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Further insight into the convergence of the Fletcher-Reeves method

Science China Mathematics 1999年第9期42卷 905-916页

作者：戴彧虹 1. State Key Laboratory of Scientific and Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing Chinese Academy of Sciences 100080 Beijing China

The convergence properties of the Fletcher-Reeves method for unconstrained optimization are further studied with the technique of generalized line search. Two conditions are given which guarantee the global convergence of the Fletcher-Reeves method using generalized Wolfe line searches or generalized Arjimo line searches, whereas an example is constructed showing that the conditions cannot be relaxed in certain senses.

关键词： unconstrained optimization conjugate gradient Fletcher-Reeves method generalid line search global convergence

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Convergence properties of Beale-Powell restart algorithm

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Science China Mathematics 1998年第11期41卷 1142-1150页

作者：戴彧虹袁亚湘 1. State Key Laboratory of Scientific and Engineering Computing ICMSEC Chinese Academy of Sciences 100080 Beijing China

The Beale\|Powell restart algorithm is highly useful for large\|scale unconstrained optimization. An example is taken to show that the algorithm may fail to converge. The global convergence of a slightly modified algo... 详细信息

关键词： unconstrained optimization conjugate gradient restart line search global convergence.

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Sparse two-sided rank-one updates for nonlinear equations

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Science China Mathematics 2010年第11期53卷 2907-2915页

作者： CHENG MingHou & DAI YuHong state {1. 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 1.01.0, China 1. 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 100190 China

The two-sided rank-one (TR1. update method was introduced by Griewank and Walther (2002) for solving nonlinear equations. It generates dense approximations of the Jacobian and thus is not applicable to large-scale sparse problems. To overcome this difficulty, we propose sparse extensions of the TR1.update and give some convergence analysis. The numerical experiments show that some of our extensions are superior to the TR1.update method. Some convergence analysis is also presented.

关键词： TR1 update Broyden rank-one update sparsity

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Several splittings for non-Hermitian linear systems

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Science China Mathematics 2008年第8期51卷 1339-1348页

作者： BAI Zhong-Zhi state {1. laboratory of scientific/engineering computing,Institute of Computational Mathematics and scientific/engineering computing,Academy of Mathematics and Systems Science,Chinese Academy of Sciences,P.O.Box 271.,Beijing 1.0080,China 1. State Key Laboratory of Scientific/Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences P.O. Box 2719 Beijing 100080 China

For large sparse non-Hermitian positive definite system of linear equations, we present several variants of the Hermitian and skew-Hermitian splitting (HSS) about the coefficient matrix and establish correspondingly several HSS-based iterative schemes. Theoretical analyses show that these methods are convergent unconditionally to the exact solution of the referred system of linear equations, and they may show advantages on problems that the HSS method is ineffective.

关键词： Hermitian and skew-Hermitian splitting non-Hermitian linear system splitting iterative scheme convergence 65F1. 65F1. 65F50 65N22

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Numerical simulation of Richtmyer-Meshkov instability

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Science China Mathematics 2004年第z1期47卷 234-244页

作者： FU Dexun, MA Yanwen, ZHANG Linbo & TIAN Baolinstate {1. laboratory of Nonlinear Mechanics, Institute of Mechanics, Chinese Academy of Sciences, Beijing 1.0080, China state {1. laboratory of scientific and engineering computing, Institute of Computational Mathematics, Chinese Academy of Sciences, Beijing 1.0080, China 1. State Key Laboratory of Nonlinear Mechanics Institute of Mechanics Chinese Academy of Sciences 100080 Beijing China2. State Key Laboratory of Scientific and Engineering Computing Institute of Computational Mathematics Chinese Academy of Sciences 100080 Beijing China

The compressible Navier-Stokes equations discretized with a fourth order accurate compact finite difference scheme with group velocity control are used to simulate the Richtmyer-Meshkov (R-M) instability problem produced by cylindrical shock-cylindrical material interface with shock Mach number Ms=1.2 and density ratio 1.20 (interior density/outer density). Effect of shock refraction, reflection, interaction of the reflected shock with the material interface, and effect of initial perturbation modes on R-M instability are investigated numerically. It is noted that the shock refraction is a main physical mechanism of the initial phase changing of the material surface. The multiple interactions of the reflected shock from the origin with the interface and the R-M instability near the material interface are the reason for formation of the spike-bubble structures. Different viscosities lead to different spike-bubble structure characteristics. The vortex pairing phenomenon is found in the initial double mode simulation. The mode interaction is the main factor of small structures production near the interface.

关键词： R-M instability, direct numerical simulation, shock-interface interaction.

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A new trust region algorithm for image restoration

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Science China Mathematics 2005年第2期48卷 169-184页

作者： WEN Zaiwen & WANG Yanfei state {1. laboratory of scientific engineering computing, Institute of Computational Mathematics and scientific/engineering computing, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 1.0080, China National {1. laboratory on Remote Sensing Science, Institute of Remote Sensing Applications, Chinese Academy of Sciences, Beijing 1.01.1. China 1. State Key Laboratory of Scientific Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and System Sciences Chinese Academy of Sciences 100080 Beijing China 2. National Key Laboratory on Remote Sensing Science Institute of Remote Sensing Applications Chinese Academy of Sciences 100101 Beijing China

The image restoration problems play an important role in remote sensing and astronomical image analysis. One common method for the recovery of a true image from corrupted or blurred image is the least squares error (LSE) method. But the LSE method is unstable in practical applications. A popular way to overcome instability is the Tikhonov regularization. However, difficulties will encounter when adjusting the so-called regularization parameter a. Moreover, how to truncate the iteration at appropriate steps is also challenging. In this paper we use the trust region method to deal with the image restoration problem, meanwhile, the trust region subproblem is solved by the truncated Lanczos method and the preconditioned truncated Lanczos method. We also develop a fast algorithm for evaluating the Kronecker matrix-vector product when the matrix is banded. The trust region method is very stable and robust, and it has the nice property of updating the trust region automatically. This releases us from tedious finding the regularization parameters and truncation levels. Some numerical tests on remotely sensed images are given to show that the trust region method is promising.

关键词： trust region algorithm, image restoration, Lanczos method, Kronecker matrix-vector product, preconditioning.

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Sensing With Communication Signals: From Information Theory to Signal Processing

arXiv

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arXiv 2025年

作者： Liu, Fan Liu, Ya-Feng Cui, Yuanhao Masouros, Christos Xu, Jie Han, Tony Xiao Buzzi, Stefano Eldar, Yonina C. Jin, Shi National Mobile Communications Research Laboratory School of Information Science and Engineering Southeast University Nanjing210096 China State Key Laboratory of Scientific and Engineering Computing Institute of Computational Mathematicsand Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China Department of Communication Engineering Beijing University of Posts and Telecommunications Beijing100876 China Department of Electronic and Electrical Engineering University College London LondonWC1E 7JE United Kingdom Guangdong518172 China Huawei Technologies Co. Ltd China Department of Electrical and Information Engineering University of Cassino and Southern Lazio CassinoI-03043 Italy Department of Electronics Information and Bioengineering Politecnico di Milano MilanoI-20133 Italy ParmaI-43124 Italy Faculty of Mathematics and Computer Science Weizmann Institute of Science Rehovot Israel

The Integrated Sensing and Communications (ISAC) paradigm is anticipated to be a cornerstone of the upcoming 6G networks. In order to optimize the use of wireless resources, 6G ISAC systems need to harness the communication data payload signals, which are inherently random, for both sensing and communication (S&C) purposes. This tutorial paper provides a comprehensive technical overview of the fundamental theory and signal processing methodologies for ISAC transmission with random communication signals. We begin by introducing the deterministic-random tradeoff (DRT) between S&C from an information-theoretic perspective, emphasizing the need for specialized signal processing techniques tailored to random ISAC signals. Building on this foundation, we review the core signal models and processing pipelines for communication-centric ISAC systems, and analyze the average squared auto-correlation function (ACF) of random ISAC signals, which serves as a fundamental performance metric for multi-target ranging tasks. Drawing insights from these theoretical results, we outline the design principles for the three key components of communication-centric ISAC systems: modulation schemes, constellation design, and pulse shaping filters. The goal is to either enhance sensing performance without compromising communication efficiency or to establish a scalable tradeoff between the two. We then extend our analysis from a single-antenna ISAC system to its multi-antenna counterpart, discussing recent advancements in multiinput multi-output (MIMO) precoding techniques specifically designed for random ISAC signals. We conclude by highlighting several open challenges and future research directions in the field of sensing with communication signals. Copyright © 2025, The Authors. All rights reserved.

关键词： Signal modulation

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Convergence analysis of the adaptive finite element method with the red-green refinement

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Science China Mathematics 2010年第2期53卷 499-512页

作者： ZHAO XuYing 1. 3 , HU Jun 2,? & SHI ZhongCi 1.1.state {1. 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 1.01.0, China 2 {1. laboratory of Mathematics and Applied Mathematics (Peking University), Ministry of Education and School of Mathematical Sciences, Peking University, Beijing 1.0871. China 3 Graduate University of Chinese Academy of Sciences, Beijing 1.01.0, China 1. 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 100190 China3. Graduate University of Chinese Academy of Sciences Beijing 100190 China2. Key Laboratory of Mathematics and Applied Mathematics (Peking University) Ministry of Education and School of Mathematical Sciences Peking University Beijing 100871 China

In this paper, we analyze the convergence of the adaptive conforming P 1.element method with the red-green refinement. Since the mesh after refining is not nested into the one before, the Galerkin-orthogonality does not hold for this case. To overcome such a difficulty, we prove some quasi-orthogonality instead under some mild condition on the initial mesh (Condition A). Consequently, we show convergence of the adaptive method by establishing the reduction of some total error. To weaken the condition on the initial mesh, we propose a modified red-green refinement and prove the convergence of the associated adaptive method under a much weaker condition on the initial mesh (Condition B).

关键词： red-green refinement adaptive finite element method convergence non-nested local refinement

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Extended multivariate public key cryptosystems with secure encryption function

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Science China(Information Sciences) 2011年第6期54卷 1161-1171页

作者： WANG HouZhen 1.2 , ZHANG HuanGuo 1.2 , WANG ZhangYi 1.2 & TANG Ming 1.2 1.The key {1. of Aerospace Information Security and Trusted computing, Ministry of Education, School of Computer, Wuhan University, Wuhan 430079, China 2 state key {1. of Software engineering, Wuhan University, Wuhan 430072, China 1. The Key Laboratory of Aerospace Information Security and Trusted Computing Ministry of Education School of Computer Wuhan University Wuhan 430079 China2. State Key Laboratory of Software Engineering Wuhan University Wuhan 430072 China

Advances in quantum computers pose great threats on the currently used public key cryptographic algorithms such as RSA and ECC. As a promising candidate secure against attackers equipped with quantum computational power, multivariate public key cryptosystems (MPKCs) have attracted increasing attention in recently years. Unfortunately, the existing MPKCs can only be used as a multivariate signature scheme, and it remains unknown how to construct an efficient MPKC enabling secure encryption. Furthermore, some multivariate signature schemes have been shown insecure in recent years, and it is also not trivial to build MPKC which can serve as a secure signature scheme. By employing the basic MQ-trapdoors, this paper proposes a novel MPKC and shows how it can be used as a multivariate signature scheme and a multivariate encryption scheme, respectively. The goal is achieved by incorporating our new hash authentication techniques and some modification methods such as the Shamir's minus method. Thorough analysis shows that our schemes are secure and efficient. Our MPKC gives a positive response to the challenges in multivariate public key cryptography.

关键词： cryptography post-quantum cryptography MQ problem hash function

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A trust region-CG algorithm for deblurring problem in atmospheric image reconstruction

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Science China Mathematics 2002年第6期45卷 731-740页

作者：王彦飞袁亚湘张洪超 1. State Key Laboratory of Scientific and Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and System Sciences Chinese Academy of Sciences 100080 Beijing China

In this paper we solve large scale ill-posed problems, particularly the image restoration problem in atmospheric imaging sciences, by a trust region-CG algorithm. Image restoration involves the removal or minimization of degradation (blur, clutter, noise, etc.) in an image using a priori knowledge about the degradation phenomena. Our basic technique is the so-called trust region method, while the subproblem is solved by the truncated conjugate gradient method, which has been well developed for well-posed *** trust region method, due to its robustness in global convergence, seems to be a promising way to deal with ill-posed problems.

关键词： image reconstruction, ill-posed problems, truncated CG, trust region.

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