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New characterizations for the variation of the spectrum of an arbitrary matrix

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arXiv 2017年

作者： Xu, Xuefeng 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

The celebrated Hoffman–Wielandt theorem reveals the strong stability of the spectrum of a normal matrix under perturbations. Over the past decades, some analogs of the Hoffman–Wielandt theorem have been developed to characterize the stability of the spectrum of an arbitrary matrix. In this paper, we establish new perturbation bounds to characterize the variation of the spectrum of an arbitrary matrix. The counterparts of the existing results are also given, which are sharper than the existing ones. Moreover, our results have generalized some perturbation bounds for the spectrum of a normal matrix. Copyright © 2017, The Authors. All rights reserved.

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TIGHTNESS OF A NEW AND ENHANCED SEMIDEFINITE RELAXATION FOR MIMO DETECTION

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arXiv 2017年

作者： LU, CHENG LIU, YA-FENG ZHANG, WEI-QIANG ZHANG, SHUZHONG School of Economics and Management North China Electric Power University Beijing 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 Department of Electronic Engineering Tsinghua University Beijing100084 China Department of Industrial and Systems Engineering University of Minnesota MinneapolisMN55455 United States

In this paper, we consider a fundamental problem in modern digital communications known as multiple-input multiple-output (MIMO) detection, which can be formulated as a complex quadratic programming problem subject to unit-modulus and discrete argument constraints. Various semidefinite relaxation (SDR) based algorithms have been proposed to solve the problem in the literature. In this paper, we first show that the conventional SDR is generally not tight for the problem. Then, we propose a new and enhanced SDR and show its tightness under an easily checkable condition, which essentially requires the level of the noise to be below a certain threshold. The above results have answered an open question posed by So in [35]. Numerical simulation results show that our proposed SDR significantly outperforms the conventional SDR in terms of the relaxation gap. Copyright © 2017, The Authors. All rights reserved.

关键词： Quadratic programming

A note on semidefinite programming relaxations for polynomial optimization over a single sphere

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Science China mathematics 2016年第8期59卷 1543-1560页

作者： HU Jiang JIANG Bo LIU Xin WEN ZaiWen Beijing International Center for Mathematical Research Peking University Research Center for Management Science and Data Analytics School of Information Management and EngineeringShanghai University of Finance and Economics LSEC Institute of Computational Mathematics and Scientific/Engineering ComputingAcademy of Mathematics and Systems Science Chinese Academy of Sciences

We study two instances of polynomial optimization problem over a single sphere. The first problem is to compute the best rank-1 tensor approximation. We show the equivalence between two recent semidefinite relaxations methods. The other one arises from Bose-Einstein condensates(BEC), whose objective function is a summation of a probably nonconvex quadratic function and a quartic term. These two polynomial optimization problems are closely connected since the BEC problem can be viewed as a structured fourth-order best rank-1 tensor approximation. We show that the BEC problem is NP-hard and propose a semidefinite relaxation with both deterministic and randomized rounding procedures. Explicit approximation ratios for these rounding procedures are presented. The performance of these semidefinite relaxations are illustrated on a few preliminary numerical experiments.

关键词： polynomial optimization over a single sphere semidefinite programming best rank-1 tensor ap-proximation Bose-Einstein condensates

Optimal Estimation of Sensor Biases for Asynchronous Multi-Sensor Data Fusion

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arXiv 2017年

作者： Pu, Wenqiang Liu, Ya-Feng Yan, Junkun Liu, Hongwei Luo, Zhi-Quan National Laboratory of Radar Signal Processing Collaborative Innovation Center of Information Sensing and Understanding Xidian University Xi'an710071 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 Shenzhen Research Institute of Big Data Chinese University of Hong Kong Shenzhen518100 China

An important step in a multi-sensor surveillance system is to estimate sensor biases from their noisy asynchronous measurements. This estimation problem is computationally challenging due to the highly nonlinear transformation between the global and local coordinate systems as well as the measurement asynchrony from different sensors. In this paper, we propose a novel nonlinear least squares (LS) formulation for the problem by assuming the existence of a reference target moving with an (unknown) constant velocity. We also propose an efficient block coordinate decent (BCD) optimization algorithm, with a judicious initialization, to solve the problem. The proposed BCD algorithm alternately updates the range and azimuth bias estimates by solving linear least squares problems and semidefinite programs (SDPs). In the absence of measurement noise, the proposed algorithm is guaranteed to find the global solution of the problem and the true biases. Simulation results show that the proposed algorithm significantly outperforms the existing approaches in terms of the root mean square error (RMSE). Copyright © 2017, The Authors. All rights reserved.

关键词： Mean square error

Error estimates of the Crank-Nicolson Galerkin method for the time-dependent Maxwell-Schrödinger equations under the Lorentz gauge

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arXiv 2017年

作者： Chupeng, M.A. Liqun, C.A.O. Yanping, L.I.N. Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China Department of Applied Mathematics Hong Kong Polytechnic University Hung Hom Kowloon Hong Kong

In this paper we study the numerical method and the convergence for solving the time-dependent Maxwell-Schrödinger equations under the Lorentz gauge. An alternating Crank-Nicolson finite element method for solving the problem is presented and the optimal error estimate for the numerical algorithm is obtained by a mathematical inductive method. Numerical examples are then carried out to confirm the theoretical results. Copyright © 2017, The Authors. All rights reserved.

关键词： Galerkin methods

Sample Approximation-Based Deflation Approaches for Chance SINR-Constrained Joint Power and Admission Control

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Sample Approximation-Based Deflation Approaches for Chance S...

作者： Liu, Ya-Feng Hong, Mingyi Song, Enbin State Key Laboratory of Scientific and Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing Chinese Academy of Sciences Beijing100190 China Iowa State University AmesIA50011 United States Department of Mathematics Sichuan University Chengdu610064 China

Consider the joint power and admission control (JPAC) problem for a multiuser single-input single-output (SISO) interference channel. Most existing works on JPAC assume the perfect instantaneous channel state information (CSI). In this paper, we consider the JPAC problem with the imperfect CSI, i.e., we assume that only the channel distribution information (CDI) is available. We formulate the JPAC problem into a chance (probabilistic)-constrained program, where each link's SINR outage probability is enforced to be less than or equal to a specified tolerance. To circumvent the computational difficulty of the chance SINR constraints, we propose to use the sample (scenario) approximation scheme to convert them into finitely many simple linear constraints. Furthermore, we reformulate the sample approximation of the chance SINR-constrained JPAC problem as a composite group sparse minimization problem and then approximate it by a second-order cone program (SOCP). The solution of the SOCP approximation can be used to check the simultaneous supportability of all links in the network and to guide an iterative link removal procedure (the deflation approach). We exploit the special structure of the SOCP approximation and custom-design an efficient algorithm for solving it. Finally, we illustrate the effectiveness and efficiency of the proposed sample approximation-based deflation approaches by simulations. © 2016 IEEE.

关键词： Channel state information

Low-resolution simulations of vesicle suspensions in 2D

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arXiv 2017年

作者： Kabacaoglu, Gökberk Quaife, Bryan Biros, George Department of Mechanical Engineering University of Texas at Austin AustinTX78712 United States Department of Scientific Computing Florida State University TallahasseeFL32306 United States Institute for Computational Engineering and Sciences The University of Texas at Austin AustinTX78712 United States

Vesicle suspensions appear in many biological and industrial applications. These suspensions are characterized by rich and complex dynamics of vesicles due to their interaction with the bulk fluid, and their large deformations and nonlinear elastic properties. Many existing state-of-the-art numerical schemes can resolve such complex vesicle flows. However, even when using provably optimal algorithms, these simulations can be computationally expensive, especially for suspensions with a large number of vesicles. These high computational costs can limit the use of simulations for parameter exploration, optimization, or uncertainty quantification. One way to reduce the cost is to use low-resolution discretizations in space and time. However, it is well-known that simply reducing the resolution results in vesicle collisions, numerical instabilities, and often in erroneous results. In this paper, we investigate the effect of a number of algorithmic empirical fixes (which are commonly used by many groups) in an attempt to make low-resolution simulations more stable and more predictive. Based on our empirical studies for a number of flow configurations, we propose a scheme that attempts to integrate these fixes in a systematic way. This low-resolution scheme is an extension of our previous work [49, 51]. Our low-resolution correction algorithms (LRCA) include anti-aliasing and membrane reparametrization for avoiding spurious oscillations in vesicles' membranes, adaptive time stepping and a repulsion force for handling vesicle collisions and, correction of vesicles' area and arc-length for maintaining physical vesicle shapes. We perform a systematic error analysis by comparing the low-resolution simulations of dilute and dense suspensions with their high-fidelity, fully resolved, counterparts. We observe that the LRCA enables both efficient and statistically accurate low-resolution simulations of vesicle suspensions, while it can be 10× to 100× faster. Copyright © 2017, The

关键词： Suspensions (fluids)

Fourier phase retrieval: Uniqueness and algorithms

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arXiv 2017年

作者： Bendory, Tamir Beinerty, Robert Eldarz, Yonina C. Program in Applied and Computational Mathematics Princeton University PrincetonNJ United States Institute of Mathematics and Scientific Computing University of Graz Heinrichstraße 36 Graz8010 Austria Andrew and Erna Viterbi Faculty of Electrical Engineering Technion - Israel Institute of Technology Haifa Israel

The problem of recovering a signal from its phaseless Fourier transform measurements, called Fourier phase retrieval, arises in many applications in engineering and science. Fourier phase retrieval poses fundamental theoretical and algorithmic challenges. In general, there is no unique mapping between a one-dimensional signal and its Fourier magnitude and therefore the problem is ill-posed. Additionally, while almost all multidimensional signals are uniquely mapped to their Fourier magnitude, the performance of existing algorithms is generally not well-understood. In this chapter we survey methods to guarantee uniqueness in Fourier phase retrieval. We then present different algorithmic approaches to retrieve the signal in practice. We conclude by outlining some of the main open questions in this field. Copyright © 2017, The Authors. All rights reserved.

关键词： Fourier transforms

Prediction of the low-velocity distribution from the pore structure in simple porous media

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Physical Review Fluids 2017年第12期2卷 124103-124103页

作者： Pietro de Anna Bryan Quaife George Biros Ruben Juanes Institut des Sciences de la Terre University of Lausanne Lausanne 1015 Switzerland Department of Scientific Computing Florida State University Tallahassee Florida 32306 USA Institute for Computational Engineering and Sciences The University of Texas at Austin 201 East 24th Street Austin Texas 78712 USA Massachusetts Institute of Technology 77 Massachusetts Avenue Cambridge Massachusetts 02139 USA

The macroscopic properties of fluid flow and transport through porous media are a direct consequence of the underlying pore structure. However, precise relations that characterize flow and transport from the statistics of pore-scale disorder have remained elusive. Here we investigate the relationship between pore structure and the resulting fluid flow and asymptotic transport behavior in two-dimensional geometries of nonoverlapping circular posts. We derive an analytical relationship between the pore throat size distribution fλ∼λ−β and the distribution of the low fluid velocities fu∼u−β/2, based on a conceptual model of porelets (the flow established within each pore throat, here a Hagen-Poiseuille flow). Our model allows us to make predictions, within a continuous-time random-walk framework, for the asymptotic statistics of the spreading of fluid particles along their own trajectories. These predictions are confirmed by high-fidelity simulations of Stokes flow and advective transport. The proposed framework can be extended to other configurations which can be represented as a collection of known flow distributions.

关键词： Flows in porous media Microfluidics Mixing in geophysical flows