Convergence conditions for a restarted GMRES method augmented with eigenspaces

This paper presents a class of new accelerated restarted GMRES method for calculating the stationary probability vector of an irreducible Markov chain. We focus on the mechanism of this new hybrid method by showing how to periodically combine the GMRES and vector extrapolation method into a much efficient one for improving the convergence rate in Markov chain problems. Numerical experiments are carried out to demonstrate the efficiency of our new algorithm on several typical Markov chain problems.

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A Restarted GMRES Method Augmented with Eigenvectors

SIAM Journal on Matrix Analysis and Applications ◽

10.1137/s0895479893253975 ◽

1995 ◽

Vol 16 (4) ◽

pp. 1154-1171 ◽

Cited By ~ 173

Author(s):

Ronald B. Morgan

Keyword(s):

Gmres Method ◽

Restarted Gmres

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Analysis of convergence of a restarted GMRES method augmented with eigenvectors

Numerical Mathematics and Advanced Applications ◽

10.1007/978-88-470-2089-4_90 ◽

2003 ◽

pp. 989-996

Author(s):

J. Zítko

Keyword(s):

Gmres Method ◽

Restarted Gmres

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A Factorization of Least-Squares Projection Schemes for Ill-Posed Problems

Computational Methods in Applied Mathematics ◽

10.1515/cmam-2019-0173 ◽

2020 ◽

Vol 20 (4) ◽

pp. 783-798

Author(s):

Shukai Du ◽

Nailin Du

Keyword(s):

Least Squares ◽

Rate Of Convergence ◽

Convergence Conditions ◽

Principal Angles ◽

Factorization Formula ◽

Ill Posed

AbstractWe give a factorization formula to least-squares projection schemes, from which new convergence conditions together with formulas estimating the rate of convergence can be derived. We prove that the convergence of the method (including the rate of convergence) can be completely determined by the principal angles between {T^{\dagger}T(X_{n})} and {T^{*}T(X_{n})}, and the principal angles between {X_{n}\cap(\mathcal{N}(T)\cap X_{n})^{\perp}} and {(\mathcal{N}(T)+X_{n})\cap\mathcal{N}(T)^{\perp}}. At the end, we consider several specific cases and examples to further illustrate our theorems.

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Centered error entropy Kalman filter with application to satellite attitude determination

Transactions of the Institute of Measurement and Control ◽

10.1177/01423312211019867 ◽

2021 ◽

pp. 014233122110198

Author(s):

Baojian Yang ◽

Lu Cao ◽

Dechao Ran ◽

Bing Xiao

Keyword(s):

Kalman Filter ◽

Gaussian Noise ◽

Attitude Determination ◽

Complexity Analysis ◽

Iteration Algorithm ◽

Robust Algorithms ◽

Convergence Conditions ◽

Satellite Attitude ◽

Heavy Tailed ◽

Non Gaussian

Due to unavoidable factors, heavy-tailed noise appears in satellite attitude estimation. Traditional Kalman filter is prone to performance degradation and even filtering divergence when facing non-Gaussian noise. The existing robust algorithms have limited accuracy. To improve the attitude determination accuracy under non-Gaussian noise, we use the centered error entropy (CEE) criterion to derive a new filter named centered error entropy Kalman filter (CEEKF). CEEKF is formed by maximizing the CEE cost function. In the CEEKF algorithm, the prior state values are transmitted the same as the classical Kalman filter, and the posterior states are calculated by the fixed-point iteration method. The CEE EKF (CEE-EKF) algorithm is also derived to improve filtering accuracy in the case of the nonlinear system. We also give the convergence conditions of the iteration algorithm and the computational complexity analysis of CEEKF. The results of the two simulation examples validate the robustness of the algorithm we presented.

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An iterative algorithm for the system of split mixed equilibrium problem

Demonstratio Mathematica ◽

10.1515/dema-2020-0027 ◽

2020 ◽

Vol 53 (1) ◽

pp. 309-324

Author(s):

Ibrahim Karahan ◽

Lateef Olakunle Jolaoso

Keyword(s):

Variational Inequality ◽

Iterative Algorithm ◽

Equilibrium Problems ◽

Nonexpansive Mappings ◽

Equilibrium System ◽

Convergence Conditions ◽

Mixed Equilibrium Problems ◽

Split Variational Inequality ◽

Mixed Equilibrium ◽

Split Mixed Equilibrium Problem

AbstractIn this article, a new problem that is called system of split mixed equilibrium problems is introduced. This problem is more general than many other equilibrium problems such as problems of system of equilibrium, system of split equilibrium, split mixed equilibrium, and system of split variational inequality. A new iterative algorithm is proposed, and it is shown that it satisfies the weak convergence conditions for nonexpansive mappings in real Hilbert spaces. Also, an application to system of split variational inequality problems and a numeric example are given to show the efficiency of the results. Finally, we compare its rate of convergence other algorithms and show that the proposed method converges faster.

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