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arXiv 2024年

作者： Xu, Shu Cao, Liqun School of Mathematical Sciences Peking University Beijing100871 China Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China State Key Laboratory of Scientific and Engineering Computing Beijing100190 China National Center for Mathematics and Interdisciplinary Sciences Chinese Academy of Sciences Beijing100190 China

We investigate the convergence of a backward Euler finite element discretization applied to a multi-domain and multi-scale elliptic-parabolic system, derived from the Doyle-Fuller-Newman model for lithium-ion batteries. We establish optimal-order error estimates for the solution in the norms l2(H1) and l2(L2(Hrq)), q = 0,1. To improve computational efficiency, we propose a novel solver that accelerates the solution process and controls memory usage. Numerical experiments with realistic battery parameters validate the theoretical error rates and demonstrate the significantly superior performance of the proposed solver over existing solvers. Copyright © 2024, The Authors. All rights reserved.

关键词： computational efficiency

Martingale deep learning for very high dimensional quasi-linear partial differential equations and stochastic optimal controls

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arXiv 2024年

作者： Cai, Wei Fang, Shuixin Zhang, Wenzhong Zhou, Tao Department of Mathematics Southern Methodist University United States Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences China Suzhou Institute for Advanced Research University of Science and Technology of China China

In this paper, a highly parallel and derivative-free martingale neural network learning method is proposed to solve Hamilton-Jacobi-Bellman (HJB) equations arising from stochastic optimal control problems (SOCPs), as well as general quasilinear parabolic partial differential equations (PDEs). In both cases, the PDEs are reformulated into a martingale formulation such that loss functions will not require the computation of the gradient or Hessian matrix of the PDE solution, while its implementation can be parallelized in both time and spatial domains. Moreover, the martingale conditions for the PDEs are enforced using a Galerkin method in conjunction with adversarial learning techniques, eliminating the need for direct computation of the conditional expectations associated with the martingale property. For SOCPs, a derivative-free implementation of the maximum principle for optimal controls is also introduced. The numerical results demonstrate the effectiveness and efficiency of the proposed method, which is capable of solving HJB and quasilinear parabolic PDEs accurately in dimensions as high as 10,000. © 2024, CC BY.

关键词： Galerkin methods

Generalization Error Estimates of Machine Learning Methods for Solving High Dimensional Schrödinger Eigenvalue Problems

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arXiv 2024年

作者： Yu, Hao Guo, Yixiao Ming, Pingbing 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

We propose a machine learning method for computing eigenvalues and eigenfunctions of the Schrödinger operator on a d-dimensional hypercube with Dirichlet boundary conditions. The cut-off function technique is employed to construct trial functions that precisely satisfy the homogeneous boundary conditions. This approach eliminates the error caused by the standard boundary penalty method, improves the overall accuracy of the method, as demonstrated by the typical numerical examples. Under the assumption that the eigenfunctions belong to a spectral Barron space, we derive an explicit convergence rate of the generalization error of the proposed method, which does not suffer from the curse of dimensionality. We verify the assumption by proving a new regularity shift result for the eigenfunctions when the potential function belongs to an appropriate spectral Barron space. Moreover, we extend the generalization error bound to the normalized penalty method, which is widely used in practice. Copyright © 2024, The Authors. All rights reserved.

关键词： Boundary conditions

Adaptive neural network basis methods for partial differential equations with low-regular solutions

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arXiv 2024年

作者： Huang, Jianguo Wu, Haohao Zhou, Tao School of Mathematical Sciences MOE-LSC Shanghai Jiao Tong University Shanghai200240 China Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China

This paper aims to devise an adaptive neural network basis method for numerically solving a second-order semilinear partial differential equation (PDE) with low-regular solutions in two/three dimensions. The method is obtained by combining basis functions from a class of shallow neural networks and the resulting multi-scale analogues, a residual strategy in adaptive methods and the non-overlapping domain decomposition method. At the beginning, in view of the solution residual, we partition the total domain Ω into K + 1 non-overlapping subdomains, denoted respectively as {Ωk}Kk=0, where the exact solution is smooth on subdomain Ω0 and low-regular on subdomain Ωk (1 ≤ k ≤ K). Secondly, the low-regular solutions on different subdomains Ωk (1 ≤ k ≤ K) are approximated by neural networks with different scales, while the smooth solution on subdomain Ω0 is approximated by the initialized neural network. Thirdly, we determine the undetermined coefficients by solving the linear least squares problems directly or the nonlinear least squares problem via the Gauss-Newton method. The proposed method can be extended to multi-level case naturally. Finally, we use this adaptive method for several peak problems in two/three dimensions to show its high-efficient computational performance. Copyright © 2024, The Authors. All rights reserved.

关键词： Partial differential equations

Efficient Global Algorithms for Transmit Beamforming Design in ISAC Systems

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arXiv 2024年

作者： Wu, Jiageng Wang, Zhiguo Liu, Ya-Feng Liu, Fan The School of Mathematics Jilin University Changchun130012 China The College of Mathematics Sichuan University Sichuan Chengdu610064 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 Department of Electrical and Electronic Engineering Southern University of Science and Technology Shenzhen518055 China

In this paper, we propose a multi-input multi-output transmit beamforming optimization model for joint radar sensing and multi-user communications, where the design of the beamformers is formulated as an optimization problem whose objective is a weighted combination of the sum rate and the Cramér-Rao bound, subject to the transmit power budget. Obtaining the global solution for the formulated nonconvex problem is a challenging task, since the sum-rate maximization problem itself (even without considering the sensing metric) is known to be NP-hard. The main contributions of this paper are threefold. Firstly, we derive an optimal closed-form solution to the formulated problem in the single-user case and the multi-user case where the channel vectors of different users are orthogonal. Secondly, for the general multi-user case, we propose a novel branch and bound (B&B) algorithm based on the McCormick envelope relaxation. The proposed algorithm is guaranteed to find the globally optimal solution to the formulated problem. Thirdly, we design a graph neural network (GNN) based pruning policy to determine irrelevant nodes that can be directly pruned in the proposed B&B algorithm, thereby significantly reducing the number of unnecessary enumerations therein and improving its computational efficiency. Simulation results show the efficiency of the proposed vanilla and GNN-based accelerated B&B algorithms. Copyright © 2024, The Authors. All rights reserved.

关键词： computational efficiency

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

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arXiv 2023年

作者： Fan, Xilai Liu, Ya-Feng Liu, Liang Chang, Tsung-Hui 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 Electronic and Information Engineering The Hong Kong Polytechnic University Hong Kong School of Science and Engineering The Chinese University of Hong Kong Shenzhen Research Institute of Big Data Shenzhen Shenzhen 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 rate-limited 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 finds 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 evaluating the functions in the fixed point iterations are computationally cheap. Numerical results show the global optimality and efficiency of the proposed algorithm. Copyright © 2023, The Authors. All rights reserved.

关键词： Quality of service

ANALYSIS ACCELERATED MIRROR DESCENT VIA HIGH-RESOLUTION ODES

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arXiv 2023年

作者： Yuan, Ya-Xiang Zhang, Yi Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China University of Chinese Academy of Sciences Beijing100049 China

Mirror descent plays a crucial role in constrained optimization and acceleration schemes, along with its corresponding low-resolution ordinary differential equations (ODEs) framework have been proposed. However, the low-resolution ODEs are unable to distinguish between Polyak's heavy-ball method and Nesterov's accelerated gradient method. This problem also arises with accelerated mirror descent. To address this issue, we derive high-resolution ODEs for accelerated mirror descent and propose a general Lyapunov function framework to analyze its convergence rate in both continuous and discrete time. Furthermore, we demonstrate that accelerated mirror descent can minimize the squared gradient norm at an inverse cubic rate. © 2023, CC BY.

关键词： Lyapunov functions

Physics-constrained coupled neural differential equations for one dimensional blood flow modeling

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Computers in Biology and Medicine 2025年 186卷 109644-109644页

作者： Csala, Hunor Mohan, Arvind Livescu, Daniel Arzani, Amirhossein Department of Mechanical Engineering University of Utah Salt Lake CityUT United States Scientific Computing and Imaging Institute University of Utah Salt Lake CityUT United States Computational Physics and Methods Los Alamos National Laboratory Los AlamosNM United States

Background: computational cardiovascular flow modeling plays a crucial role in understanding blood flow dynamics. While 3D models provide acute details, they are computationally expensive, especially with fluid–structure interaction (FSI) simulations. 1D models offer a computationally efficient alternative, by simplifying the 3D Navier–Stokes equations through axisymmetric flow assumption and cross-sectional averaging. However, traditional 1D models based on finite element methods (FEM) often lack accuracy compared to 3D averaged solutions. Methods: This study introduces a novel physics-constrained machine learning technique that enhances the accuracy of 1D cardiovascular flow models while maintaining computational efficiency. Our approach, utilizing a physics-constrained coupled neural differential equation (PCNDE) framework, demonstrates superior performance compared to conventional FEM-based 1D models across a wide range of inlet boundary condition waveforms and stenosis blockage ratios. A key innovation lies in the spatial formulation of the momentum conservation equation, departing from the traditional temporal approach and capitalizing on the inherent temporal periodicity of blood flow. Results: This spatial neural differential equation formulation switches space and time and overcomes issues related to coupling stability and smoothness, while simplifying boundary condition implementation. The model accurately captures flow rate, area, and pressure variations for unseen waveforms and geometries, having 3–5 times smaller error than 1D FEM, and less than 1.2% relative error compared to 3D averaged training data. We evaluate the model's robustness to input noise and explore the loss landscapes associated with the inclusion of different physics terms. Conclusion: This advanced 1D modeling technique offers promising potential for rapid cardiovascular simulations, achieving computational efficiency and accuracy. By combining the strengths of physics-based and data-d

关键词： Digital elevation model

Symplectic Discretization Approach for Developing New Proximal Point Algorithm

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arXiv 2023年

The rapid advancements in high-dimensional statistics and machine learning have increased the use of first-order methods. Many of these methods can be regarded as instances of the proximal point algorithm. Given the importance of the proximal point algorithm, there has been growing interest in developing its accelerated variants. However, some existing accelerated proximal point algorithms exhibit oscillatory behavior, which can impede their numerical convergence rate. In this paper, we first introduce an ODE system and demonstrate its o(1/t2) convergence rate and weak convergence property. Next, we apply the Symplectic Euler Method to discretize the ODE and obtain a new accelerated proximal point algorithm, which we call the Symplectic Proximal Point Algorithm. The reason for using the Symplectic Euler Method is its ability to preserve the geometric structure of the ODEs. Theoretically, we demonstrate that the Symplectic Proximal Point Algorithm achieves an o(1/k2) convergence rate and that the sequences generated by our method converge weakly to the solution set. Practically, our numerical experiments illustrate that the Symplectic Proximal Point Algorithm significantly reduces oscillatory behavior, leading to improved long-time behavior and faster numerical convergence *** Codes 47J20, 47H05, 65K15, 65Y20, 90C25, 90C52 © 2023, CC BY.

关键词： Lyapunov functions

Quantized Constant-Envelope Waveform Design for Massive MIMO DFRC Systems

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arXiv 2024年

作者： Wu, Zheyu Liu, Ya-Feng Chen, Wei-Kun Masouros, Christos 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 Mathematics and Statistics Beijing Key Laboratory on MCAACI Beijing Institute of Technology Beijing100081 China The Department of Electronic and Electrical Engineering University College London Torrington Place LondonWC1E 7JE United Kingdom

Both dual-functional radar-communication (DFRC) and massive multiple-input multiple-output (MIMO) have been recognized as enabling technologies for 6G wireless networks. This paper considers the advanced waveform design for hardware-efficient massive MIMO DFRC systems. Specifically, the transmit waveform is imposed with the quantized constant-envelope (QCE) constraint, which facilitates the employment of low-resolution digital-to-analog converters (DACs) and power-efficient amplifiers. The waveform design problem is formulated as the minimization of the mean square error (MSE) between the designed and desired beampatterns subject to the constructive interference (CI)-based communication quality of service (QoS) constraints and the QCE constraint. To solve the formulated problem, we first utilize the penalty technique to transform the discrete problem into an equivalent continuous penalty model. Then, we propose an inexact augmented Lagrangian method (ALM) algorithm for solving the penalty model. In particular, the ALM subproblem at each iteration is solved by a custom-built block successive upper-bound minimization (BSUM) algorithm, which admits closed-form updates, making the proposed inexact ALM algorithm computationally efficient. Simulation results demonstrate the superiority of the proposed approach over existing state-of-the-art ones. In addition, extensive simulations are conducted to examine the impact of various system parameters on the trade-off between communication and radar performances. Copyright © 2024, The Authors. All rights reserved.

关键词： Mean square error