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iteration algorithms for solving a system of fuzzy linear equations

FUZZY SETS AND SYSTEMS 2001年第1期119卷 121-128页

作者： Wang, XZ Zhong, ZM Ha, MH Hebei Univ Dept Math Baoding 071002 Hebei Peoples R China Logging Corp Liaohe Oilfield Ctr Data Explanat Liaoning Peoples R China

In this paper, we discuss the solution of a system of fuzzy linear equations, X = AX + U, and its iteration algorithms where A is a real n x n matrix, the unknown vector X and the constant U are all vectors consisting of n fuzzy numbers, and the addition, scale-multiplication are defined by Zadeh's extension principle. After introducing a metric between two fuzzy vectors, we prove that the system has unique solution if \\A\\(infinity) < 1 We also give the convergence and the error estimation for using simple iteration to obtain the solution. Finally, we give the convergence and the error estimation of successive iteration sequence for obtaining the solution. (C) 2001 Elsevier Science B.V. All rights reserved.

关键词： algebra numerical algebra fuzzy numbers systems of fuzzy linear equations iteration algorithms

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Fixed point approximation of multi-valued non-expansive mappings in uniformly convex Banach spaces via AR-iteration

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MODERN PHYSICS LETTERS B 2025年第13期39卷

作者： Nawaz, Sundas Rafique, Khadija Batool, Afshan Mahmood, Zafar Muhammad, Taseer Fatima Jinnah Women Univ Dept Math Sci Rawalpindi Pakistan Hazara Univ Dept Math & Stat Mansehra Pakistan King Khalid Univ Coll Sci Dept Math Abha Saudi Arabia

The aim of this paper is to introduce a novel approach for estimating the fixed points of multi-valued non-expansive mappings using the AR-iteration scheme in the context of uniformly convex Banach spaces. We establish strong and weak convergence results, providing rigorous analytical proofs and illustrating the results with a detailed example. Our approach showcases the potential of the AR-iteration scheme in solving real-world problems, particularly in addressing two-point boundary value problems. We demonstrate the applicability and effectiveness of the AR-iteration scheme in numerical analysis and computational mathematics. Our results contribute significantly to the advancement of numerical methods in solving boundary value problems, offering new insights and directions for future research in this area. Furthermore, we provide a detailed explanation of Green's function approach and its implications for various scientific and engineering applications, paving the way for future research in this area.

关键词： iteration algorithms multi-valued mapping AR-iteration non-expansive mapping boundary value problems Suzuki non-expansive mapping

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Multi-metric learning by a pair of twin-metric learning framework

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APPLIED INTELLIGENCE 2022年第15期52卷 17490-17507页

作者： Zhang, Min Yang, Liming Yuan, Chao Ren, Qiangqiang China Agr Univ Coll Sci Beijing Peoples R China China Agr Univ Coll Informat & Elect Engn Beijing Peoples R China

Multi-metric learning is important for improving classification performance since learning a single metric is usually insufficient for complex data. The existing multi-metric learning methods are based on the triplet constraints, and thus are with high computing complexity. In this work, we propose an efficient multi-metric learning framework by a pair of two-metric learning schemes (called TMML) to jointly train two local metrics and a global metric, where the distances between samples are automatically adjusted to maximize classification margin. Instead of the triplet constraints, the proposed TMML is based on the pair constraints to reduce the computational burden. Moreover, a global regularization is introduced to improve generalization and control overfitting. The proposed TMML improves the limitation of a single metric, where a pair of local metrics are interrelated to conduct adaptation for the local characteristics, while global metrics are to depict the common properties from all the data. Furthermore, we develop an alternating direction iterative algorithm to optimize the proposed TMML. The convergence of the algorithm is analyzed theoretically. Numerical experiments are carried out on different scale datasets. Under different evaluation criteria, experiments show that the proposed TMML is superior to the single metric learning methods, and achieves better performance than other state-of-the-art multi-metric learning methods in most cases.

关键词： Multi-metric learning Two-metric learning Pair constraints Classification iteration algorithms

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A fast image reconstruction algorithm based on penalized-likelihood estimate

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MEDICAL ENGINEERING & PHYSICS 2005年第8期27卷 679-686页

作者： Sheng, JH Ying, L Rush Univ Dept Med Phys Chicago IL 60607 USA Univ Wisconsin Dept Elect Engn & Comp Sci Milwaukee WI 53201 USA

Statistical iterative methods for image reconstruction like maximum likelihood expectation maximization (ML-EM) are more robust and flexible than analytical inversion methods and allow for accurately modeling the counting statistics and the photon transport during acquisition. They are rapidly becoming the standard for image reconstruction in emission computed tomography. The maximum likelihood approach provides images with superior noise characteristics compared to the conventional filtered back projection algorithm. But a major drawback of the statistical iterative image reconstruction is its high computational cost. In this paper, a fast algorithm is proposed as a modified OS-EM (MOS-EM) using a penalized function, which is applied to the least squares merit function to accelerate image reconstruction and to achieve better convergence. The experimental results show that the algorithm can provide high quality reconstructed images with a small number of iterations. (c) 2005 IPEM. Published by Elsevier Ltd. All rights reserved.

关键词： expectation maximization maximum likelihood image reconstruction iteration algorithms ordered subset penalized function

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Analysis for step-size optimisation on MPPT algorithm for photovoltaic systems

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IET POWER ELECTRONICS 2017年第13期10卷 1647-1654页

作者： Tang, Lei Xu, Wei Mu, Chaoxu Xi An Jiao Tong Univ Sch Elect Engn Xian 710049 Shaanxi Peoples R China Huazhong Univ Sci & Technol Sch Elect & Elect Engn State Key Lab Adv Electromagnet Engn & Technol Wuhan 430074 Hubei Peoples R China Tianjin Univ Sch Elect & Informat Engn Tianjin 300072 Peoples R China

As the output characteristic of photovoltaic arrays is a non-linear function of external environment, the step-size of maximum power point tracking (MPPT) algorithm should be regulated dynamically to improve the MPPT efficiency. However, the step-size, essentially determining the performance of MPPT algorithm, is often designed for a particular system with certain working conditions. To solve this problem, an optimisation algorithm is proposed in this study, which can adjust the step-size automatically. From the essence of optimisation problem, the tracking process can be converted into a maximum sequence which can be solved by several iteration algorithms. This study provides a simple and effective way to obtain the step-size. Once the iteration method is selected, the performance of the MPPT algorithm has been decided. The analysis of convergence, convergence order and stability are presented in this study. Comprehensive simulation and experimental results demonstrate that this proposed algorithm has a good stable and dynamic characteristic.

关键词： photovoltaic power systems maximum power point trackers optimisation iterative methods convergence photovoltaic systems convergence order iteration algorithms step-size optimisation MPPT algorithm maximum power point tracking algorithm photovoltaic arrays

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An improved maximum likelihood approach to image reconstruction using ordered subsets and data subdivisions

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IEEE TRANSACTIONS ON NUCLEAR SCIENCE 2004年第1期51卷 130-135页

作者： Sheng, JH Liu, DR Univ Illinois Dept Elect & Comp Engn Chicago IL 60607 USA

Iterative algorithms such as maximum likelihood expectation maximization (ML-EM) algorithm are rapidly becoming the standard for image reconstruction in emission computed tomography. The maximum likelihood approach provides images with superior noise characteristics compared to conventional filtered backprojection algorithm. A major drawback of the iterative image reconstruction methods is their high computational cost. In this paper, we develop a new algorithm called the improved ordered subset expectation maximization (IOS-EM) algorithm. This algorithm modifies the number of projections in each subset and the step size (i.e., the relaxation factor) for each iteration in order to recover various frequency components in early iteration steps. In the method presented in this paper, the number of projections in a subset increases and the step size decreases after each iteration. In addition, pixel data are grouped into subdivisions to accelerate image reconstruction. Experimental results show that the IOS-EM algorithm can provide high quality reconstructed images at a small number of iterations.

关键词： expectation maximization filtered backprojection image reconstruction iteration algorithms maximum likelihood ordered subset

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A robust alternative to examine data dependency of fixed points of quasi-contractive operators: an efficient approach that relies on the collage theorem

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COMPUTATIONAL & APPLIED MATHEMATICS 2024年第4期43卷 1-29页

作者： Gursoy, Faik Adiyaman Univ Dept Math TR-02040 Adiyaman Turkiye

Usurelu et al. (Int J Comput Math 98:1049-1068, 2021) presented stability and data dependence results for a TTP (Thakur-Thakur-Postolache) iteration algorithm associated with quasi-strictly contractive mappings and contraction mappings, respectively, but these results were subject to strong conditions on the parametric control sequences used in the TTP iteration algorithm. This article aims to expand those results conducting a thorough analysis of the convergence of TTP and S iteration algorithms and improve those results by removing the restrictions on the parametric control sequences. Additionally, a data dependence result for the TTP iteration algorithm of quasi-strictly contractive mappings is established and several collage theorems are introduced to offer new insights into the data dependence of fixed points of quasi-strictly contractive mappings and to solve related inverse problems. In order to exhibit the dependability and effectiveness of all the results discussed in this work, a multitude of numerical examples are furnished, encompassing both linear and nonlinear differential equations (DEs) and partial differential equations (PDEs). This work can be viewed as an important refinement and complement to the study by Usurelu et al. (Int J Comput Math 98:1049-1068, 2021).

关键词： iteration algorithms Quasi-strictly contractive mappings Data dependency Collage theorems

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Strong convergence theorems for a common zero of a finite family of H-accretive operators in Banach space

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SPRINGERPLUS 2016年第1期5卷 1-10页

作者： He, Huimin Liu, Sanyang Chen, Rudong Xidian Univ Sch Math & Stat Xian 710071 Peoples R China Tianjin Polytech Univ Dept Math Tianjin 300160 Peoples R China

The aim of this paper is to study a finite family of H-accretive operators and prove common zero point theorems of them in Banach space. The results presented in this paper extend and improve the corresponding results of Zegeye and Shahzad (Nonlinear Anal 66: 1161-1169, 2007), Liu and He (J Math Anal Appl 385: 466-476, 2012) and the related results.

关键词： H-accretive operators Resolvent operator iteration algorithms Strong convergence

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Image definition evaluation functions for X-ray crystallography: a new perspective on the phase problem

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ACTA CRYSTALLOGRAPHICA A-FOUNDATION AND ADVANCES 2015年第5期71卷 526-533页

作者： Li, Hui He, Meng Zhang, Ze Beijing Univ Technol Beijing 100124 Peoples R China Natl Ctr Nanosci & Technol CAS Key Lab Nanosyst & Hierarch Fabricat Beijing 100190 Peoples R China Zhejiang Univ Hangzhou 310014 Zhejiang Peoples R China

The core theme of X-ray crystallography is reconstructing the electron-density distribution of crystals under the constraints of observed diffraction data. Nevertheless, reconstruction of the electron-density distribution by straightforward Fourier synthesis is usually hindered due to the well known phase problem and the finite resolution of diffraction data. In analogy with optical imaging systems, the reconstructed electron-density map may be regarded as the image of the real electron-density distribution in crystals. Inspired by image definition evaluation functions applied in the auto-focusing process, two evaluation functions are proposed for the reconstructed electron-density images. One of them is based on the atomicity of the electron-density distribution and properties of Fourier synthesis. Tests were performed on synthetic data of known structures, and it was found that this evaluation function can distinguish the correctly reconstructed electron-density image from wrong ones when diffraction data of atomic resolution are available. An algorithm was established based on this evaluation function and applied in reconstructing the electron-density image from the synthetic data of known structures. The other evaluation function, which is based on the positivity of electron density and constrained power spectrum entropy maximization, was designed for cases where only diffraction data of rather limited resolution are available. Tests on the synthetic data indicate that this evaluation function may identify the correct phase set even for a data set with resolution as low as 3.5 angstrom. Though no algorithm for structure solution has been figured out based on the latter function, the results presented here provide a new perspective on the phase problem.

关键词： phase problem image definition evaluation functions power spectrum entropy iteration algorithms charge-density maps

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Resolvent Methods for Solving System of General Variational Inclusions

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2011年第2期148卷 422-430页

作者： Noor, M. A. Noor, K. I. King Saud Univ Coll Sci Dept Math Riyadh 11451 Saudi Arabia COMSATS Inst Informat Technol Dept Math Islamabad Pakistan

In this paper, we introduce and consider some new systems of extended general variational inclusions involving seven different operators. Using the resolvent operator technique, we show that the new systems of extended general variational inclusions are equivalent to the fixed point problems. This equivalent formulation is used to suggest and analyze some new iterative methods for this system of extended general variational inclusions. We also study the convergence analysis of the new iterative method under certain mild conditions. Several special cases are also discussed. Results obtained in this paper can be viewed as pure mathematical contribution to variational analysis.

关键词： Variational inclusions iteration algorithms System of general variational inclusions Convergence criteria Hilbert spaces

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