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Register a new userOn the unique solution of the generalized absolute value equation
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Added by
Shi-Liang Wu
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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In this paper, some useful necessary and sufficient conditions for the unique solution of the generalized absolute value equation (GAVE) $Ax-B|x|=b$ with $A, Bin mathbb{R}^{ntimes n}$ from the optimization field are first presented, which cover the fundamental theorem for the unique solution of the linear system $Ax=b$ with $Ain mathbb{R}^{ntimes n}$. Not only that, some new sufficient conditions for the unique solution of the GAVE are obtained, which are weaker than the previous published works.
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In Wang et al. (J. Optim. Theory Appl., textbf{181}: 216--230, 2019), a class of effective modified Newton-tpye (MN) iteration methods are proposed for solving the generalized absolute value equations (GAVE) and it has been found that the MN iteration method involves the classical Picard iteration method as a special case. In the present paper, it will be claimed that a Douglas-Rachford splitting method for AVE is also a special case of the MN method. In addition, a class of inexact MN (IMN) iteration methods are developed to solve GAVE. Linear convergence of the IMN method is established and some specific sufficient conditions are presented for symmetric positive definite coefficient matrix. Numerical results are given to demonstrate the efficiency of the IMN iteration method.
The SOR-like iteration method for solving the absolute value equations~(AVE) of finding a vector $x$ such that $Ax - |x| - b = 0$ with $ u = |A^{-1}|_2 < 1$ is investigated. The convergence conditions of the SOR-like iteration method proposed by Ke and Ma ([{em Appl. Math. Comput.}, 311:195--202, 2017]) are revisited and a new proof is given, which exhibits some insights in determining the convergent region and the optimal iteration parameter. Along this line, the optimal parameter which minimizes $|T_ u(omega)|_2$ with $$T_ u(omega) = left(begin{array}{cc} |1-omega| & omega^2 u |1-omega| & |1-omega| +omega^2 u end{array}right)$$ and the approximate optimal parameter which minimizes $eta_{ u}(omega) =max{|1-omega|, uomega^2}$ are explored. The optimal and approximate optimal parameters are iteration-independent and the bigger value of $ u$ is, the smaller convergent region of the iteration parameter $omega$ is. Numerical results are presented to demonstrate that the SOR-like iteration method with the optimal parameter is superior to that with the approximate optimal parameter proposed by Guo, Wu and Li ([{em Appl. Math. Lett.}, 97:107--113, 2019]). In some situation, the SOR-like itration method with the optimal parameter performs better, in terms of CPU time, than the generalized Newton method (Mangasarian, [{em Optim. Lett.}, 3:101--108, 2009]) for solving the AVE.
We develop a numerical method for solving the boundary value problem of The Linear Seventh Ordinary Boundary Value Problem by using seventh degree B-Spline function. Formulation is based on particular terms of order of seventh order boundary value problem. We obtain Septic B-Spline formulation and the Collocation B-spline Method is formulated as an approximation solution. We apply the presented method to solve an example of seventh-order boundary value problem which the results show that there is an agreement between approximate solutions and exact solutions. Resulting low absolute errors show that the presented numerical method is effective for solving high order boundary value problems. Finally, a general conclusion has been included.
We consider a stabilized finite element method based on a spacetime formulation, where the equations are solved on a global (unstructured) spacetime mesh. A unique continuation problem for the wave equation is considered, where data is known in an interior subset of spacetime. For this problem, we consider a primal-dual discrete formulation of the continuum problem with the addition of stabilization terms that are designed with the goal of minimizing the numerical errors. We prove error estimates using the stability properties of the numerical scheme and a continuum observability estimate, based on the sharp geometric control condition by Bardos, Lebeau and Rauch. The order of convergence for our numerical scheme is optimal with respect to stability properties of the continuum problem and the interpolation errors of approximating with polynomial spaces. Numerical examples are provided that illustrate the methodology.
This paper proposes a computer-assisted solution existence verification method for the stationary Navier-Stokes equation over general 3D domains. The proposed method verifies that the exact solution as the fixed point of the Newton iteration exists around the approximate solution through rigorous computation and error estimation. The explicit values of quantities required by applying the fixed point theorem are obtained by utilizing newly developed quantitative error estimation for finite element solutions to boundary value problems and eigenvalue problems of the Stokes equation.
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