements. Coupling between metallic gammadions that gives rise to the scattered electromagnetic field with polarization components different from the polarization state of the incident light can be directly identified. We have also investigated far-field effects in two-dimensional gratings consisting of regular patterns of dielectric gammadions on a silicon substrate.

关键词： Lighting control Optical polarization Electromagnetic wave polarization Dielectric substrates Visualization Dielectric measurements Electromagnetic measurements Electromagnetic coupling Optical coupling Electromagnetic scattering

Self-assembly of the simple cubic lattice with an isotropic potential

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Physical Review E 2006年第2期74卷 021404-021404页

作者： Mikael C. Rechtsman Frank H. Stillinger Salvatore Torquato Department of Physics Princeton University Princeton New Jersey 08544 USA Department of Chemistry Princeton University Princeton New Jersey 08544 USA Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA Princeton Institute for the Science and Technology Materials Princeton University Princeton New Jersey 08544 USA Princeton Center for Theoretical Physics Princeton University Princeton New Jersey 08544 USA

Conventional wisdom presumes that low-coordinated crystal ground states require directional interactions. Using our recently introduced optimization procedure to achieve self-assembly of targeted structures [M. C. Rechtsman et al., Phys. Rev. Lett. 95, 228301 (2005); Phys. Rev. E 73, 011406 (2006)], we present an isotropic pair potential V(r) for a three-dimensional many-particle system whose classical ground state is the low-coordinated simple cubic lattice. This result is part of an ongoing pursuit by the authors to develop analytical and computational tools to solve statistical-mechanical inverse problems for the purpose of achieving targeted self-assembly. The purpose of these methods is to design interparticle interactions that cause self-assembly of technologically important target structures for applications in photonics, catalysis, separation, sensors, and electronics. We also show that standard approximate integral-equation theories of the liquid state that utilize pair correlation function information cannot be used in the reverse mode to predict the correct simple cubic potential. We report in passing optimized isotropic potentials that yield the body-centered-cubic and simple hexagonal lattices, which provide other examples of non-close-packed structures that can be assembled using isotropic pair interactions.

关键词： CRYSTALS CLUSTERS DENSE

Non-Euclidean Differentially Private Stochastic Convex Optimization: Optimal Rates in Linear Time

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arXiv 2021年

作者： Bassily, Raef Guzmán, Cristóbal Nandi, Anupama The Ohio State University United States Department of Applied Mathematics University of Twente and Institute for Mathematical and Computational Engineering Pontificia Universidad Católica de Chile Chile Department of Computer Science & Engineering The Ohio State University United States

Differentially private (DP) stochastic convex optimization (SCO) is a fundamental problem, where the goal is to approximately minimize the population risk with respect to a convex loss function, given a dataset of n i.i.d. samples from a distribution, while satisfying differential privacy with respect to the dataset. Most of the existing works in the literature of private convex optimization focus on the Euclidean (i.e., 2) setting, where the loss is assumed to be Lipschitz (and possibly smooth) w.r.t. the 2norm over a constraint set with bounded 2diameter. Algorithms based on noisy stochastic gradient descent (SGD) are known to attain the optimal excess risk in this setting. In this work, we conduct a systematic study of DP-SCO for p-setups under a standard smoothness assumption on the loss. For 1 1-setup also extends to general polyhedral norms and feasible sets. Moreover, we show that the excess risk bounds resulting from our algorithms for 1 ≤ p ≤ 2 are attained with high probability. For 2 © 2021, CC BY.

关键词： Convex optimization

Accuracy of a one-dimensional reduction of dynamical systems on networks

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Physical Review E 2022年第2期105卷 024305-024305页

作者： Prosenjit Kundu Hiroshi Kori Naoki Masuda Department of Mathematics State University of New York at Buffalo Buffalo New York 14260-2900 USA Department of Complexity Science and Engineering The University of Tokyo Chiba 277-8561 Japan Computational and Data-Enabled Science and Engineering Program State University of New York at Buffalo Buffalo New York 14260-5030 USA

Resilience is an ability of a system with which the system can adjust its activity to maintain its functionality when it is perturbed. To study resilience of dynamics on networks, Gao et al. [Nature (London) 530, 307 (2016)] proposed a theoretical framework to reduce dynamical systems on networks, which are high dimensional in general, to one-dimensional dynamical systems. The accuracy of this one-dimensional reduction relies on three approximations in addition to the assumption that the network has a negligible degree correlation. In the present study, we analyze the accuracy of the one-dimensional reduction assuming networks without degree correlation. We do so mainly through examining the validity of the individual assumptions underlying the method. Across five dynamical system models, we find that the accuracy of the one-dimensional reduction hinges on the spread of the equilibrium value of the state variable across the nodes in most cases. Specifically, the one-dimensional reduction tends to be accurate when the dispersion of the node's state is small. We also find that the correlation between the node's state and the node's degree, which is common for various dynamical systems on networks, is unrelated to the accuracy of the one-dimensional reduction.

关键词： Complex systems Dynamics of networks Network resilience

Nonequilibrium static diverging length scales on approaching a prototypical model glassy state

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Physical Review E 2012年第2期86卷 021505-021505页

作者： Adam B. Hopkins Frank H. Stillinger Salvatore Torquato Department of Chemistry Princeton University Princeton New Jersey 08544 USA Princeton Institute for the Science and Technology of Materials Princeton University Princeton New Jersey 08544 USA Department of Physics Princeton University Princeton New Jersey 08544 USA Princeton Center for Theoretical Science Princeton University Princeton New Jersey 08544 USA Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA

Maximally random jammed states of hard spheres are prototypical glasses. We study the small wavenumber k behavior of the structure factor S(k) of overcompressed million-sphere packings as a function of density up to the jammed state. We find both a precursor to the glassy jammed state evident long before the jamming density is reached and two associated growing length scales, one extracted from the volume integral of the direct correlation function c(r) and the other from the small-k behavior of the structure factor S(k), that can diverge at the “critical” jammed state. We also define a nonequilibrium index X and use it to demonstrate that the packings studied are intrinsically nonequilibrium in nature well before the critical state is reached. The extension of the results reported in the present work to supercooled atomic-liquid models in which the atoms interact with both repulsive and attractive forces is also discussed.

关键词： PROTOTYPES GLASS FACTOR structure DENSITY functionals NONEQUILIBRIUM thermodynamics MATHEMATICAL models CORRELATION (Statistics) CRITICAL point

Numerical aspect of large-scale electronic state calculation for flexible device material

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arXiv 2018年

作者： Hoshi, Takeo Imachi, Hiroto Kuwata, Akiyoshi Kakuda, Kohsuke Fujita, Takatoshi Matsui, Hiroyuki Department of Applied Mathematics and Physics Tottori University 4-101 Koyama-Minami Tottori680-8552 Japan Department of Theoretical and Computational Molecular Science Institute for Molecular Science Japan Research Center of Organic Electronics Yamagata University Japan

Numerical aspects of large-scale electronic state calculation are explored on flexible organic device materials. Physical theory, numerical method and real application studies are discussed in the context of application-algorithmarchitecture co-design. An application study was carried out for disordered organic thin film. Participation ratio, a measure for the spatial extension of electronic wavefunction is focused on, since it is crucial for device property. A data scientific research is reported for a classification problem of disordered organic polymers, in which participation ratio is used as descriptor. These application studies indicate the potential need of purpose-specific solvers for internal eigenpairs. Copyright © 2018, The Authors. All rights reserved.

关键词： Numerical methods