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ON SHRINKING TARGETS FOR LINEAR EXPANDING AND HYPERBOLIC TORAL ENDOMORPHISMS

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arXiv 2024年

作者： Hu, Zhang-Nan Persson, Tomas Wu, Wanlou Zhang, Yiwei College of Science China University of Petroleum Beijing102249 China Centre for Mathematical Sciences Lund University Box 118 Lund221 00 Sweden School of Mathematics and Statistics Jiangsu Normal University Xuzhou221116 China School of Mathematics and Statistics Center for Mathematical Sciences Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China

Let A be an invertible d × d matrix with integer elements. Then A determines a self-map T of the d-dimensional torus Td = Rd/Zd. Given a real number τ > 0, and a sequence {zn} of points in Td, let Wτ be the set of points x ∈ Td such that Tn(x) ∈ B(zn, e−nτ) for infinitely many n ∈ . The Hausdorff dimension of Wτ has previously been studied by Hill-Velani and Li-Liao-Velani-Zorin. We provide complete results on the Hausdorff dimension of Wτ for any expanding matrix. For hyperbolic matrices, we compute the dimension of Wτ only when A is a 2 × 2 matrix. We give counterexamples to a natural candidate for a dimension formula for general dimension *** Codes 37C45, 37E05 Copyright © 2024, The Authors. All rights reserved.

关键词： Matrix algebra

Efficient Estimation of Sensor Biases for the 3-Dimensional Asynchronous Multi-Sensor System

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arXiv 2021年

作者： Pu, Wenqiang Liu, Ya-Feng Luo, Zhi-Quan Shenzhen Research Institute of Big Data The Chinese University of Hong Kong Shenzhen518172 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

An important preliminary procedure in multi-sensor data fusion is sensor registration, and the key step in this procedure is to estimate sensor biases from their noisy measurements. There are generally two difficulties in this bias estimation problem: one is the unknown target states which serve as the nuisance variables in the estimation problem, and the other is the highly nonlinear coordinate transformation between the local and global coordinate systems of the sensors. In this paper, we focus on the 3-dimensional asynchronous multi-sensor scenario and propose a weighted nonlinear least squares (NLS) formulation by assuming that there is a target moving with a nearly constant velocity. We propose two possible choices of the weighting matrix in the NLS formulation, which correspond to classical and weighted NLS estimation and maximum likelihood (ML) estimation, respectively. To address the intrinsic nonlinearity, we propose a block coordinate descent (BCD) algorithm for solving the formulated problem, which alternately updates different kinds of bias estimates. Specifically, the proposed BCD algorithm involves solving linear LS problems and nonconvex quadratically constrained quadratic program (QCQP) problems with special structures. Instead of adopting the semidefinite relaxation technique, we develop a much more computationally efficient algorithm based on the alternating direction method of multipliers (ADMM) to solve the nonconvex QCQP subproblems. The convergence of the ADMM to the global solution of the QCQP subproblems is established under mild conditions. The effectiveness and efficiency of the proposed BCD algorithm are demonstrated via numerical simulations. Copyright © 2021, The Authors. All rights reserved.

关键词： Maximum likelihood estimation

Freezing dynamics of wetting droplet under a uniform electric field

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arXiv 2024年

作者： Huang, Jiangxu Li, Hanqing Che, Jiaqi Chai, Zhenhua Wang, Lei Shi, Baochang School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Institute of Interdisciplinary Research for Mathematics and Applied Science Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China Marine Design & Research Institute of China China The State Key Laboratory of Digital Manufacturing Equipment and Technology Huazhong University of Science and Technology Wuhan430074 China School of Mathematics and Physics China University of Geosciences Wuhan430074 China

Electrofreezing is a powerful technique that employs the electric field to control and enhance the freezing process. In this work, a phase-field-based lattice Boltzmann (LB) method is developed to study the electrofreezing process of sessile droplet on a cooled substrate. The accuracy of the present LB method is first validated through performing some simulations of the three-phase Stefan problem, the droplet freezing on a cold wall, and the droplet deformation under a uniform electric field. Then it is used to investigate the effect of an electric field on the freezing of a wetting droplet on a cold substrate, and the numerical results show that the electric field has a significant influence on the freezing time of the droplet mainly through changing the morphology of the droplet. In particular, under the effect of the electric field, the freezing time is increased for the droplet with a prolate pattern, while the freezing time of the droplet with an oblate pattern is decreased. These numerical results bring some new insights on the electrofreezing and provide a valuable guidance for the precise regulation of droplet freezing. Copyright © 2024, The Authors. All rights reserved.

关键词： Wetting

A Progressive Image Restoration Network for High-order Degradation Imaging in Remote Sensing

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arXiv 2024年

作者： Feng, Yujie Yang, Yin Fan, Xiaohong Zhang, Zhengpeng Bu, Lijing Zhang, Jianping School of Mathematics and Computational Science Xiangtan University Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory for Intelligent Computing and Information Processing of the Ministry of Education Xiangtan411105 China School of Mathematics and Computational Science Xiangtan University National Center for Applied Mathematics in Hunan Hunan International Scientific and Technological Innovation Cooperation Base of Computational Science Xiangtan411105 China College of Mathematical Medicine Zhejiang Normal University Jinhua321004 China School of Automation and Electronic Information Xiangtan University Xiangtan411105 China

Recently, deep learning methods have gained remarkable achievements in the field of image restoration for remote sensing (RS). However, most existing RS image restoration methods focus mainly on conventional first-order degradation models, which may not effectively capture the imaging mechanisms of remote sensing images. Furthermore, many RS image restoration approaches that use deep learning are often criticized for their lacks of architecture transparency and model interpretability. To address these problems, we propose a novel progressive restoration network for high-order degradation imaging (HDI-PRNet), to progressively restore different image degradation. HDI-PRNet is developed based on the theoretical framework of degradation imaging, offering the benefit of mathematical interpretability within the unfolding network. The framework is composed of three main components: a module for image denoising that relies on proximal mapping prior learning, a module for image deblurring that integrates Neumann series expansion with dual-domain degradation learning, and a module for super-resolution. Extensive experiments demonstrate that our method achieves superior performance on both synthetic and real remote sensing images. Copyright © 2024, The Authors. All rights reserved.

关键词： Image reconstruction

Achievable Rate Maximization Pattern Design for Reconfigurable MIMO Antenna Array

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arXiv 2022年

作者： Wang, Haonan Li, Ang Liu, Ya-Feng Qin, Qibo Song, Lingyang Li, Yonghui The School of Information and Communications Engineering Faculty of Electronic and Information Engineering Xi'an Jiaotong University Shaanxi Xi'An710049 China The State Key Laboratory of Integrated Services Networks Xidian University Shaanxi Xi'An China 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 Wireless Network RAN Research Department Shanghai Huawei Technologies Co. Ltd. Shanghai201206 China The School of Electrical Engineering and Computer Science Peking University Beijing100871 China The School of Electrical and Information Engineering The University of Sydney SydneyNSW2006 Australia

Reconfigurable multiple-input multiple-output can provide performance gains over traditional MIMO by reshaping the channels, i.e., introducing more channel realizations. In this paper, we focus on the achievable rate maximization pattern design for reconfigurable MIMO systems. Firstly, we introduce the matrix representation of pattern reconfigurable MIMO (PR-MIMO), based on which a pattern design problem is formulated. To further reveal the effect of the radiation pattern on the wireless channel, we consider pattern design for both the single-pattern case where the optimized radiation pattern is the same for all the antenna elements, and the multi-pattern case where different antenna elements can adopt different radiation patterns. For the single-pattern case, we show that the pattern design is equivalent to a redistribution of gains among all scattering paths, and an eigenvalue optimization based solution is obtained. For the multi-pattern case, we propose a sequential optimization framework with manifold optimization and eigenvalue decomposition to obtain near-optimal solutions. Numerical results validate the superiority of PR-MIMO systems over traditional MIMO in terms of achievable rate, and also show the effectiveness of the proposed solutions. Copyright © 2022, The Authors. All rights reserved.

关键词： MIMO systems

Convergence of physics-informed neural networks applied to linear second-order elliptic interface problems

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arXiv 2022年

作者： Wu, Sidi Zhu, Aiqing Tang, Yifa Lu, Benzhuo State Key Laboratory of Scientific and Engineering Computing National Center for Mathematics and Interdisciplinary Sciences Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China

With the remarkable empirical success of neural networks across diverse scientific disciplines, rigorous error and convergence analysis are also being developed and enriched. However, there has been little theoretical work focusing on neural networks in solving interface problems. In this paper, we perform a convergence analysis of physics-informed neural networks (PINNs) for solving second-order elliptic interface problems. Specifically, we consider PINNs with domain decomposition technologies and introduce gradient-enhanced strategies on the interfaces to deal with boundary and interface jump conditions. It is shown that the neural network sequence obtained by minimizing a Lipschitz regularized loss function converges to the unique solution to the interface problem in H2 as the number of samples increases. Numerical experiments are provided to demonstrate our theoretical analysis. Copyright © 2022, The Authors. All rights reserved.

关键词： Domain decomposition methods

Asymptotic Behavior of the Principal Eigenvalue of a Linear Second Order Elliptic Operator with Large Advection and General Boundary Conditions

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SSRN

SSRN 2023年

作者： Peng, Rui Zhang, Guanghui Zhou, Maolin School of Mathematical Sciences Zhejiang Normal University Zhejiang Jinhua321004 China School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China Chern Institute of Mathematics LPMC Nankai University Tianjin300071 China

In this paper, we consider asymptotic behavior of the principal eigenvalue of some second order elliptic operator with general boundary conditions. For $N=1$, we provide a complete characterization of the asymptotic behavior, assuming that $m'$ changes sign at most finitely many times. Our findings not only improve upon the previous work in \cite{BHN2005,CL2008,PZ2018}, but also partially address some of the open questions posed in \cite{BHN2005}. Furthermore, our results elucidate the novel influence of boundary conditions on such asymptotics. To achieve these results, we develop several sophisticated techniques to overcome the new difficulties posed by the problems under consideration. © 2023, The Authors. All rights reserved.

关键词： Advection

ASYMPTOTIC BEHAVIOR OF THE PRINCIPAL EIGENVALUE OF A LINEAR SECOND ORDER ELLIPTIC OPERATOR WITH LARGE ADVECTION AND GENERAL BOUNDARY CONDITIONS

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arXiv 2023年

Consider the eigenvalue problem of a linear second order elliptic operator: − D∆ − 2α∇m(x) · ∇ + V (x) = λ in Ω, complemented by the Dirichlet boundary condition or the following general Robin boundary condition: (Formula Presented) is a bounded smooth domain, n(x) is the unit exterior normal to ∂Ω at x ∈ ∂Ω, D > 0 and α > 0 are, respectively, the diffusion and advection coefficients, m ∈ C2(Ω), V ∈ C(Ω), β ∈ C(∂Ω) are given functions, and β allows to be positive, sign-changing or negative. In [41], the asymptotic behavior of the principal eigenvalue of the above eigenvalue problem as D → 0 or D → ∞ was studied. In this paper, when N ≥ 2, under proper conditions on the advection function m, we establish the asymptotic behavior of the principal eigenvalue as α → ∞, and when N = 1, we obtain a complete characterization for such asymptotic behavior provided m0 changes sign at most finitely many times. Our results complement or improve those in [4, 7, 44] and also partially answer some questions raised in [4]. © 2023, CC0.

关键词： Advection

The macroscopic finite-difference scheme and modified equations of the general propagation multiple-relaxation-time lattice Boltzmann model

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arXiv 2023年

作者： Chen, Ying Liu, Xi Chai, Zhenhua Shi, Baochang School of Mathematics and Statistics Huazhong University of Science and Technology Hubei Wuhan430074 China Institute of Interdisciplinary Research for Mathematics and Applied Science Huazhong University of Science and Technology Organization Hubei Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Organization Hubei Wuhan430074 China

In this paper, we first present the general propagation multiple-relaxation-time lattice Boltzmann (GPMRT-LB) model and obtain the corresponding macroscopic finite-difference (GPMFD) scheme on conservative moments. Then based on the Maxwell iteration method, we conduct the analysis on the truncation errors and modified equations (MEs) of the GPMRT-LB model and GPMFD scheme at both diffusive and acoustic scalings. For the nonlinear anisotropic convection-diffusion equation (NACDE) and Navier-Stokes equations (NSEs), we also derive the first- and second-order MEs of the GPMRT-LB model and GPMFD scheme. In particular, for the one-dimensional convection-diffusion equation (CDE) with the constant velocity and diffusion coefficient, we can develop a fourth-order GPMRT-LB (F-GPMRT-LB) model and the corresponding fourth-order GPMFD (F-GPMFD) scheme at the diffusive scaling. Finally, two benchmark problems, Gauss hill problem and Poiseuille flow in two-dimensional space, are used to test the GPMRT-LB model and GPMFD scheme, and it is found that the numerical results are not only in good agreement with corresponding analytical solutions, but also have a second-order convergence rate in space. Additionally, a numerical study on one-dimensional CDE also demonstrates that the F-GPMRT-LB model and F-GPMFD scheme can achieve a fourth-order accuracy in space, which is consistent with our theoretical analysis. Copyright © 2023, The Authors. All rights reserved.

关键词： Navier Stokes equations