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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 of the second-order BDF scheme with variable steps for the molecular beam epitaxial model without slope selection

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Science China mathematics 2021年第5期64卷 887-902页

作者： Hong-Lin Liao Xuehua Song Tao Tang Tao Zhou Department of Mathematics Nanjing University of Aeronautics and AstronauticsNanjing 211101China Department of Mathematics and International Center for Mathematics Southern University of Science and TechnologyShenzhen 518055China Division of Science and Technology BNU-HKBU United International CollegeZhuhai 519087China NCMIS&LSEC Institute of Computational Mathematics and Scientific Engineering ComputingAcademy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China

In this work,we are concerned with the stability and convergence analysis of the second-order backward difference formula(BDF2)with variable steps for the molecular beam epitaxial model without slope *** first show that the variable-step BDF2 scheme is convex and uniquely solvable under a weak time-step *** we show that it preserves an energy dissipation law if the adjacent time-step ratios satisfy r_(k):=τ_(k)/τ_(k-1)<***,with a novel discrete orthogonal convolution kernels argument and some new estimates on the corresponding positive definite quadratic forms,the L^(2)norm stability and rigorous error estimates are established,under the same step-ratio constraint that ensures the energy stability,i.e.,0

关键词： molecular beam epitaxial growth variable-step BDF2 scheme discrete orthogonal convolution kernels energy stability convergence analysis

ADAPTIVE IMPORTANCE SAMPLING FOR DEEP RITZ

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arXiv 2023年

作者： Wan, Xiaoliang Zhou, Tao Zhou, Yuancheng Department of Mathematics Center for Computation and Technology Louisiana State University Baton Rouge70803 United States LSEC Institute of Computational Mathematics and Scientific/Engineering Computing AMSS Chinese Academy of Sciences Beijing China

We introduce an adaptive sampling method for the Deep Ritz method aimed at solving partial differential equations(PDEs). Two deep neural networks are used. One network is employed to approximate the solution of PDEs, while the other one is a deep generative model used to generate new collocation points to refine the training set. The adaptive sampling procedure consists of two main steps. The first step is solving the PDEs using the Deep Ritz method by minimizing an associated variational loss discretized by the collocation points in the training set. The second step involves generating a new training set, which is then used in subsequent computations to further improve the accuracy of the current approximate solution. We treat the integrand in the variational loss as an unnormalized probability density function(PDF) and approximate it using a deep generative model called bounded KRnet. The new samples and their associated PDF values are obtained from the bounded KRnet. With these new samples and their associated PDF values, the variational loss can be approximated more accurately by importance sampling. Compared to the original Deep Ritz method, the proposed adaptive method improves accuracy, especially for problems characterized by low regularity and high dimensionality. We demonstrate the effectiveness of our new method through a series of numerical experiments. © 2023, CC BY.

关键词： Importance sampling

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

Convergence of machine learning methods for feedback control laws: averaged feedback learning scheme and data driven method

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arXiv 2024年

作者： Kunisch, Karl Vásquez-Varas, Donato Radon Institute for Computational and Applied Mathematics Austrian Academy of Sciences Altenbergerstraße 69 Upper Austria Linz4040 Austria Institute of Mathematics and Scientific Computing University of Graz Heinrichstraße 36 Styria Graz8010 Austria

This work addresses the synthesis of optimal feedback control laws via machine learning. In particular, the Averaged Feedback Learning Scheme (AFLS) and a data driven method are considered. Hypotheses for each method ensuring the convergence of the evaluation of the objective function of the underlying control problem at the obtained feedback-laws towards the optimal value function are provided. These hypotheses are connected to the regularity of the value function and the stability of the dynamics. In the case of AFLS these hypotheses only require Hölder continuity of the value function, whereas for the data driven method the value function must be at least C2. It is demonstrated that these methods are connected via their optimality conditions. Additionally, numerical experiments are provided by applying both methods to a family control problems, parameterized by a positive real number which controls the regularity of the value function. For small parameters the value function is smooth and in contrast for large parameters it is non-differentiable, but semi-concave. The results of the experiments indicate that both methods have a similar performance for the case that the value function is smooth. On the other hand, if the value function is not differentiable, AFLS has a better performance which is consistent with the obtained convergence *** Codes 49L12, 49J15, 49N35, 68Q32, 93B52 © 2024, CC BY.

关键词： Machine learning

A New Adaptive Balanced Augmented Lagrangian Method with Application to ISAC Beamforming Design

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arXiv 2024年

作者： Wu, Jiageng Jiang, Bo Li, Xinxin Liu, Ya-Feng Yuan, Jianhua School of Mathematics Jilin University Changchun130012 China Ministry of Education Key Laboratory for NSLSCS School of Mathematical Sciences Nanjing Normal University Nanjing210023 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 Beijing100190 China School of Sciences Beijing University of Posts and Telecommunications Beijing100876 China

In this paper, we consider a class of convex programming problems with linear equality constraints, which finds broad applications in machine learning and signal processing. We propose a new adaptive balanced augmented Lagrangian (ABAL) method for solving these problems. The proposed ABAL method adaptively selects the stepsize parameter and enjoys a low per-iteration complexity, involving only the computation of a proximal mapping of the objective function and the solution of a linear equation. These features make the proposed method wellsuited to large-scale problems. We then custom-apply the ABAL method to solve the ISAC beamforming design problem, which is formulated as a nonlinear semidefinite program in a previous work. This customized application requires careful exploitation of the problem's special structure such as the property that all of its signal-to-interference-and-noise-ratio (SINR) constraints hold with equality at the solution and an efficient computation of the proximal mapping of the objective function. Simulation results demonstrate the efficiency of the proposed ABAL method. © 2024, CC BY.

关键词： Signal to noise ratio

Smooth Approximation of Feedback Laws for Infinite Horizon Control Problems with Non-Smooth Value Functions

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SSRN

SSRN 2024年

作者： Vásquez-Varas, Donato maximiliano Kunisch, Karl Johann Radon Institute for Computational and Applied Mathematics Austrian Academy of Sciences Altenbergerstraße 69 Upper Austria Linz4040 Austria Institute of Mathematics and Scientific Computing University of Graz Heinrichstraße 36 Styria Graz8010 Austria

In this work the synthesis of approximate optimal and smooth feedback laws for infinite horizon optimal control problems is addressed. In this regards, $L^{p}$ type error bounds of the approximating smooth feedback laws are derived, depending on regularity of the value function. These error bounds combined with the existence of a Lyapunov type function are used to prove the existence of an approximate optimal sequence of smooth feedback laws. Moreover, we extend this result to the Hölder continuous case by a diagonalization argument combined with the Moreau envelope. It is foreseen that these error bounds could be applied to study the convergence of synthesis of feedback laws via data driven machine learning methods. We provide an example for which the value functions is non-differentiable but Lipschitz continuous. We point out that in this example no restrictions on either the controls or the trajectories are assumed. © 2024, The Authors. All rights reserved.

关键词： Learning systems

PML-based boundary integral equation method for electromagnetic scattering problems in a layered-medium

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arXiv 2024年

作者： Bao, Gang Lu, Wangtao Yin, Tao Zhang, Lu School of Mathematical Sciences Zhejiang University Zhejiang310027 China 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 Zhejiang University Hangzhou310027 China

This paper proposes a new boundary integral equation (BIE) methodology based on the perfectly matched layer (PML) truncation technique for solving the electromagnetic scattering problems in a multi-layered medium. Instead of using the original PML stretched fields, artificial fields which are also equivalent to the solutions in the physical region are introduced. This significantly simplifies the study of the proposed methodology to derive the PML problem. Then some PML transformed layer potentials and the associated boundary integral operators (BIOs) are defined and the corresponding jump relations are shown. Under the assumption that the fields vanish on the PML boundary, the solution representations, as well as the related BIEs and regularization of the hyper-singular operators, in terms of the current density functions on the truncated interface, are derived. Numerical experiments are presented to demonstrate the efficiency and accuracy of the method. Copyright © 2024, The Authors. All rights reserved.

关键词： Boundary integral equations

Mixed-integer linearity in nonlinear optimization: a trust region approach

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arXiv 2023年

作者： De Marchi, Alberto University of the Bundeswehr Munich Department of Aerospace Engineering Institute of Applied Mathematics and Scientific Computing Neubiberg85577 Germany

Bringing together nonlinear optimization with polyhedral and integrality constraints enables versatile modeling, but poses significant computational challenges. We investigate a method to address these problems based on sequential mixed-integer linearization with trust region safeguard, computing feasible iterates via calls to a generic mixed-integer linear solver. Convergence to critical, possibly suboptimal, feasible points is established for arbitrary starting points. Finally, we present numerical applications in nonsmooth optimal control and optimal network design and *** Codes 65K05, 90C06, 90C11, 90C30 © 2023, CC BY.

关键词： Nonlinear programming