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

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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

A riemannian exponential augmented lagrangian method for computing the projection robust Wasserstein distance 23

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A riemannian exponential augmented lagrangian method for com...

Proceedings of the 37th International Conference on Neural Information Processing Systems

作者： Bo Jiang Ya-Feng Liu Ministry of Education Key Laboratory of NSLSCS School of Mathematical Sciences Nanjing Normal University Nanjing 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 Beijing China

Projection robust Wasserstein (PRW) distance is recently proposed to efficiently mitigate the curse of dimensionality in the classical Wasserstein distance. In this paper, by equivalently reformulating the computation of the PRW distance as an optimization problem over the Cartesian product of the Stiefel manifold and the Euclidean space with additional nonlinear inequality constraints, we propose a Riemannian exponential augmented Lagrangian method (REALM) for solving this problem. Compared with the existing Riemannian exponential penalty-based approaches, REALM can potentially avoid too small penalty parameters and exhibit more stable numerical performance. To solve the subproblems in REALM efficiently, we design an inexact Riemannian Barzilai-Borwein method with Sinkhorn iteration (iRBBS), which selects the stepsizes adaptively rather than tuning the step-sizes in efforts as done in the existing methods. We show that iRBBS can return an ε-stationary point of the original PRW distance problem within O(ε-3) iterations, which matches the best known iteration complexity result. Extensive numerical results demonstrate that our proposed methods outperform the state-of-the-art solvers for computing the PRW distance.

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Numerical analysis of a hybridized discontinuous Galerkin method for the Cahn-Hilliard problem

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arXiv 2023年

作者： Kirk, Keegan L.A. Masri, Rami Riviere, Beatrice Department of Computational and Applied Mathematics Rice University Houston United States Department of Numerical Analysis and Scientific Computing Simula Research Laboratory Oslo Norway

The mixed form of the Cahn-Hilliard equations is discretized by the hybridizable discontinuous Galerkin method. For any chemical energy density, existence and uniqueness of the numerical solution is obtained. The scheme is proved to be unconditionally stable. Convergence of the method is obtained by deriving a priori error estimates that are valid for the Ginzburg-Lindau chemical energy density and for convex domains. The paper also contains discrete functional tools, namely discrete Agmon and Gagliardo-Nirenberg inequalities, which are proved to be valid in the hybridizable discontinuous Galerkin *** Codes 65M15 (Primary) 65M60 (Secondary) Copyright © 2023, The Authors. All rights reserved.

关键词： Numerical methods

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

CPAFT: A Consistent Parallel Advancing Front Technique for Unstructured Triangular/Tetrahedral Mesh Generation

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arXiv 2024年

作者： Ma, Chengdi Huang, Jizu Luo, Hao Yang, Chao 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 PKU-Changsha Institute for Computing and Digital Economy Peking University Hunan410006 China

Compared with the remarkable progress made in parallel numerical solvers of partial differential equations, the development of algorithms for generating unstructured triangular/tetrahedral meshes has been relatively sluggish. In this paper, we propose a novel, consistent parallel advancing front technique (CPAFT) by combining the advancing front technique, the domain decomposition method based on space-filling curves, the distributed forest-of-overlapping-trees approach, and the consistent parallel maximal independent set algorithm. The newly proposed CPAFT algorithm can mathematically ensure that the generated unstructured triangular/tetrahedral meshes are independent of the number of processors and the implementation of domain decomposition. Several numerical tests are conducted to validate the parallel consistency and outstanding parallel efficiency of the proposed algorithm, which scales effectively up to two thousand processors. This is, as far as we know, the first parallel unstructured triangular/tetrahedral mesh generator with scalability to O(1,000) CPU *** Codes 65M50, 65M55, 68W10 Copyright © 2024, The Authors. All rights reserved.

关键词： Domain decomposition methods

STABILIZABILITY OF PARABOLIC EQUATIONS BY SWITCHING CONTROLS BASED ON POINT ACTUATORS

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arXiv 2024年

作者： Azmi, Behzad Kunisch, Karl Rodrigues, Sérgio S. Department of Mathematics and Statistics University of Konstanz KonstanzD-78457 Germany Johann Radon Institute for Computational and Applied Mathematics ÖAW Altenbergerstrasse 69 Linz4040 Austria Institute for Mathematics and Scientific Computing University of Graz Heinrichstrasse 36 Graz8010 Austria

MSC Codes 93C05, 93C20, 93D20, 35Q93, 49M05It is shown that a switching control involving a finite number of Dirac delta actuators is able to steer the state of a general class of nonautonomous parabolic equations to zero as time increases to infinity. The strategy is based on a recent feedback stabilizability result, which utilizes control forces given by linear combinations of appropriately located Dirac delta distribution actuators. Then, the existence of a stabilizing switching control with no more than one actuator active at each time instant is established. For the implementation in practice, the stabilization problem is formulated as an infinite horizon optimal control problem, with cardinality-type control constraints enforcing the switching property. Subsequently, this problem is tackled using a receding horizon framework. Its suboptimality and stabilizabilizing properties are analyzed. Numerical simulations validate the approach, illustrating its stabilizing and switching properties. Copyright © 2024, The Authors. All rights reserved.

关键词： Actuators

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

Asymptotic Estimates for Spectral Estimators of Rotationally Invariant Matrices

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Asymptotic Estimates for Spectral Estimators of Rotationally...

IEEE International Symposium on Information Theory

作者： Zhuohang He Xiaojun Yuan Junjie Ma National Key Laboratory of Wireless Communications University of Electronic Science and Technology of China Chengdu China Institute of Computational Mathematics and Scientific/Engineering Computing AMSS Chinese Academy of Sciences Beijing China

ISBN: (数字)9798350382846

ISBN: (纸本)9798350382853

In this paper, we consider the recovery of low-rank matrices from noisy observations using spectral denoisers, where the singular values are denoised through an identical scalar smoothing function. We explore the asymptotic mean squared error (AMSE) of these denoisers within a framework where the rank of the matrix to be recovered grows linearly with the matrix size. We demonstrate that, under arbitrary i.i.d. noise and some mild regularity assumptions, the AMSE converges in probability to a deterministic function of the noise power. Our results are applicable to commonly used denoisers, including the best-rank-r denoiser, the singular-value soft-threshold denoiser, and the singular-value hard-threshold denoiser. To the best of our knowledge, this is the first study to establish an analytical expression for the asymptotic MSE under arbitrary i.i.d. noise. The derived analytical expression depends solely on the empirical distribution of the singular values of the low-rank matrix and the specific form of the spectral denoiser employed.

关键词： Smoothing methods Limiting Noise Noise reduction Noise measurement Information theory

GRADIENT-ENHANCED DEEP NEURAL NETWORK APPROXIMATIONS

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arXiv 2022年

作者： Feng, Xiaodong Zeng, Li LSEC Institute of Computational Mathematics and Scientific/Engineering Computing AMSS Chinese Academy of Sciences Beijing China

We propose in this work the gradient-enhanced deep neural networks (DNNs) approach for function approximations and uncertainty quantification. More precisely, the proposed approach adopts both the function evaluations and the associated gradient information to yield enhanced approximation accuracy. In particular, the gradient information is included as a regularization term in the gradient-enhanced DNNs approach, for which we present similar posterior estimates (by the two-layer neural networks) as those in the path-norm regularized DNNs approximations. We also discuss the application of this approach to gradient-enhanced uncertainty quantification, and present several numerical experiments to show that the proposed approach can outperform the traditional DNNs approach in many cases of interests. Copyright © 2022, The Authors. All rights reserved.

关键词： Deep neural networks