scholarly journals A Fast Alternating Minimization Algorithm for Nonlocal Vectorial Total Variational Multichannel Image Denoising

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Rubing Xi ◽  
Zhengming Wang ◽  
Xia Zhao ◽  
Meihua Xie
Keyword(s):  
Complexity Analysis ◽  
Iterative Algorithms ◽  
Linear Convergence ◽  
Convergence Properties ◽  
Theoretical Derivation ◽  
Variational Models ◽  
Fixed Point Algorithm ◽  
Alternating Minimization Algorithm ◽  
Uniqueness Of The Solution ◽  
Nonlocal Regularization

The variational models with nonlocal regularization offer superior image restoration quality over traditional method. But the processing speed remains a bottleneck due to the calculation quantity brought by the recent iterative algorithms. In this paper, a fast algorithm is proposed to restore the multichannel image in the presence of additive Gaussian noise by minimizing an energy function consisting of anl2-norm fidelity term and a nonlocal vectorial total variational regularization term. This algorithm is based on the variable splitting and penalty techniques in optimization. Following our previous work on the proof of the existence and the uniqueness of the solution of the model, we establish and prove the convergence properties of this algorithm, which are the finite convergence for some variables and theq-linear convergence for the rest. Experiments show that this model has a fabulous texture-preserving property in restoring color images. Both the theoretical derivation of the computation complexity analysis and the experimental results show that the proposed algorithm performs favorably in comparison to the widely used fixed point algorithm.

An Iterative algorithm for $\eta$-(anti)-Hermitian least-squares solutions of quaternion matrix equations

2015 ◽  
Vol 30 ◽  
Author(s):  
Fatemeh Beik ◽  
Salman Ahmadi-Asl
Keyword(s):  
Least Squares ◽  
Iterative Algorithm ◽  
Numerical Experiments ◽  
Iterative Algorithms ◽  
Iterative Approach ◽  
Matrix Equations ◽  
Quaternion Matrix ◽  
Hermitian Matrices ◽  
Convergence Properties ◽  
Least Squares Problems

Recently, some research has been devoted to finding the explicit forms of the η-Hermitian and η-anti-Hermitian solutions of several kinds of quaternion matrix equations and their associated least-squares problems in the literature. Although exploiting iterative algorithms is superior than utilizing the explicit forms in application, hitherto, an iterative approach has not been offered for finding η-(anti)-Hermitian solutions of quaternion matrix equations. The current paper deals with applying an efficient iterative manner for determining η-Hermitian and η-anti-Hermitian least-squares solutions corresponding to the quaternion matrix equation AXB + CY D = E. More precisely, first, this paper establishes some properties of the η-Hermitian and η-anti-Hermitian matrices. These properties allow for the demonstration of how the well-known conjugate gradient least- squares (CGLS) method can be developed for solving the mentioned problem over the η-Hermitian and η-anti-Hermitian matrices. In addition, the convergence properties of the proposed algorithm are discussed with details. In the circumstance that the coefficient matrices are ill-conditioned, it is suggested to use a preconditioner for accelerating the convergence behavior of the algorithm. Numerical experiments are reported to reveal the validity of the elaborated results and feasibility of the proposed iterative algorithm and its preconditioned version.

Nonlinear Multiparameter Eigenvalue Problems in Aeroelasticity

2019 ◽  
Vol 19 (05) ◽  
pp. 1941008 ◽  
Author(s):  
Arion Pons ◽  
Stefanie Gutschmidt
Keyword(s):  
Significant Influence ◽  
Eigenvalue Problems ◽  
Iterative Algorithms ◽  
System Behavior ◽  
Convergence Properties ◽  
Computational Costs ◽  
Solution Algorithms ◽  
Aeroelastic Flutter ◽  
Restriction Method ◽  
Definition Of

In this work, we devise solution algorithms for nonlinear multiparameter eigenvalue problems arising in the analysis of aeroelastic flutter. Two iterative algorithms are devised, as well as a restriction method for simplifying the system behavior away from the desired flutter points. The algorithms are tested on a sectional model and on the Goland wing. Both are found to have fast and reliable convergence properties, yielding flutter point solutions which are validated by the literature. The definition of the eigenvalues is found to have a significant influence on the convergence properties of the algorithm; preferable choices for eigenvalue definitions are noted. The computational costs of the algorithms are tested and discussed; they are found to be favorable relative to other approaches from the literature. Opportunities for extending these methods are also tested and discussed.

scholarly journals Ball Convergence of a Parametric Efficient Family of Iterative Methods for Solving Nonlinear Equations

Foundations ◽  
2021 ◽  
Vol 1 (1) ◽  
pp. 23-31
Author(s):  
Samundra Regmi ◽  
Christopher I. Argyros ◽  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Linear Convergence ◽  
Error Tolerance ◽  
First Derivative ◽  
Uniqueness Of The Solution ◽  
Ball Convergence ◽  
Space Case ◽  
Local Results ◽  
Lipschitz Conditions ◽  
Theoretical Results ◽  
General Banach Space

The goal is to extend the applicability of Newton-Traub-like methods in cases not covered in earlier articles requiring the usage of derivatives up to order seven that do not appear in the methods. The price we pay by using conditions on the first derivative that actually appear in the method is that we show only linear convergence. To find the convergence order is not our intention, however, since this is already known in the case where the spaces coincide with the multidimensional Euclidean space. Note that the order is rediscovered by using ACOC or COC, which require only the first derivative. Moreover, in earlier studies using Taylor series, no computable error distances were available based on generalized Lipschitz conditions. Therefore, we do not know, for example, in advance, how many iterates are needed to achieve a predetermined error tolerance. Furthermore, no uniqueness of the solution results is available in the aforementioned studies, but we also provide such results. Our technique can be used to extend the applicability of other methods in an analogous way, since it is so general. Finally note that local results of this type are important, since they demonstrate the difficulty in choosing initial points. Our approach also extends the applicability of this family of methods from the multi-dimensional Euclidean to the more general Banach space case. Numerical examples complement the theoretical results.

scholarly journals An Extremal Problem with Applications to Renewable Energy Production

2021 ◽  
Author(s):  
Thomas Ashley ◽  
Emilio Carrizosa ◽  
Enrique Fernández-Cara
Keyword(s):  
Energy Production ◽  
Iterative Algorithms ◽  
Optimal Solution ◽  
Engineering Problem ◽  
Convergence Properties ◽  
Numerical Illustration ◽  
Dynamic Optimisation ◽  
Solar Power Tower ◽  
Future Work ◽  
Realistic Data

Dynamic optimisation provides complex challenges for optimal solution, but greatly in- creases applicability when considering time dependent situations. In this work, a constrained dynamic optimisation problem is analysed and subsequently applied to the resolution of a real-world engineering problem concerning Solar Power Tower plants. We study the ex- istence of solutions and deduce an appropriate optimality characterisation in this applied framework. Two iterative algorithms are presented, convergence properties are discussed and a numerical illustration is given utilising realistic data. Finally, conclusions are drawn on the considered model and some ideas for future work are discussed.

scholarly journals Weak, Strong, and Linear Convergence of a Double-Layer Fixed Point Algorithm

2017 ◽  
Vol 27 (3) ◽  
pp. 1431-1458 ◽  
Author(s):  
Victor I. Kolobov ◽  
Simeon Reich ◽  
Rafał Zalas
Keyword(s):  
Fixed Point ◽  
Double Layer ◽  
Linear Convergence ◽  
Fixed Point Algorithm

scholarly journals Local Convergence and Dynamical Analysis of a Third and Fourth Order Class of Equation Solvers

2021 ◽  
Vol 5 (2) ◽  
pp. 27
Author(s):  
Debasis Sharma ◽  
Ioannis K. Argyros ◽  
Sanjaya Kumar Parhi ◽  
Shanta Kumari Sunanda
Keyword(s):  
Convergence Analysis ◽  
Banach Spaces ◽  
Fourth Order ◽  
Local Convergence ◽  
Taylor Expansion ◽  
Iterative Algorithms ◽  
Continuity Condition ◽  
Dynamical Analysis ◽  
Local Analysis ◽  
Uniqueness Of The Solution

In this article, we suggest the local analysis of a uni-parametric third and fourth order class of iterative algorithms for addressing nonlinear equations in Banach spaces. The proposed local convergence is established using an ω-continuity condition on the first Fréchet derivative. In this way, the utility of the discussed schemes is extended and the application of Taylor expansion in convergence analysis is removed. Furthermore, this study provides radii of convergence balls and the uniqueness of the solution along with the calculable error distances. The dynamical analysis of the discussed family is also presented. Finally, we provide numerical explanations that show the suggested analysis performs well in the situation where the earlier approach cannot be implemented.

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