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A Dynamical Variable-separation Method for Parameter-dependent Dynamical Systems

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arXiv 2025年

作者： Chen, Liang Chen, Yaru Li, Qiuqi Zhou, Tao School of Mathematics Hunan University Changsha410082 China LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China

This paper proposes a dynamical Variable-separation method for solving parameter-dependent dynamical systems. To achieve this, we establish a dynamical low-rank approximation for the solutions of these dynamical systems by successively enriching each term in the reduced basis functions via a greedy algorithm. This enables us to reformulate the problem to two decoupled evolution equations at each enrichment step, of which one is a parameter-independent partial differential equation and the other one is a parameter-dependent ordinary differential equation. These equations are directly derived from the original dynamical system and the previously separate representation terms. Moreover, the computational process of the proposed method can be split into an offline stage, in which the reduced basis functions are constructed, and an online stage, in which the efficient low-rank representation of the solution is employed. The proposed dynamical Variable-separation method is capable of offering reduced computational complexity and enhanced efficiency compared to many existing low-rank separation techniques. Finally, we present various numerical results for linear/nonlinear parameter-dependent dynamical systems to demonstrate the effectiveness of the proposed *** Codes 35R60, 37M99, 65P99 Copyright © 2025, The Authors. All rights reserved.

关键词： Ordinary differential equations

An optimal piecewise cubic nonconforming finite element scheme for the planar biharmonic equation on general triangulations

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Science China mathematics 2021年第11期64卷 2579-2602页

作者： Shuo Zhang LSEC Institute of Computational Mathematics and Scientific/Engineering ComputingAcademy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China School of Mathematical Sciences University of Chinese Academy of SciencesBeijing 100049China

This paper presents a nonconforming finite element scheme for the planar biharmonic equation,which applies piecewise cubic polynomials(P_(3))and possesses O(h^(2))convergence rate for smooth solutions in the energy norm on general shape-regular *** Dirichlet and Navier type boundary value problems are *** basis for the scheme is a piecewise cubic polynomial space,which can approximate the H^(4) functions with O(h^(2))accuracy in the broken H^(2) ***,a discrete strengthened Miranda-Talenti estimate(▽^(2)_(h)·,▽^(2)_(h)·)=(Δh·,Δh·),which is usually not true for nonconforming finite element spaces,is *** finite element space does not correspond to a finite element defined with Ciarlet’s triple;however,it admits a set of locally supported basis functions and can thus be implemented by the usual *** notion of the finite element Stokes complex plays an important role in the analysis as well as the construction of the basis functions.

关键词： biharmonic equation optimal cubic finite element scheme general triangulation discretized Stokes complex discrete strengthened Miranda-Talenti estimate

Precise Analysis of Covariance Identifiability for Activity Detection in Grant-Free Random Access

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arXiv 2024年

作者： Luo, Shengsong Ma, Junjie Xu, Chongbin Wang, Xin the Department of Communication Science and Engineering Fudan University Shanghai200433 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

We consider the identifiability issue of maximum-likelihood based activity detection in massive MIMO-based grant-free random access. An intriguing observation by Chen et al. [1] indicates that the identifiability undergoes a phase transition for commonly-used random user signatures as L2, N and K tend to infinity with fixed ratios, where L, N and K denote the user signature length, the total number of users, and the number of active users, respectively. In this letter, we provide a precise analytical characterization of the phase transition based on a spectral universality conjecture. Numerical results demonstrate excellent agreement between our theoretical predictions and the empirical phase transitions. Copyright © 2024, The Authors. All rights reserved.

关键词： Maximum likelihood

QoS-Based Beamforming and Compression Design for Cooperative Cellular Networks via Lagrangian Duality

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IEEE Transactions on Signal Processing 2025年

作者： Fan, Xilai Liu, Ya-Feng Liu, Liang Chang, Tsung-Hui Chinese Academy of Sciences State Key Laboratory of Scientific and Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Beijing100190 China The Hong Kong Polytechnic University Department of Electronic and Information Engineering Hong Kong Hong Kong The Chinese University of HongKong School of Science and Engineering Shenzhen Shenzhen China Shenzhen Research Institute of Big Data China

This paper considers the quality-of-service (QoS)-based joint beamforming and compression design problem in the downlink cooperative cellular network, where multiple relay-like base stations (BSs), connected to the central processor via ratelimited fronthaul links, cooperatively transmit messages to the users. The problem of interest is formulated as the minimization of the total transmit power of the BSs, subject to all users' signal-to-interference-plus-noise ratio (SINR) constraints and all BSs' fronthaul rate constraints. In this paper, we first show that there is no duality gap between the considered joint optimization problem and its Lagrangian dual by showing the tightness of its semidefinite relaxation (SDR). Then, we propose an efficient algorithm based on the above duality result for solving the considered problem. The proposed algorithm judiciously exploits the special structure of an enhanced Karush-Kuhn-Tucker (KKT) conditions of the considered problem and approaches the solution that satisfies the enhanced KKT conditions via two fixed point iterations. Two key features of the proposed algorithm are: (1) it is able to detect whether the considered problem is feasible or not and find its globally optimal solution when it is feasible;(2) it is highly efficient because both of the fixed point iterations in the proposed algorithm are linearly convergent and function evaluations in the fixed point iterations are computationally cheap. Numerical results show the global optimality and efficiency of the proposed algorithm. © 1991-2012 IEEE.

关键词： Lagrange multipliers

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

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The Journal of Machine Learning Research

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

A New Function Space from Barron Class and Application to Neural Network Approximation

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Communications in computational Physics 2022年第10期32卷 1361-1400页

作者： Yan Meng Pingbing Ming School of Mathematics Renmin University of ChinaBeijing 100872China LSEC Institute of Computational Mathematics and Scientific/Engineering ComputingAMSSChinese Academy of SciencesNo.55East Road Zhong-Guan-CunBeijing 100190China School of Mathematical Sciences University of Chinese Academy of SciencesBeijing 100049China.

We introduce a new function space,dubbed as the Barron spectrum space,which arises from the target function space for the neural network *** give a Bernstein type sufficient condition for functions in this space,and clarify the embedding among the Barron spectrum space,the Bessel potential space,the Besov space and the Sobolev ***,the unexpected smoothness and the decaying behavior of the radial functions in the Barron spectrum space have been *** an application,we prove a dimension explicit L^(q) error bound for the two-layer neural network with the Barron spectrum space as the target function space,the rate is dimension independent.

关键词： Fourier transform Besov space Sobolev space radial function neural network

Quasiperiodic Symmetric Tilt FCC Grain Boundaries

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arXiv 2024年

作者： Zou, Wenwen Zhang, Juan Xu, Jie Jiang, Kai Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China LSEC NCMIS Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China

In this work, we investigate [110] symmetric tilt FCC grain boundaries (GBs) by a recently developed approach for quasiperiodic interfaces using the Landau-Brazovskii model. On special tilt angles associated with quadratic algebraic numbers, quasiperiodic GBs exhibit generalized Fibonacci substitution rules. The transition mechanism from quasiperiodic GBs to periodic GBs is explored through sphere offsets and spectral coalescence. We also propose an accurate method to calculate the GB energy for arbitrary tilt angle and analyze the factors affecting the GB energy. The revealing GB energy change is continue along with tilt angle except periodic GBs. © 2024, CC BY-NC-SA.

关键词： Grain boundaries

An inexact augmented lagrangian algorithm for training leaky ReLU neural network with group sparsity

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The Journal of Machine Learning Research

The Journal of Machine Learning Research 2023年第1期24卷 10097-10139页

作者： Wei Liu Xin Liu Xiaojun Chen Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China Department of Applied Mathematics The Hong Kong Polytechnic University Hung Hom Kowloon Hong Kong

The leaky ReLU network with a group sparse regularization term has been widely used in the recent years. However, training such network yields a nonsmooth nonconvex optimization problem and there exists a lack of approaches to compute a stationary point deterministically. In this paper, we first resolve the multi-layer composite term in the original optimization problem by introducing auxiliary variables and additional constraints. We show the new model has a nonempty and bounded solution set and its feasible set satisfies the Mangasarian-Fromovitz constraint qualification. Moreover, we show the relationship between the new model and the original problem. Remarkably, we propose an inexact augmented Lagrangian algorithm for solving the new model, and show the convergence of the algorithm to a KKT point. Numerical experiments demonstrate that our algorithm is more efficient for training sparse leaky ReLU neural networks than some well-known algorithms.

关键词： sparse neural network leaky ReLU group sparsity penalty method inexact augmented Lagrangian method

A DISCONTINUOUS PLANE WAVE NEURAL NETWORK METHOD FOR HELMHOLTZ EQUATION AND TIME-HARMONIC MAXWELL'S EQUATIONS

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arXiv 2023年

作者： Yuan, Long Hu, Qiya College of Mathematics and Systems Science Shandong University of Science and Technology Qingdao266590 China LSEC Institute of Computational Mathematics and Scientic/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

In this paper we propose a discontinuous plane wave neural network (DPWNN) method with hp−refinement for approximately solving Helmholtz equation and time-harmonic Maxwell equations. In this method, we define a quadratic functional as in the plane wave least square (PWLS) method with h−refinement and introduce new discretization sets spanned by element-wise neural network functions with a single hidden layer, where the activation function on each element is chosen as a complex-valued exponential function like the plane wave function. The desired approximate solution is recursively generated by iteratively solving a quasi-minimization problem associated with the functional and the sets described above, which is defined by a sequence of approximate minimizers of the underlying residual functionals, where plane wave direction angles and activation coefficients are alternatively computed by iterative algorithms. For the proposed DPWNN method, the plane wave directions are adaptively determined in the iterative process, which is different from that in the standard PWLS method (where the plane wave directions are preliminarily given). Numerical experiments will confirm that this DPWNN method can generate approximate solutions with higher accuracy than the PWLS method. Copyright © 2023, The Authors. All rights reserved.

关键词： Wave propagation