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On the solvability of a unilateral obstacle problem on the boundary

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2004年第1期162卷 193-199页

作者： Addou, A Zahi, J Univ Mohammed 1 Fac Sci Dept Math & Informat Oujda Morocco

关键词： obstacle penalization projection algorithm

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A parameter adaptive controller which provides exponential stability: The first order case

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SYSTEMS & CONTROL LETTERS 2017年 103卷 23-31页

作者： Miller, Daniel E. Univ Waterloo Elect & Comp Engn Dept Waterloo ON Canada

Classical discrete-time adaptive controllers provide asymptotic stabilization. While the original adaptive controllers did not handle noise or unmodelled dynamics well, redesigned versions were proven to have some tolerance;however, neither exponential stabilization nor a bounded noise gain is typically proven. Here we consider the first order case and prove that if the original, ideal, projection algorithm is used in the estimation process (subject to the common assumption that the plant parameters lie in a convex, compact set and that the parameter estimates are restricted to that set), then it guarantees linear-like convolution bounds on the closed loop behaviour, which implies exponential stability and a bounded noise gain, as well an easily proven tolerance to unmodelled dynamics and plant parameter variation. (C) 2017 Elsevier B.V. All rights reserved.

关键词： Adaptive control projection algorithm projection Exponential stability Bounded gain

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Constraining by a family of strictly nonexpansive idempotent functions with applications in image reconstruction

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BIT NUMERICAL MATHEMATICS 2013年第2期53卷 527-544页

作者： Pantelimon, Ioana Popa, Constantin Ovidius Univ Constanta Constanta 900527 Romania Gheorghe Mihoc Caius Iacob Inst Stat Math & Appl Bucharest 050711 Romania

When solving an image reconstruction problem a previous knowledge concerning the original image may lead to various constraining strategies. A convergence result has been previously proved for a constrained version of the Kaczmarz projection algorithm with a single strictly nonexpansive idempotent function with a closed image. In this paper we consider a more general projection based iterative method and a family of such constraining functions with some additional hypotheses in order to better use the a priori information for every approximation calculated. We present a particular family of box-constraining functions which satisfies our assumptions and we design an adaptive algorithm that uses an iteration-dependent family of constraining functions for some numerical experiments of image reconstruction on Tomographic Particle Image Velocimetry.

关键词： Constraining strategy Strictly nonexpansive functions projection algorithm Least squares problems Image reconstruction

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A convergence result on random products of mappings in metric spaces

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FIXED POINT THEORY AND APPLICATIONS 2012年第1期2012卷 1-7页

作者： Khamsi, Mohamed Amine Louhichi, Issam Univ Texas El Paso Dept Math Sci El Paso TX 79968 USA King Fahd Univ Petr & Minerals Dept Math & Stat Dhahran 31261 Saudi Arabia

Let be a metric space and {T-1, ..., T-N} be a finite family of mappings defined on D subset of X. Let r: N -> {1,..., N} be a map that assumes every value infinitely often. The purpose of this article is to establish the convergence of the sequence (x(n)) defined by x(0) is an element of D;and x(n+1) = T-r(n)(x(n)), for all n >= 0. In particular, we extend the study of Bauschke [1] from the linear case of Hilbert spaces to metric spaces. Similarly we show that the examples of convergence hold in the absence of compactness. These type of methods have been used in areas like computerized tomography and signal processing.

关键词： computerized tomography convex feasibility problem convex programming Fejer monotone sequence image reconstruction image recovery innate bounded regularity Kaczmarz's method nonexpansive mapping projection algorithm projective mapping random product signal processing unrestricted iteration unrestricted product'

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A convergence result on random products of mappings in metric trees

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FIXED POINT THEORY AND APPLICATIONS 2012年第1期2012卷 1-10页

作者： Al-Mezel, Saleh Abdullah Khamsi, Mohamed Amine King Abdulaziz Univ Dept Math Jeddah 21589 Saudi Arabia Univ Texas El Paso Dept Math Sci El Paso TX 79968 USA

Let X be a metric space and {T (1), ..., T (N) } be a finite family of mappings defined on D aS, X. Let r : a"center dot -> {1,..., N} be a map that assumes every value infinitely often. The purpose of this article is to establish the convergence of the sequence (x (n) ) defined by x(0) is an element of D;and x(n+1) = T-r(n)(x(n)), for all n >= 0. In particular we prove Amemiya and Ando's theorem in metric trees without compactness assumption. This is the first attempt done in metric spaces. These type of methods have been used in areas like computerized tomography and signal processing.

关键词： computerized tomography convex feasibility problem convex programming metric tree nonexpansive mapping projection algorithm projective mapping random product signal processing unrestricted iteration unrestricted product

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A NORM CONVERGENCE RESULT ON RANDOM PRODUCTS OF RELAXED projectionS IN HILBERT-SPACE

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TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY 1995年第4期347卷 1365-1373页

作者： BAUSCHKE, HH Centre for Experimental and Constructive Mathematics Simon Fraser University Burnaby BC V5A 1S6 Canada

Suppose X is a Hilbert space and C-1, ..., C-N are closed convex intersecting subsets with projections P-1, ..., P-N. Suppose further r is a mapping from N onto {1, ..., N} that assumes every value infinitely often. We prove (a more general version of) the following result: If the N-tuple (C-1, ..., C-N) is ''innately boundedly regular'', then the sequence (x(n)), defined by x(0) is an element of X arbitrary, x(n+1) := P-r(n)x(n), for all n greater than or equal to 0, converges in norm to some point in boolean AND(i=1)(N) C-i. Examples without the usual assumptions on compactness are given. Methods of this type have been used in areas like computerized tomography and signal processing.

关键词： BANACH CONTRACTION COMPUTERIZED TOMOGRAPHY CONVEX FEASIBILITY PROBLEM CONVEX PROGRAMMING CONVEX SET FEJER MONOTONE SEQUENCE HILBERT SPACE IMAGE RECONSTRUCTION IMAGE RECOVERY INNATE BOUNDED REGULARITY KACZMARZS METHOD NONEXPANSIVE MAPPING ORTHOGONAL projection projection algorithm projection METHOD PROJECTIVE MAPPING RANDOM PRODUCT RELAXATION METHOD RELAXED projection SIGNAL PROCESSING UNRESTRICTED ITERATION UNRESTRICTED PRODUCT

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Solving the split equality problem without prior knowledge of operator norms

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OPTIMIZATION 2015年第9期64卷 1887-1906页

作者： Dong, Qiao-Li He, Songnian Zhao, Jing Civil Aviat Univ China Coll Sci Tianjin Peoples R China Civil Aviat Univ China Tianjin Key Lab Adv Signal Proc Tianjin Peoples R China

The split equality problem has extraordinary utility and broad applicability in many areas of applied mathematics. Recently, Moudafi proposed an alternating CQ algorithm and its relaxed variant to solve it. However, to employ Moudafi's algorithms, one needs to know a priori norm (or at least an estimate of the norm) of the bounded linear operators (matrices in the finite-dimensional framework). To estimate the norm of an operator is very difficult, but not an impossible task. It is the purpose of this paper to introduce a projection algorithm with a way of selecting the stepsizes such that the implementation of the algorithm does not need any priori information about the operator norms. We also practise this way of selecting stepsizes for variants of the projection algorithm, including a relaxed projection algorithm where the two closed convex sets are both level sets of convex functions, and a viscosity algorithm. Both weak and strong convergence are investigated.

关键词： split equality problem projection algorithm relaxed projection algorithm viscosity algorithm

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Two algorithms for solving single-valued variational inequalities and fixed point problems

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JOURNAL OF FIXED POINT THEORY AND APPLICATIONS 2016年第1期18卷 27-43页

作者： Fang, Changjie Wang, Ying Yang, Shaokang Chongqing Univ Posts & Telecommun Coll Sci Chongqing 400065 Peoples R China

In this paper, we suggest two new iterative methods for finding a common element of the solution set of a variational inequality problem and the set of fixed points of a contraction mapping in Hilbert space. We also present weak and strong convergence theorems for these new methods, provided that the fixed point mapping is a theta-strict pseudocontraction and the mapping associated with the variational inequality problem is monotone. The results presented in this paper improve and unify important recent results announced by many authors.

关键词： Variational inequality problem fixed point problem projection algorithm monotone theta-strict pseudocontraction

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Quadratically Convergent algorithms and One-Dimensional Search Schemes

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1973年第2期11卷 175-188页

作者： Huang, H. Y. Chambliss, J. P. Rice Univ Dept Mech & Aerosp Engn & Mat Sci Houston TX 77251 USA

In this paper, the performances of three quadratically convergent algorithms coupled with four one-dimensional search schemes are studied through several nonquadratic examples. The algorithms are the rank-one algorithm (algorithm I), the projection algorithm (algorithm II), and the Fletcher-Reeves algorithm (algorithm Ill). The search schemes are the exact quadratic search (EQS), the exact cubic search (ECS), the approximate quadratic search (AQS), and the approximate cubic search (ACS). The performances are analyzed in terms of number of iterations and number of equivalent function evaluations for convergence. From the numerical experiments, the following conclusions are found: (a) while the number of iterations generally increases by relaxing the search stopping condition, the number of equivalent function evaluations decreases;therefore, approximate searches should be preferred to exact searches;(b) the numbers of iterations for ACS, ECS, and EQS are about the same;therefore, the use of more sophisticated, higher order search schemes is not called for;the present ACS scheme, modified so that only the function, instead of the gradient, is used in bracketing the minimal point, could prove to be most desirable in terms of the number of equivalent function evaluations;(c) for algorithm I, ACS and AQS yield almost identical results;it is believed that further improvements in efficiency are possible if one uses a fixed stepsize approach, thus bypassing the one-dimensional search completely;(d) the combination of algorithm II and ACS exhibits high efficiency in treating functions whose order is higher than two and whose Hessian at the minimal point is singular;and (f) algorithm III, even with the best search scheme, is inefficient in treating functions with flat bottoms;it is doubtful that the simplicity of its update will compensate for its inefficiency in such pathological cases.

关键词： Function Evaluation Search Scheme projection algorithm Equivalent Function Convergent algorithm

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Applying adaptive prediction to sea-water quality measurements

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EXPERT SYSTEMS WITH APPLICATIONS 2009年第3期36卷 6773-6779页

作者： Hatzikos, E. Hatonen, J. Bassiliades, N. Vlahavas, I. Fournou, E. Aristotle Univ Thessaloniki Dept Informat Thessaloniki 54124 Greece Inst Educ Technol Dept Automat Thessaloniki Greece Univ Sheffield Dept Automat Control & Syst Engn Sheffield S10 2TN S Yorkshire England Inst Educ Technol Dept Gen Sci Thessaloniki Greece

This study explores the possibility of using adaptive filters to predict sea-water quality indicators such as water temperature, pH and dissolved oxygen based on measurements produced by an under-water measurement set-up. Two alternative adaptive approaches are tested, namely a projection algorithm and a least squares algorithm. These algorithms were chosen for comparison because they are widely used prediction algorithms. The results indicate that if the measurements remain reasonably stationary, it is possible to make one-day ahead predictions, which perform better than the prediction that the value of a certain quality variable tomorrow is going to be equal to the value today. (C) 2008 Elsevier Ltd. All rights reserved.

关键词： projection algorithm Least square algorithm One-day ahead prediction

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