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Projected gradient descent algorithm for ab initio crystal structure relaxation under a fixed unit cell volume

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arXiv 2024年

作者： Hu, Yukuan Yin, Junlei Gao, Xingyu Liu, Xin Song, Haifeng 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 School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China State Key Laboratory of Solidification Processing Northwestern Polytechnical University Shaanxi Xi’an710072 China Innovation Center NPU Chongqing Chongqing401135 China Laboratory of Computational Physics Institute of Applied Physics and Computational Mathematics Beijing100088 China

This paper is concerned with ab initio crystal structure relaxation under a fixed unit cell volume, which is a step in calculating the static equations of state and forms the basis of thermodynamic property calculations for materials. The task can be formulated as an energy minimization with a determinant constraint. Widely used line minimization-based methods (e.g., conjugate gradient method) lack both efficiency and convergence guarantees due to the nonconvex nature of the feasible region as well as the significant differences in the curvatures of the potential energy surface with respect to atomic and lattice components. To this end, we propose a projected gradient descent algorithm named PANBB. It is equipped with (i) search direction projections onto the tangent spaces of the nonconvex feasible region for lattice vectors, (ii) distinct curvature-aware initial trial step sizes for atomic and lattice updates, and (iii) a nonrestrictive line minimization criterion as the stopping rule for the inner loop. It can be proved that PANBB favors theoretical convergence to equilibrium states. Across a benchmark set containing 223 structures from various categories, PANBB achieves average speedup factors of approximately 1.41 and 1.45 over the conjugate gradient method and direct inversion in the iterative subspace implemented in off-the-shelf simulation software, respectively. Moreover, it normally converges on all the systems, manifesting its unparalleled robustness. As an application, we calculate the static equations of state for the high-entropy alloy AlCoCrFeNi, which remains elusive owing to 160 atoms representing both chemical and magnetic disorder and the strong local lattice distortion. The results are consistent with the previous calculations and are further validated by experimental thermodynamic data. © 2024, CC BY.

关键词： Conjugate gradient method

Optimal-rate error estimates and a twice decoupled solver for a backward Euler finite element scheme of the Doyle-Fuller-Newman model of lithium-ion batteries

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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

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

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

LEAST H2 NORM UPDATING QUADRATIC INTERPOLATION MODEL FUNCTION FOR DERIVATIVE-FREE TRUST-REGION ALGORITHMS

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arXiv 2023年

作者： Xie, Pengcheng Yuan, Ya-Xiang Institute of Computational Math and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences University of Chinese Academy of Sciences China

Derivative-free optimization methods are numerical methods for optimization problems in which no derivative information is used. Such optimization problems are widely seen in many real applications. One particular class of derivative-free optimization algorithms is trust-region algorithms based on quadratic models given by underdetermined interpolation. Different techniques in updating the quadratic model from iteration to iteration will give different interpolation models. In this paper, we propose a new way to update the quadratic model by minimizing the H2 norm of the difference between neighbouring quadratic models. Motivation for applying the H2 norm is given, and theoretical properties of our new updating technique are also presented. Projection in the sense of H2 norm and interpolation error analysis of our model function are proposed. We obtain the coefficients of the quadratic model function by using the KKT conditions. Numerical results show advantages of our model, and the derivative-free algorithms based on our least H2 norm updating quadratic model functions can solve test problems with fewer function evaluations than algorithms based on least Frobenius norm *** Codes 90C56, 90C30, 65K05, 90C90 Copyright © 2023, The Authors. All rights reserved.

关键词： Interpolation

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

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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

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

Time parallelization for hyperbolic and parabolic problems

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arXiv 2025年

作者： Gander, Martin J. Wu, Shu-Lin Zhou, Tao Department of Mathematics University of Geneva CP64 Geneva 41211 Switzerland School of Mathematics and Statistics Northeast Normal University Changchun130024 China Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences China

Time parallelization, also known as PinT (Parallel-in-Time) is a new research direction for the development of algorithms used for solving very large scale evolution problems on highly parallel computing architectures. Despite the fact that interesting theoretical work on PinT appeared as early 1964, it was not until 2004, when processor clock speeds reached their physical limit, that research in PinT took off. A distinctive characteristic of parallelization in time is that information flow only goes forward in time, meaning that time evolution processes seem necessarily to be sequential. Nevertheless, many algorithms have been developed over the last two decades to do PinT computations, and they are often grouped into four basic classes according to how the techniques work and are used: shooting-type methods;waveform relaxation methods based on domain decomposition;multigrid methods in space-time;and direct time parallel methods. However, over the past few years, it has been recognized that highly successful PinT algorithms for parabolic problems struggle when applied to hyperbolic problems. We focus in this survey therefore on this important aspect, by first providing a summary of the fundamental differences between parabolic and hyperbolic problems for time parallelization. We then group PinT algorithms into two basic groups: the first group contains four effective PinT techniques for hyperbolic problems, namely Schwarz Waveform Relaxation with its relation to Tent Pitching;Parallel Integral Deferred Correction;ParaExp;and ParaDiag. While the methods in the first group also work well for parabolic problems, we then present PinT methods especially designed for parabolic problems in the second group: Parareal: the Parallel Full Approximation Scheme in Space-Time;Multigrid Reduction in Time;and Space-Time Multigrid. We complement our analysis with numerical illustrations using four time-dependent PDEs: the heat equation;the advection-diffusion equation;Burgers’ equa

关键词： Wave equations