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Finite Element Method for a Nonlinear PML Helmholtz Equation with High Wave Number

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arXiv 2022年

作者： Jiang, Run Li, Yonglin Wu, Haijun Zou, Jun Department of Mathematics Nanjing University Jiangsu210093 China Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China Department of Applied Mathematics The Hong Kong Polytechnic University Hung Hom Hong Kong Department of Mathematics The Chinese University of Hong Kong N.T. Shatin Hong Kong

A nonlinear Helmholtz equation (NLH) with high wave number and Sommerfeld radiation condition is approximated by the perfectly matched layer (PML) technique and then discretized by the linear finite element method (FEM). Wave-number-explicit stability and regularity estimates and the exponential convergence are proved for the nonlinear truncated PML problem. Preasymptotic error estimates are obtained for the FEM, where the logarithmic factors in h required by the previous results for the NLH with impedance boundary condition are removed in the case of two dimensions. Moreover, local quadratic convergences of the Newton's methods are derived for both the NLH with PML and its FEM. Numerical examples are presented to verify the accuracy of the FEM, which demonstrate that the pollution errors may be greatly reduced by applying the interior penalty technique with proper penalty parameters to the FEM. The nonlinear phenomenon of optical bistability can be successfully simulated. Copyright © 2022, The Authors. All rights reserved.

关键词： Finite element method

A Compact Simple Hweno Scheme with Ader Time Discretization for Hyperbolic Conservation Laws I: Structured Meshes

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SSRN

SSRN 2023年

作者： Luo, Dongmi Li, Shiyi Qiu, Jianxian Zhu, Jun Chen, Yibing Institute of Applied Physics and Computational Mathematics Beijing100088 China School of Mathematical Sciences Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing Xiamen University Fujian Xiamen361005 China MIIT Nanjing University of Aeronautics and Astronautics Jiangsu Nanjing210016 China

In this paper, a compact and high order ADER (Arbitrary high order using DERivatives) scheme using the simple HWENO method (ADER-SHWENO) is proposed for hyperbolic conservation laws. The newly-developed method employs the Lax-Wendroff procedure to convert time derivatives to spatial derivatives, which provides the time evolution of the variables at the cell interfaces. This information is required for the simple HWENO reconstructions, which take advantages of the simple WENO and the classic HWENO. Compared with the original Runge-Kutta HWENO method (RK-HWENO), the new method has two advantages. Firstly, RK-HWENO method must solve the additional equations for reconstructions, which is avoided for the new method. Secondly, the SHWENO reconstruction is performed once with one stencil and is different from the classic HWENO methods, in which both the function and its derivative values are reconstructed with two different stencils, respectively. Thus the new method is more efficient than the RK-HWENO method. Moreover, the new method is more compact than the existing ADER-WENO method. Besides, the new method makes the best use of the information in the ADER method. Thus, the time evolution of the cell averages of the derivatives is simpler than that developed in the work [Li et. al., 447 (2021), 110661.]. Numerical tests indicate that the new method can achieve high order for smooth solutions both in space and time, keep non-oscillatory at discontinuities. © 2023, The Authors. All rights reserved.

关键词： Runge Kutta methods

PowerNet: Efficient Representations of Polynomials and Smooth Functions by Deep Neural Networks with Rectified Power Units

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数学研究 2020年第2期53卷 159-191页

作者： Bo Li Shanshan Tang Haijun Yu NCMIS&LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Beijing 100190 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing 100049 China China Justice Big Data Institute Beijing 100043 China

Deep neural network with rectified linear units (ReLU) is getting more and more popular recently. However, the derivatives of the function represented by a ReLU network are not continuous, which limit the usage of ReLU network to situations only when smoothness is not required. In this paper, we construct deep neural networks with rectified power units (RePU), which can give better approximations for smooth functions. Optimal algorithms are proposed to explicitly build neural networks with sparsely connected RePUs, which we call PowerNets, to represent polynomials with no approximation error. For general smooth functions, we first project the function to their polynomial approximations, then use the proposed algorithms to construct cor-responding PowerNets. Thus, the error of best polynomial approximation provides an upper bound of the best RePU network approximation error. For smooth functions in higher dimensional Sobolev spaces, we use fast spectral transforms for tensor-product grid and sparse grid discretization to get polynomial approximations. Our construc-tive algorithms show clearly a close connection between spectral methods and deep neural networks: PowerNets with n hidden layers can exactly represent polynomials up to degree sn, where s is the power of RePUs. The proposed PowerNets have po-tential applications in the situations where high-accuracy is desired or smoothness is required.

关键词： Deep neural network rectified linear unit rectified power unit sparse grid Power-Net

An Adaptive Finite Element DtN Method for the Acoustic-Elastic Interaction Problem in Periodic Structures

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arXiv 2025年

作者： Lin, Lei Lv, Junliang School of Mathematics Jilin University Qianjin Street Jilin Province Changchun130012 China LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China

Consider a time-harmonic acoustic plane wave incident onto an elastic body with an unbounded periodic surface. The medium above the surface is supposed to be filled with a homogeneous compressible inviscid air/fluid of constant mass density, while the elastic body is assumed to be isotropic and linear. By introducing the Dirichlet-to-Neumann (DtN) operators for acoustic and elastic waves simultaneously, the model is formulated as an acoustic-elastic interaction problem in periodic structures. Based on a duality argument, an a posteriori error estimate is derived for the associated truncated finite element approximation. The a posteriori error estimate consists of the finite element approximation error and the truncation error of two different DtN operators, where the latter decays exponentially with respect to the truncation parameter. Based on the a posteriori error, an adaptive finite element algorithm is proposed for solving the acoustic-elastic interaction problem in periodic structures. Numerical experiments are presented to demonstrate the effectiveness of the proposed *** Codes 65N12, 65N15, 65N30 © 2025, CC BY-NC-ND.

关键词： Acoustic noise

Adaptive finite element approximations of the first eigenpair associated with p-Laplacian

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arXiv 2024年

作者： Li, Guanglian Li, Jing Merten, Julie Xu, Yifeng Zhu, Shengfeng Department of Mathematics The University of Hong Kong Hong Kong School of Mathematical Sciences East China Normal University Shanghai200241 China Computational and Numerical Mathematics Bernoulli Institute University of Groningen Netherlands Department of Mathematics Scientific Computing Key Laboratory of Shanghai Universities Shanghai Normal University Shanghai200234 China Key Laboratory of MEA Ministry of Education Shanghai Key Laboratory of PMMP School of Mathematical Sciences East China Normal University Shanghai200241 China

In this paper, we propose an adaptive finite element method for computing the first eigenpair of the p-Laplacian problem. We prove that starting from a fine initial mesh our proposed adaptive algorithm produces a sequence of discrete first eigenvalues that converges to the first eigenvalue of the continuous problem and the distance between discrete eigenfunctions and the normalized eigenfunction set corresponding to the first eigenvalue in W1,p-norm also tends to zero. Extensive numerical examples are provided to show the effectiveness and *** Codes 65N12, 65N25, 65N30, 65N50, 35P30 © 2024, CC BY-NC-SA.

关键词： Discrete element methods

A lowest-degree strictly conservative finite element scheme for incompressible stokes problem on general triangulations

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arXiv 2021年

作者： Liu, Wenjia Zhang, Shuo LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and System Sciences Chinese Academy of Sciences Beijing100190 China University of Chinese Academy of Sciences Beijing100049 China

In this paper, we propose a finite element pair for incompressible Stokes problem. The pair uses a slightly enriched piecewise linear polynomial space for velocity and piecewise constant space for pressure, and is illustrated to be a lowest-degree conservative stable pair for the Stokes problem on general triangulations. Copyright © 2021, The Authors. All rights reserved.

关键词： Navier Stokes equations

Optimality conditions for homogeneous polynomial optimization on the unit sphere

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arXiv 2021年

作者： Huang, Lei Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing China

In this note, we prove that for homogeneous polynomial optimization on the sphere, if the objective f is generic in the input space, all feasible points satisfying the first order and second order necessary optimality conditions are local minimizers, which addresses an issue raised in the recent work by Lasserre (Optimization Letters, 2021). As a corollary, this implies that Lasserre’s hierarchy has finite convergence when f is generic. Copyright © 2021, The Authors. All rights reserved.

关键词： Spheres

Covariance-Based Joint Device Activity and Delay Detection in Asynchronous mMTC

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arXiv 2021年

作者： Wang, Zhaorui Liu, Ya-Feng Liu, Liang The Department of Information Engineering The Chinese University of Hong Kong Hong Kong 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 Department of Electronic and Information Engineering The Hong Kong Polytechnic University Hong Kong

In this letter, we study the joint device activity and delay detection problem in asynchronous massive machine-type communications (mMTC), where all active devices asynchronously transmit their preassigned preamble sequences to the base station (BS) for device identification and delay detection. We first formulate this joint detection problem as a maximum likelihood estimation problem, which depends on the received signal only through its sample covariance, and then propose efficient coordinate descent type of algorithms to solve the formulated problem. Our proposed covariance-based approach is sharply different from the existing compressed sensing (CS) approach for the same problem. Numerical results show that our proposed covariance-based approach significantly outperforms the CS approach in terms of the detection performance since our proposed approach can make better use of the BS antennas than the CS approach. © 2021, CC BY.

关键词： Compressed sensing

A compact simple HWENO scheme with ADER time discretization for hyperbolic conservation laws I: structured meshes

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arXiv 2023年

关键词： Runge Kutta methods