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A BROKEN HARDY INEQUALITY ON FINITE ELEMENT SPACE AND APPLICATION TO STRAIN GRADIENT ELASTICITY

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arXiv 2025年

作者： Liao, Yulei Ming, Pingbing Department of Mathematics Faculty of Science National University of Singapore 10 Lower Kent Ridge Road Singapore119076 Singapore LSEC Institute of Computational Mathematics and Scientific/Engineering Computing AMSS Chinese Academy of Sciences Beijing100190 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China

We illustrate a broken Hardy inequality on discontinuous finite element spaces, blowing up with a logarithmic factor with respect to the meshes size. This is motivated by numerical analysis for the strain gradient elasticity with natural boundary conditions. A mixed finite element pair is employed to solve this model with nearly incompressible materials. This pair is quasi-stable with a logarithmic factor, which is not significant in the approximation error, and converges robustly in the incompressible limit and uniformly in the microscopic material parameter. Numerical results back up that the theoretical predictions are nearly optimal. Moreover, the regularity estimates for the model over a smooth domain have been proved with the aid of the Agmon-Douglis-Nirenberg theory. Copyright © 2025, The Authors. All rights reserved.

关键词： Set theory

NUMERICAL ANALYSIS OF THE PARALLEL ORBITAL-UPDATING APPROACH FOR EIGENVALUE PROBLEMS∗

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arXiv 2024年

作者： Dai, Xiaoying Li, Yan Yang, Bin Zhou, Aihui 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 School of Statistics and Mathematics Central University of Finance and Economics Beijing102206 China

The parallel orbital-updating approach is an orbital/eigenfunction iteration based approach for solving eigenvalue problems when many eigenpairs are required. It has been proven to be efficient, for instance, in electronic structure calculations. In this paper, based on the investigation of a quasi-orthogonality, we present the numerical analysis of the parallel orbital-updating approach for linear eigenvalue problems, including convergence and error estimates of the numerical approximations. Copyright © 2024, The Authors. All rights reserved.

关键词： Eigenvalues and eigenfunctions

A Survey of Recent Advances in Optimization Methods for Wireless Communications

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arXiv 2024年

作者： Liu, Ya-Feng Chang, Tsung-Hui Hong, Mingyi Wu, Zheyu So, Anthony Man-Cho Jorswieck, Eduard A. Yu, Wei 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 School of Science and Engineering The Chinese University of Hong Kong Shenzhen Shenzhen China Shenzhen Research Institute of Big Data China The Department of Electrical and Computer Engineering University of Minnesota MinneapolisMN55455 United States The Department of Systems Engineering and Engineering Management The Chinese University of Hong Kong Hong Kong The Institute for Communication Technology Technische Universitat Braunschweig Germany The Edward S. Rogers Sr. Department of Electrical and Computer Engineering University of Toronto TorontoONM5S 3G4 Canada

mathematical optimization is now widely regarded as an indispensable modeling and solution tool for the design of wireless communications systems. While optimization has played a significant role in the revolutionary progress in wireless communication and networking technologies from 1G to 5G and onto the future 6G, the innovations in wireless technologies have also substantially transformed the nature of the underlying mathematical optimization problems upon which the system designs are based and have sparked significant innovations in the development of methodologies to understand, to analyze, and to solve those problems. In this paper, we provide a comprehensive survey of recent advances in mathematical optimization theory and algorithms for wireless communication system design. We begin by illustrating common features of mathematical optimization problems arising in wireless communication system design. We discuss various scenarios and use cases and their associated mathematical structures from an optimization perspective. We then provide an overview of recently developed optimization techniques in areas ranging from nonconvex optimization, global optimization, and integer programming, to distributed optimization and learning-based optimization. The key to successful solution of mathematical optimization problems is in carefully choosing or developing suitable algorithms (or neural network architectures) that can exploit the underlying problem structure. We conclude the paper by identifying several open research challenges and outlining future research directions. Copyright © 2024, The Authors. All rights reserved.

关键词： Beamforming

New semidefinite relaxations for a class of complex quadratic programming problems

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arXiv 2023年

作者： Xu, Yingzhe Lu, Cheng Deng, Zhibin Liu, Ya-Feng School of Economics and Management North China Electric Power University Beijing102206 China School of Economics and Management University of Chinese Academy of Sciences MOE Social Science Laboratory of Digital Economic Forecasts and Policy Simulation at UCAS Beijing100190 China 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

In this paper, we propose some new semidefinite relaxations for a class of nonconvex complex quadratic programming problems, which widely appear in the areas of signal processing and power system. By deriving new valid constraints to the matrix variables in the lifted space, we derive some enhanced semidefinite relaxations of the complex quadratic programming problems. Then, we compare the proposed semidefinite relaxations with existing ones, and show that the newly proposed semidefinite relaxations could be strictly tighter than the previous ones. Moreover, the proposed semidefinite relaxations can be applied to more general cases of complex quadratic programming problems, whereas the previous ones are only designed for special cases. Numerical results indicate that the proposed semidefinite relaxations © 2023, CC BY.

关键词： Approximation algorithms

A Reissner-Mindlin plate formulation using symmetric Hu-Zhang elements via polytopal transformations

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arXiv 2023年

作者： Sky, Adam Neunteufel, Michael Hale, Jack S. Zilian, Andreas Institute of Computational Engineering and Sciences Department of Engineering Faculty of Science Technology and Medicine University of Luxembourg 6 Avenue de la Fonte Esch-sur-AlzetteL-4362 Luxembourg Institute of Analysis and Scientific Computing Technische Universität Wien Wiedner Hauptstr. 8-10 Wien1040 Austria

In this work we develop new finite element discretisations of the shear-deformable Reissner-Mindlin plate problem based on the Hellinger-Reissner principle of symmetric stresses. Specifically, we use conforming Hu-Zhang elements to discretise the bending moments in the space of symmetric square integrable fields with a square integrable divergence M ∈ HZ ⊂ Hsym(Div). The latter results in highly accurate approximations of the bending moments M and in the rotation field being in the discontinuous Lebesgue space φ ∈ [L2]2, such that the Kirchhoff-Love constraint can be satisfied for t → 0. In order to preserve optimal convergence rates across all variables for the case t → 0, we present an extension of the formulation using Raviart-Thomas elements for the shear stress q ∈ RT ⊂ H(div). We prove existence and uniqueness in the continuous setting and rely on exact complexes for inheritance of well-posedness in the discrete setting. This work introduces an efficient construction of the Hu-Zhang base functions on the reference element via the polytopal template methodology and Legendre polynomials, making it applicable to hp-FEM. The base functions on the reference element are then mapped to the physical element using novel polytopal transformations, which are suitable also for curved geometries. The robustness of the formulations and the construction of the Hu-Zhang element are tested for shearlocking, curved geometries and an L-shaped domain with a singularity in the bending moments M. Further, we compare the performance of the novel formulations with the primal-, MITC- and recently introduced TDNNS methods. © 2023, CC BY-NC-ND.

关键词： Mindlin plates

Non-Iterative Recovery from Nonlinear Observations using Generative Models

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arXiv 2022年

作者： Liu, Jiulong Liu, Zhaoqiang LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and System Sciences Chinese Academy of Sciences 100190 China Department of Computer Science National University of Singapore Singapore

In this paper, we aim to estimate the direction of an underlying signal from its nonlinear observations following the semi-parametric single index model (SIM). Unlike conventional compressed sensing where the signal is assumed to be sparse, we assume that the signal lies in the range of an L-Lipschitz continuous generative model with bounded k-dimensional inputs. This is mainly motivated by the tremendous success of deep generative models in various real applications. Our reconstruction method is non-iterative (though approximating the projection step may use an iterative procedure) and highly efficient, and it is shown to attain the near-optimal statistical rate of order √(k log L)/m, where m is the number of measurements. We consider two specific instances of the SIM, namely noisy 1-bit and cubic measurement models, and perform experiments on image datasets to demonstrate the efficacy of our method. In particular, for the noisy 1-bit measurement model, we show that our non-iterative method significantly outperforms a state-of-the-art iterative method in terms of both accuracy and efficiency. Copyright © 2022, The Authors. All rights reserved.

关键词： Iterative methods

An extended plane wave framework for the electronic structure calculations of twisted bilayer material systems

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arXiv 2023年

作者： Dai, Xiaoying Zhou, Aihui Zhou, Yuzhi 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 Beijing100088 China

In this paper, we propose an extended plane wave framework to make the electronic structure calculations of the twisted bilayer 2D material systems practically feasible. Based on the foundation in [Y. Zhou, H. Chen, A. Zhou, J. Comput. Phys. 384, 99 (2019)], following extensions take place: (1) an tensor-producted basis set, which adopts PWs in the incommensurate dimensions, and localized basis in the interlayer dimension, (2) a practical application of a novel cutoff techniques we have recently developed, and (3) a quasi-band structure picture under the small twisted angles and weak interlayer coupling limits. With (1) and (2) now the dimensions of Hamiltonian matrix are reduced by about 2 orders of magnitude compared with the original framework. And (3) enables us to better organize the calculations and understand the results. For numerical examples, we study the electronic structures of the linear bilayer graphene lattice system with the magic twisted angle (∼ 1.05◦). The famous flat bands have been reproduced with their features in quantitative agreement with those from experiments and other theoretical calculations. Moreover, the extended framework has much less computational cost compared to the commensurate cell approximations, and is more extendable compared to the traditional model hamiltonians and tight binding models. Finally this framework can readily accommodate nonlinear models thus will laid the foundations for more effective yet accurate Density Functional Theory (DFT) calculations. © 2023, CC BY.

关键词： Electronic structure

Localization in the Incommensurate Systems: A Plane Wave Study via Effective Potentials

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arXiv 2023年

作者： Wang, Ting Zhou, Yuzhi Zhou, Aihui LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China CAEP Software Center for High Performance Numerical Simulation Beijing100088 China Institute of Applied Physics and Computational Mathematics Beijing100094 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China

In this paper, we apply the effective potentials in the localization landscape theory (Filoche et al., 2012, Arnold et al., 2016) to study the spectral properties of the incommensurate systems. We uniquely develop a plane wave method for the effective potentials of the incommensurate systems and utilize that, the localization of the electron density can be inferred from the effective potentials. Moreover, we show that the spectrum distribution can also be obtained from the effective potential version of Weyl’s law. We perform some numerical experiments on some typical incommensurate systems, showing that the effective potential provides an alternative tool for investigating the localization and spectrum distribution of the systems. Copyright © 2023, The Authors. All rights reserved.

关键词： Elastic waves

ORDER REDUCED METHODS FOR QUAD-CURL EQUATIONS WITH NAVIER TYPE BOUNDARY CONDITIONS

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Journal of computational mathematics 2020年第4期38卷 565-579页

作者： Weifeng Zhang Shuo Zhang Department of Engineering Mechanics and Department of Applied Mathematics Kunming University of Science and TechnologyKunming 650500China Shuo Zhang LSEC Institute of Computational Mathematics and Scientific/Engineering ComputingAcademy of Mathematics and System SciencesChinese Academy of SciencesBeijing 100190China

Quad-curl equations with Navier type boundary conditions are studied in this *** order reduced formulations equivalent to the model problems are presented,and finite element discretizations are *** convergence rates a... 详细信息

关键词： Quad-curl equation Order reduced scheme Regularity analysis Finite element method