On the Use of the Conjugate Gradient Method for the Numerical Solution of First-Kind Integral Equations in Two Variables

2002 ◽  
pp. 51-56 ◽  
Author(s):  
Barbara Bertram ◽  
Haiyan Cheng
Keyword(s):  
Numerical Solution ◽  
Integral Equations ◽  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method

Construction of a Preconditioner for General Elliptic Problems Using Riesz Map

2020 ◽  
Vol 18 (01) ◽  
pp. 2050031
Author(s):  
Raghia El Hanine ◽  
Said Raghay ◽  
Hassane Sadok
Keyword(s):  
Hilbert Space ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Solution ◽  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Two Dimensional ◽  
General Elliptic ◽  
Symmetric Systems

The current work aspires to design and study the construction of an efficient preconditioner for linear symmetric systems in a Hilbert space setting. Compliantly to Josef Málek and Zdeněk Strakoš’s work [Preconditioning and the Conjugate Gradient Method in the Context of Solving[Formula: see text] PDEs, Vol. 1 (SIAM, USA).], we shed new light on the dependence of algebraic preconditioners with the resolution steps of partial differential equations (PDEs) and describe their impact on the final numerical solution. The numerical strength and efficiency of the proposed approach is demonstrated on a two-dimensional examples.

scholarly journals Riemannian Gradient Algorithm for the Numerical Solution of Linear Matrix Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Xiaomin Duan ◽  
Huafei Sun ◽  
Xinyu Zhao
Keyword(s):  
Numerical Solution ◽  
Objective Function ◽  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Geodesic Distance ◽  
Convergence Speed ◽  
Gradient Algorithm ◽  
Linear Matrix ◽  
Geodesic Curve

A Riemannian gradient algorithm based on geometric structures of a manifold consisting of all positive definite matrices is proposed to calculate the numerical solution of the linear matrix equationQ=X+∑i=1mAiTXAi. In this algorithm, the geodesic distance on the curved Riemannian manifold is taken as an objective function and the geodesic curve is treated as the convergence path. Also the optimal variable step sizes corresponding to the minimum value of the objective function are provided in order to improve the convergence speed. Furthermore, the convergence speed of the Riemannian gradient algorithm is compared with that of the traditional conjugate gradient method in two simulation examples. It is found that the convergence speed of the provided algorithm is faster than that of the conjugate gradient method.

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