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Estimation of Spectral Bounds in gradient algorithms

ACTA APPLICANDAE MATHEMATICAE 2013年第1期127卷 117-136页

作者： Pronzato, L. Zhigljavsky, A. Bukina, E. Univ Nice Sophia Antipolis CNRS Lab I3S F-06903 Sophia Antipolis France Cardiff Univ Sch Math Cardiff CF24 4YH S Glam Wales

We consider the solution of linear systems of equations Ax=b, with A a symmetric positive-definite matrix in a"e (nxn) , through Richardson-type iterations or, equivalently, the minimization of convex quadratic functions (1/2)(Ax,x)-(b,x) with a gradient algorithm. The use of step-sizes asymptotically distributed with the arcsine distribution on the spectrum of A then yields an asymptotic rate of convergence after k < n iterations, k -> a, that coincides with that of the conjugate-gradient algorithm in the worst case. However, the spectral bounds m and M are generally unknown and thus need to be estimated to allow the construction of simple and cost-effective gradient algorithms with fast convergence. It is the purpose of this paper to analyse the properties of estimators of m and M based on moments of probability measures nu (k) defined on the spectrum of A and generated by the algorithm on its way towards the optimal solution. A precise analysis of the behavior of the rate of convergence of the algorithm is also given. Two situations are considered: (i) the sequence of step-sizes corresponds to i.i.d. random variables, (ii) they are generated through a dynamical system (fractional parts of the golden ratio) producing a low-discrepancy sequence. In the first case, properties of random walk can be used to prove the convergence of simple spectral bound estimators based on the first moment of nu (k) . The second option requires a more careful choice of spectral bounds estimators but is shown to produce much less fluctuations for the rate of convergence of the algorithm.

关键词： Estimation of leading eigenvalues Arcsine distribution gradient algorithms Conjugate gradient Fibonacci numbers

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Common Lyapunov functions and gradient algorithms

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL 2004年第6期49卷 990-994页

作者： Liberzon, D Tempo, R Univ Illinois Coordinated Sci Lab Urbana IL 61801 USA Politecn Torino CNR IEIIT I-10129 Turin Italy

This note is concerned with the problem of finding a quadratic common Lyapunov function for a large family of stable linear systems. We present gradient iteration algorithms which give deterministic convergence for fi... 详细信息

关键词： common Lyapunov functions gradient algorithms randomized algorithms switched linear systems

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Comparison of Adaptive Antenna Arrays Controlled by gradient algorithms

Comparison of Adaptive Antenna Arrays Controlled by Gradient...

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作者： Raida, Zbyněk Brno University of Technology

The paper presents the Simple Kalman filter (SKF) that has been designed for the control of digital adaptive antenna arrays. The SKF has been applied to the pilot signal system and the steering vector one. The above systems based on the SKF are compared with adaptive antenna arrays controlled by the classical LMS and the Variable Step Size (VSS) LMS algorithms and by the pure Kalman filter. It is shown that the pure Kalman filter is the most convenient for the control of the adaptive arrays because it does not require any a priori information about noise statistics and excels in high rate of convergence and low misadjustment. Extremely high computational requirements are drawback of this filter. Hence, if low computational power of signal processors is at the disposal, the SKF is recommended to be used. Computational requirements of the SKF are of the same order as the classical LMS algorithm exhibits. On the other hand, all the important features of the pure Kalman filter are inherited by the SKF. The paper shows that presented Kalman filters can be regarded as special gradient algorithms. That is why they can be compared with the LMS family.

关键词： adaptive antenna array array signal processing gradient algorithms LMS algorith Kalman filter

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Discrete-time adaptive state tracking control schemes using gradient algorithms☆

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AUTOMATICA 2024年 169卷

作者： Tao, Gang Univ Virginia Dept Elect & Comp Engn Charlottesville VA 22904 USA

This paper revisits a classical adaptive control problem: adaptive state tracking control of a state- space plant model, and solves the open discrete-time state tracking model reference adaptive control problem. Adaptive state tracking control schemes for continuous-time systems have been reported in the literature, using a Lyapunov-algorithm based design and analysis procedure. Such a procedure has not been successfully applied to the discrete-time adaptive state tracking control problem which has remained open. In this paper, new adaptive state tracking control schemes are developed for discrete-time systems, using gradient algorithms for updating the controller parameters. Both direct and indirect adaptive designs are derived, which have the desired parameter adaptation properties and closed-loop system stability and state tracking properties. Such a new gradient-algorithm based framework is also applicable to the continuous-time adaptive state tracking control problem to develop new solutions, as compared with the traditional Lyapunov-algorithm based solutions. (c) 2024 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

关键词： Adaptive control Discrete-time systems gradient algorithms Stability analysis State tracking

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SUPPLEMENTARY OPTIMALITY PROPERTIES OF THE RESTORATION PHASE OF SEQUENTIAL gradient-RESTORATION algorithms FOR OPTIMAL-CONTROL PROBLEMS

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1983年第1期41卷 169-184页

作者： MIELE, A WANG, T Professor of Astronautics and Mathematical Sciences Rice University Houston Texas Graduate Student Department of Mechanical Engineering Rice University Houston Texas

In this paper, sequential gradient-restoration algorithms for optimal control problems are considered, and attention is focused on the restoration phase. It is shown that the Lagrange multipliers associated with the restoration phase not only solve the auxiliary minimization problem of the restoration phase, but are also endowed with a supplementary optimality property: they minimize a special functional, quadratic in the multipliers, subject to the multiplier differential equations and boundary conditions, for given state, control, and parameter.

关键词： Numerical methods computing methods optimal control optimality properties supplementary optimality properties gradient algorithms gradient-restoration algorithms sequential gradient-restoration algorithms restoration algorithms general boundary conditions nondifferential constraints

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Supplementary Optimality Properties of the Family of gradient-Restoration algorithms for Optimal Control Problems

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IFAC Proceedings Volumes 1983年第8期16卷 109-120页

作者： A. Miele T. Wang Departments of Mechanical Engineering and Mathematical Sciences Rice University Houston Texas Department of Mechanical Engineering Rice University Houston Texas

The problem of minimizing a functional subject to differential constraints, nondifferential constraints, initial constraints, and final constraints is considered within the frame of the family of gradient-restoration algorithms for optimal control problems. This family includes sequential gradient-restoration algorithms (SGRA) and combined gradient-restoration algorithms (CGRA). The system of Lagrange multipliers associated with (i) the gradient phase of SGRA, (ii) the restoration phase of SGRA, and (iii) the combined gradient-restoration phase of CGRA is examined. It is shown that, in each case, the Lagrange multipliers are endowed with a suppiementary optimality property: they minimize a special functional, quadratic in the multipliers, subject to the multiplier differential equations and boundary conditions, for given state, control, and parameter. These supplementary optimality properties have considerable computational implications: they allow one to reduce the study of an iteration of (i), (ii), (iii) to a mathematical programming problem involving a finite number of parameters as unknowns.

关键词： Numerical methods computing methods mathematical programming optimal control optimality properties suppiementary optimality properties restoration algorithms gradient algorithms gradient-restoration algorithms sequential gradient-restoration algorithms combined gradient-restoration algorithms general boundary conditions nondifferential constraints

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An asymptotically optimal gradient algorithm for quadratic optimization with low computational cost

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OPTIMIZATION LETTERS 2013年第6期7卷 1047-1059页

作者： Zhigljavsky, Anatoly Pronzato, Luc Bukina, Elena Cardiff Univ Sch Math Cardiff CF24 4YH S Glam Wales CNRS UNS Lab I3S F-06903 Sophia Antipolis France

We consider gradient algorithms for minimizing a quadratic function in with large n. We suggest a particular sequence of step-sizes and demonstrate that the resulting gradient algorithm has a convergence rate comparable with that of Conjugate gradients and other methods based on the use of Krylov spaces. When the matrix is large and sparse, the proposed algorithm can be more efficient than the Conjugate gradient algorithm in terms of computational cost, as k iterations of the proposed algorithm only require the computation of O(log k) inner products.

关键词： Quadratic optimization gradient algorithms Conjugate gradient Arcsine distribution Fibonacci numbers Estimation of leading eigenvalues

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OPTIMAL COATINGS, BANG-BANG CONTROLS, AND gradient TECHNIQUES

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OPTIMAL CONTROL APPLICATIONS & METHODS 1987年第1期8卷 1-20页

作者： HAGER, WW ROSTAMIAN, R UNIV MARYLAND BALTIMORE CTY DEPT MATH CATONSVILLE MD 21228 USA

AbstractA model for a viscoelastic coating is derived and the coating reflectivity is determined. A coating which minimizes the reflection of sound waves is partly bang‐bang and partly singular. A new gradient type algorithm is presented for computing the optimal control. Formulae for the gradient of the cost in optimal control are derived in various setting

关键词： Viscoelastic coatings Sound wave reflection Bang‐bang control Singular arc problems gradient algorithms

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The analysis of gradient algorithm effectiveness - two dimensional heat transfer problem

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ARCHIVES OF THERMODYNAMICS 2010年第4期31卷 37-50页

作者： Lopata, Stanislaw Oclon, Pawel Cracow Univ Technol Div Power Engn Al Jana Pawla 2 37 PL-31864 Krakow Poland

The analysis of effectiveness of the gradient algorithm for the two-dimension steady state heat transfer problems is being performed. The three gradient algorithms - the BCG (biconjugate gradient algorithm), the BICGSTAB (biconjugate gradient stabilized algorithm), and the CGS (conjugate gradient squared algorithm) are implemented in a computer code. Because the first type boundary conditions are imposed, it is possible to compare the results with the analytical solution. Computations are carried out for different numerical grid densities. Therefore it is possible to investigate how the grid density influences the efficiency of the gradient algorithms. The total computational time, residual drop and the iteration time for the gradient algorithms are additionally compared with the performance of the SOR (successive over-relaxation) method.

关键词： Heat transfer Control volume method gradient algorithms

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Class separability estimation and incremental learning using boundary methods

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NEUROCOMPUTING 2000年第1-4期35卷 3-26页

作者： Sancho, JL Pierson, WE Ulug, B Figueiras-Vidal, AR Ahalt, SC Univ Carlos III Leganes Madrid Escuela Politecn Super ATSC DI Madrid Spain Ohio State Univ Dept Elect Engn Columbus OH 43210 USA

In this paper we discuss the use of boundary methods (BMs) for distribution analysis. We view these methods as tools which can be used to extract useful information from sample distributions. We believe that the information thus extracted has utility for a number of applications, but in particular we discuss the use of BMs as a mechanism for class separability estimation and as an aid to constructing robust and efficient neural networks (NNs) to solve classification problems. In the first case, BMs can establish the utility of a data set for classification. We demonstrate experimentally that the derived ranking is consistent with alternative ranking techniques based on Bayes error (epsilon). Finally, BMs are used as sample selection (SS) mechanism to train NN by means of gradient algorithms. In particular, elliptic BMs (EBMs) are used to select samples so that the initial partial training set is linearly separable. In a progressive way, new samples are added to the training set solving the problem in an incremental manner. Multi-layer perceptrons (MLPs) and radial basis functions (RBFs) have been used in this work. Our results show that the probability of being trapped in a local minimum is clearly reduced when EBMs are used, making the training independent of the initial weight values. Also, the effect of the very noisy samples and outliers is reduced when the SS-EBM algorithm is employed, so we propose this method as a robust procedure to train NNs by means of a gradient learning rule. (C) 2000 Elsevier Science B.V. All rights reserved.

关键词： classification neural systems class separability estimation Bayes' error estimation gradient algorithms sample selection strategies

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