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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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SCALING OPTIMIZED HERMITE APPROXIMATION METHODS

arXiv

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arXiv 2024年

作者： Hu, Hao Yu, Haijun School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China NCMIS & LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Beijing100190 China

Hermite polynomials and functions have extensive applications in scientific and engineering problems. While it is recognized that employing the scaled Hermite functions rather than the standard ones can remarkably enhance the approximation performance, the understanding of the scaling factor remains insufficient. Due to the lack of theoretical analysis, some literature still cast doubts on whether the Hermite spectral method is inferior to other methods. To dispel these doubts, we show in this article that the inefficiency of the Hermite spectral method comes from the imbalance in the decay speed of the objective function within the spatial and frequency domains. Moreover, proper scaling can render the Hermite spectral methods comparable to other methods. To illustrate this idea in more detail, we propose a novel error analysis framework for the scaled Hermite approximation. Taking the L2 projection error as an example, our framework illustrates that there are three different components of errors: the spatial truncation error, the frequency truncation error, and the Hermite spectral approximation error. Through this perspective, finding the optimal scaling factor is equivalent to balancing the spatial and frequency truncation error. As applications, we show that geometric convergence can be recovered by proper scaling for a class of functions. Furthermore, we show that proper scaling can double the convergence order for smooth functions with algebraic decay. The perplexing pre-asymptotic sub-geometric convergence when approximating algebraic decay functions can be perfectly explained by this *** Codes 65N35, 33C45, 42C05 © 2024, CC BY-SA.

关键词： Error analysis

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

arXiv

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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

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Energetic Spectral-Element Time Marching Methods for Phase-Field Nonlinear Gradient Systems

arXiv

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arXiv 2024年

作者： Liu, Shiqin Yu, Haijun School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China LSEC & NCMIS Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Beijing100190 China

We propose two efficient energetic spectral-element methods in time for marching nonlinear gradient systems with the phase-field Allen–Cahn equation as an example: one fully implicit nonlinear method and one semi-implicit linear method. Different from other spectral methods in time using spectral Petrov-Galerkin or weighted Galerkin approximations, the presented implicit method employs an energetic variational Galerkin form that can maintain the mass conservation and energy dissipation property of the continuous dynamical system. Another advantage of this method is its superconvergence. A high-order extrapolation is adopted for the nonlinear term to get the semi-implicit method. The semi-implicit method does not have superconvergence, but can be improved by a few Picard-like iterations to recover the superconvergence of the implicit method. Numerical experiments verify that the method using Legendre elements of degree three outperforms the 4th-order implicit-explicit backward differentiation formula and the 4th-order exponential time difference Runge-Kutta method, which were known to have best performances in solving phase-field equations. In addition to the standard Allen–Cahn equation, we also apply the method to a conservative Allen–Cahn equation, in which the conservation of discrete total mass is verified. The applications of the proposed methods are not limited to phase-field Allen–Cahn equations. They are suitable for solving general, large-scale nonlinear dynamical systems. © 2024, CC BY-NC-SA.

关键词： Nonlinear dynamical systems

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Quasiperiodic Symmetric Tilt FCC Grain Boundaries

arXiv

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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

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

arXiv

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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

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Efficient Global Algorithms for Transmit Beamforming Design in ISAC Systems

arXiv

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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

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Improved time integration for phase-field crystal models of solidification

Improved time integration for phase-field crystal models of ...

引用

92nd Annual Meeting of the International Association of Applied mathematics and Mechanics (GAMM)

作者： Punke, Maik Wise, Steven M. Voigt, Axel Salvalaglio, Marco Institute of Scientific Computing TU Dresden Dresden01062 Germany Department of Mathematics The University of Tennessee KnoxvilleTN37996 United States Dresden Center for Computational Materials Science TU Dresden Dresden01062 Germany

We optimize a numerical time-stabilization routine for a class of phase-field crystal (PFC) models of solidification. By numerical experiments, we demonstrate that our simple approach can improve the accuracy of underlying time integration schemes by a few orders of magnitude. We investigate different time integration schemes. Moreover, as a prototypical example for applications, we extend our numerical approach to a PFC model of solidification with an explicit temperature coupling. © 2023 The Authors. Proceedings in Applied mathematics & Mechanics published by Wiley-VCH GmbH.

关键词： Solidification

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

arXiv

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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

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Learning optimal feedback operators and their sparse polynomial approximations

The Journal of Machine Learning Research

引用

The Journal of Machine Learning Research 2023年第1期24卷 14274-14311页

作者： Karl Kunisch Donato Vásquez-Varas Daniel Walter Radon Institute for Computational and Applied Mathematics Austrian Academy of Sciences and Institute of Mathematics and Scientific Computing University of Graz Graz Austria Radon Institute for Computational and Applied Mathematics Austrian Academy of Sciences Linz Austria Institut für Mathematik Humboldt-Universität zu Berlin Berlin Germany

A learning based method for obtaining feedback laws for nonlinear optimal control problems is proposed. The learning problem is posed such that the open loop value function is its optimal solution. This infinite dimensional, function space, problem, is approximated by a polynomial ansatz and its convergence is analyzed. An ℓ1 penalty term is employed, which combined with the proximal point method, allows to find sparse solutions for the learning problem. The approach requires multiple evaluations of the elements of the polynomial basis and of their derivatives. In order to do this efficiently a graph-theoretic algorithm is devised. Several examples underline that the proposed methodology provides a promising approach for mitigating the curse of dimensionality which would be involved in case the optimal feedback law was obtained by solving the Hamilton Jacobi Bellman equation.

关键词： optimal feedback control nonlinear systems learning theory Hamilton Jacobi Bellman equation polynomial based approximation

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