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Energy Stable Nodal DG Methods for Maxwell’s Equations of Mixed-Order Form in Nonlinear Optical Media

Communications on Applied Mathematics and Computation 2024年第1期6卷 30-63页

作者： Maohui Lyu Vrushali A.Bokil Yingda Cheng Fengyan Li State Key Laboratory of Scientific and Engineering Computing(LSEC) Academy of Mathematics and System SciencesChinese Academy of SciencesBeijing100190China Department of Mathematics College of ScienceOregon State UniversityCorvallisOR97331USA Department of Mathematics Michigan State UniversityEast LansingMI48824USA Department of Computational Mathematics Science and EngineeringMichigan State UniversityEast LansingMI48824USA Department of Mathematical Sciences Rensselaer Polytechnic InstituteTroyNY12180USA

In this work,we develop energy stable numerical methods to simulate electromagnetic waves propagating in optical media where the media responses include the linear Lorentz dispersion,the instantaneous nonlinear cubic Kerr response,and the nonlinear delayed Raman molecular vibrational *** the first-order PDE-ODE governing equations considered previously in Bokil et al.(J Comput Phys 350:420–452,2017)and Lyu et al.(J Sci Comput 89:1–42,2021),a model of mixed-order form is adopted here that consists of the first-order PDE part for Maxwell’s equations coupled with the second-order ODE part(i.e.,the auxiliary differential equations)modeling the linear and nonlinear dispersion in the *** main contribution is a new numerical strategy to treat the Kerr and Raman nonlinearities to achieve provable energy stability property within a second-order temporal discretization.A nodal discontinuous Galerkin(DG)method is further applied in space for efficiently handling nonlinear terms at the algebraic level,while preserving the energy stability and achieving high-order *** with d_(E)as the number of the components of the electric field,only a d_(E)×d_(E)nonlinear algebraic system needs to be solved at each interpolation node,and more importantly,all these small nonlinear systems are completely decoupled over one time step,rendering very high parallel *** evaluate the proposed schemes by comparing them with the methods in Bokil et al.(2017)and Lyu et al.(2021)(implemented in nodal form)regarding the accuracy,computational efficiency,and energy stability,by a parallel scalability study,and also through the simulations of the soliton-like wave propagation in one dimension,as well as the spatial-soliton propagation and two-beam interactions modeled by the two-dimensional transverse electric(TE)mode of the equations.

关键词： Maxwell’s equations Kerr and Raman Discontinuous Galerkin method Energy stability

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

The Journal of Machine Learning Research

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The Journal of Machine Learning Research 2023年第1期24卷 1990-2038页

作者： Xiaoyu Wang Ya-xiang Yuan Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China and University of Chinese Academy of Sciences Beijing 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 Beijing China

We first propose a general step-size framework for the stochastic gradient descent(SGD) method: bandwidth-based step sizes that are allowed to vary within a banded region. The framework provides efficient and flexible step size selection in optimization, including cyclical and non-monotonic step sizes (e.g., triangular policy and cosine with restart), for which theoretical guarantees are rare. We provide state-of-the-art convergence guarantees for SGD under mild conditions and allow a large constant step size at the beginning of training. Moreover, we investigate the error bounds of SGD under the bandwidth step size where the boundary functions are in the same order and different orders, respectively. Finally, we propose a 1/t up-down policy and design novel non-monotonic step sizes. Numerical experiments demonstrate these bandwidth-based step sizes' efficiency and significant potential in training regularized logistic regression and several large-scale neural network tasks.

关键词： stochastic gradient descent bandwidth-based step size non-asymptotic convergence non-monotonic step size machine learning

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Spectral Galerkin Approximation of Fractional Optimal Control Problems with Fractional Laplacian

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Advances in Applied Mathematics and Mechanics 2023年第6期15卷 1631-1654页

作者： Jiaqi Zhang Yin Yang Zhaojie Zhou School of Mathematics and Computational Science Xiangtan UniversityHunan Key Laboratory for Computation and Simulation in Science and EngineeringKey Laboratory of Intelligent Computing and Information Processing of Ministry of EducationXiangtan 411105HunanChina School of Mathematics and Computational Science Xiangtan UniversityNational Center for Applied Mathematics in HunanHunan International Scientific and Technological Innovation Cooperation Base of Computational ScienceXiangtan 411105HunanChina School of Mathematics and Statistics Shandong Normal UniversityJinan 250014ShandongChina

In this paper spectral Galerkin approximation of optimal control problem governed by fractional elliptic equation is *** deal with the nonlocality of fractional Laplacian operator the Caffarelli-Silvestre extension is *** first order optimality condition of the extended optimal control problem is derived.A spectral Galerkin discrete scheme for the extended problem based on weighted Laguerre polynomials is developed.A priori error estimates for the spectral Galerkin discrete scheme is *** experiments are presented to show the effectiveness of our methods and to verify the theoretical findings.

关键词： Fractional Laplacian optimal control problem Caffarelli-Silvestre extension weighted Laguerre polynomials

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A Diffuse-Interface Lattice Boltzmann Method for the Dendritic Growth with Thermosolutal Convection

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Communications in Computational Physics 2023年第4期33卷 1164-1188页

作者： Chengjie Zhan Zhenhua Chai Baochang Shi Ping Jiang Shaoning Geng Dongke Sun School of Mathematics and Statistics Huazhong University of Science and TechnologyWuhan430074P.R.China Institute of Interdisciplinary Research for Mathematics and Applied Science Huazhong University of Science and TechnologyWuhan 430074P.R.China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and TechnologyWuhan430074P.R.China The State Key Laboratory of Digital Manufacturing Equipment and Technology School of Mechanical Science and EngineeringHuazhong University of Science and TechnologyWuhan 430074P.R.China School of Mechanical Engineering Southeast UniversityNanjing 211189P.R.China

In this work,we proposed a diffuse-interface model for the dendritic growth with thermosolutal *** this model,the sharp boundary between the fluid and solid dendrite is firstly replaced by a thin but nonzero thickness diffuse interface,which is described by the order parameter,and the diffuse-interface based governing equations for the dendritic growth are *** solve the model for the dendritic growth with thermosolutal convection,we also developed a diffuse-interface multirelaxation-time lattice Boltzmann(LB)*** this method,the order parameter in the phase-field equation is combined into the force caused by the fluid-solid interaction,and the treatment on the complex fluid-solid interface can be *** addition,four LB models are designed for the phase-field equation,concentration equation,temperature equation and the Navier-Stokes equations in a unified ***,we performed some simulations of the dendritic growth to test the present diffuse-interface LB method,and found that the numerical results are in good agreements with some previous works.

关键词： Dendritic growth diffuse interface lattice Boltzmann method phase-field method.

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Joint Design of Radar Receive Filter and Unimodular ISAC Waveform with Sidelobe Level Control

arXiv

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arXiv 2025年

作者： Zhang, Kecheng Liu, Ya-Feng Wang, Zhongbin Yuan, Weijie Keskin, Musa Furkan Wymeersch, Henk Xia, Shuqiang School of System Design and Intelligent Manufacturing The Shenzhen Key Laboratory of Robotics and Computer Vision Southern University of Science and Technology Shenzhen518055 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 Department of Electrical Engineering Chalmers University of Technology Gothenburg41296 Sweden ZTE Corporation The State Key Laboratory of Mobile Network and Mobile Multimedia Technology Shenzhen518055 China

Integrated sensing and communication (ISAC) has been considered a key feature of next-generation wireless networks. This paper investigates the joint design of the radar receive filter and dual-functional transmit waveform for the multiple-input multiple-output (MIMO) ISAC system. While optimizing the mean square error (MSE) of the radar receive spatial response and maximizing the achievable rate at the communication receiver, besides the constraints of full-power radar receiving filter and unimodular transmit sequence, we control the maximum range sidelobe level, which is often overlooked in existing ISAC waveform design literature, for better radar imaging performance. To solve the formulated optimization problem with convex and nonconvex constraints, we propose an inexact augmented Lagrangian method (ALM) algorithm. For each subproblem in the proposed inexact ALM algorithm, we custom-design a block successive upper-bound minimization (BSUM) scheme with closed-form solutions for all blocks of variable to enhance the computational efficiency. Convergence analysis shows that the proposed algorithm is guaranteed to provide a stationary and feasible solution. Extensive simulations are performed to investigate the impact of different system parameters on communication and radar imaging performance. Comparison with the existing works shows the superiority of the proposed algorithm. © 2025, CC BY-NC-SA.

关键词： Mean square error

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QoS-Aware and Routing-Flexible Network Slicing for Service-Oriented Networks

arXiv

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arXiv 2024年

作者： Chen, Wei-Kun Liu, Ya-Feng Dai, Yu-Hong Luo, Zhi-Quan The School of Mathematics and Statistics Beijing Key Laboratory on MCAACI Beijing Institute of Technology Beijing100081 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 The Shenzhen Research Institute of Big Data The Chinese University of Hong Kong Shenzhen518172 China

In this paper, we consider the network slicing (NS) problem which attempts to map multiple customized virtual network requests (also called services) to a common shared network infrastructure and manage network resources to meet diverse quality of service (QoS) requirements. We propose a mixed-integer nonlinear programming (MINLP) formulation for the considered NS problem that can flexibly route the traffic flow of the services on multiple paths and provide end-to-end delay and reliability guarantees for all services. To overcome the computational difficulty due to the intrinsic nonlinearity in the MINLP formulation, we transform the MINLP formulation into an equivalent mixed-integer linear programming (MILP) formulation and further show that their continuous relaxations are equivalent. In sharp contrast to the continuous relaxation of the MINLP formulation which is a nonconvex nonlinear programming problem, the continuous relaxation of the MILP formulation is a polynomial-time solvable linear programming problem, which significantly facilitates the algorithmic design. Based on the newly proposed MILP formulation, we develop a customized column generation (cCG) algorithm for solving the NS problem. The proposed cCG algorithm is a decomposition-based algorithm and is particularly suitable for solving large-scale NS problems. Numerical results demonstrate the efficacy of the proposed formulations and the proposed cCG algorithm. Copyright © 2024, The Authors. All rights reserved.

关键词： Mixed-integer linear programming

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Asymptotic SEP Analysis and Optimization of Linear-Quantized Precoding in Massive MIMO Systems

arXiv

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arXiv 2023年

作者： Wu, Zheyu Ma, Junjie Liu, Ya-Feng Swindlehurst, A. Lee 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 Center for Pervasive Communications and Computing University of California IrvineCA92697 United States

A promising approach to deal with the high hardware cost and energy consumption of massive MIMO transmitters is to use low-resolution digital-to-analog converters (DACs) at each antenna element. This leads to a transmission scheme where the transmitted signals are restricted to a finite set of voltage levels. This paper is concerned with the analysis and optimization of a low-cost quantized precoding strategy, referred to as linear-quantized precoding, for a downlink massive MIMO system under Rayleigh fading. In linear-quantized precoding, the signals are first processed by a linear precoding matrix and subsequently quantized component-wise by the DAC. In this paper, we analyze both the signal-to-interference-plus-noise ratio (SINR) and the symbol error probability (SEP) performances of such linear-quantized precoding schemes in an asymptotic framework where the number of transmit antennas and the number of users grow large with a fixed ratio. Our results provide a rigorous justification for the heuristic arguments based on the Bussgang decomposition that are commonly used in prior works. Based on the asymptotic analysis, we further derive the optimal precoder within a class of linear-quantized precoders that includes several popular precoders as special cases. Our numerical results demonstrate the excellent accuracy of the asymptotic analysis for finite systems and the optimality of the derived precoder. Copyright © 2023, The Authors. All rights reserved.

关键词： MIMO systems

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Riemannian Optimization for Holevo Capacity

arXiv

引用

arXiv 2025年

作者： Zhu, Chengkai Peng, Renfeng Gao, Bin Wang, Xin Guangzhou China State Key Laboratory of Scientific and Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China University of Chinese Academy of Sciences Beijing China

computing the classical capacity of a noisy quantum channel is crucial for understanding the limits of communication over quantum channels. However, its evaluation remains challenging due to the difficulty of computing the Holevo capacity and the even greater difficulty of regularization. In this work, we formulate the computation of the Holevo capacity as an optimization problem on a product manifold constructed from probability distributions and their corresponding pure input states for a quantum channel. A Riemannian gradient descent algorithm is proposed to solve the problem, providing lower bounds on the classical capacity of general quantum channels and outperforming existing methods in numerical experiments in both efficiency and scale. © 2025, CC BY.

关键词： Probability distributions

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Covariance-Based Device Activity Detection with Massive MIMO for Near-Field Correlated Channels

arXiv

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arXiv 2024年

作者： Wang, Ziyue Li, Yang Liu, Ya-Feng Ma, Junjie 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 Shenzhen Research Institute of Big Data Shenzhen518172 China The School of Science and Engineering The Chinese University of Hong Kong Shenzhen518172 China

This paper studies the device activity detection problem in a massive multiple-input multiple-output (MIMO) system for near-field communications (NFC). In this system, active devices transmit their signature sequences to the base station (BS), which detects the active devices based on the received signal. In this paper, we model the near-field channels as correlated Rician fading channels and formulate the device activity detection problem as a maximum likelihood estimation (MLE) problem. Compared to the traditional uncorrelated channel model, the correlation of channels complicates both algorithm design and theoretical analysis of the MLE problem. On the algorithmic side, we propose two computationally efficient algorithms for solving the MLE problem: an exact coordinate descent (CD) algorithm and an inexact CD algorithm. The exact CD algorithm solves the one-dimensional optimization subproblem exactly using matrix eigenvalue decomposition and polynomial root-finding. By approximating the objective function appropriately, the inexact CD algorithm solves the one-dimensional optimization subproblem inexactly with lower complexity and more robust numerical performance. Additionally, we analyze the detection performance of the MLE problem under correlated channels by comparing it with the case of uncorrelated channels. The analysis shows that when the overall number of devices N is large or the signature sequence length L is small, the detection performance of MLE under correlated channels tends to be better than that under uncorrelated channels. Conversely, when N is small or L is large, MLE performs better under uncorrelated channels than under correlated ones. Simulation results demonstrate the computational efficiency of the proposed algorithms and verify the correctness of the analysis. Copyright © 2024, The Authors. All rights reserved.

关键词： Eigenvalues and eigenfunctions

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Dendritic Integration Inspired Artificial Neural Networks Capture Data Correlation 38

Dendritic Integration Inspired Artificial Neural Networks Ca...

引用

38th Conference on Neural Information Processing Systems, NeurIPS 2024

作者： Liu, Chongming Ma, Jingyang Li, Songting Zhou, Douglas School of Mathematical Sciences Shanghai Jiao Tong University Shanghai China Institute of Natural Sciences Shanghai Jiao Tong University Shanghai China Ministry of Education Key Laboratory of Scientific and Engineering Computing Shanghai Jiao Tong University Shanghai China Shanghai Frontier Science Center of Modern Analysis Shanghai Jiao Tong University Shanghai China

Incorporating biological neuronal properties into Artificial Neural Networks (ANNs) to enhance computational capabilities is under active investigation in the field of deep learning. Inspired by recent findings indicating that dendrites adhere to quadratic integration rule for synaptic inputs, this study explores the computational benefits of quadratic neurons. We theoretically demonstrate that quadratic neurons inherently capture correlation within structured data, a feature that grants them superior generalization abilities over traditional neurons. This is substantiated by few-shot learning experiments. Furthermore, we integrate the quadratic rule into Convolutional Neural Networks (CNNs) using a biologically plausible approach, resulting in innovative architectures-Dendritic integration inspired CNNs (Dit-CNNs). Our Dit-CNNs compete favorably with state-of-the-art models across multiple classification benchmarks, e.g., ImageNet-1K, while retaining the simplicity and efficiency of traditional CNNs. All source code are available at https://***/liuchongming1999/Dendritic-integration-inspired-CNN-NeurIPS-2024. © 2024 Neural information processing systems foundation. All rights reserved.

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