PDF] Steepest Descent and Conjugate Gradient Methods with Variable Preconditioning
Por um escritor misterioso
Last updated 06 julho 2024
![PDF] Steepest Descent and Conjugate Gradient Methods with Variable Preconditioning](https://d3i71xaburhd42.cloudfront.net/a0174a41c7d682aeb1d7e7fa1fbd2404e037a638/11-Figure8.1-1.png)
It is shown that the CG method with variable preconditioning under this assumption may not give improvement, compared to the steepest descent (SD) method, and a new elegant geometric proof of the SD convergence rate bound is given. We analyze the conjugate gradient (CG) method with variable preconditioning for solving a linear system with a real symmetric positive definite (SPD) matrix of coefficients $A$. We assume that the preconditioner is SPD on each step, and that the condition number of the preconditioned system matrix is bounded above by a constant independent of the step number. We show that the CG method with variable preconditioning under this assumption may not give improvement, compared to the steepest descent (SD) method. We describe the basic theory of CG methods with variable preconditioning with the emphasis on “worst case” scenarios, and provide complete proofs of all facts not available in the literature. We give a new elegant geometric proof of the SD convergence rate bound. Our numerical experiments, comparing the preconditioned SD and CG methods, not only support and illustrate our theoretical findings, but also reveal two surprising and potentially practically important effects. First, we analyze variable preconditioning in the form of inner-outer iterations. In previous such tests, the unpreconditioned CG inner iterations are applied to an artificial system with some fixed preconditioner as a matrix of coefficients. We test a different scenario, where the unpreconditioned CG inner iterations solve linear systems with the original system matrix $A$. We demonstrate that the CG-SD inner-outer iterations perform as well as the CG-CG inner-outer iterations in these tests. Second, we compare the CG methods using a two-grid preconditioning with fixed and randomly chosen coarse grids, and observe that the fixed preconditioner method is twice as slow as the method with random preconditioning.
![PDF] Steepest Descent and Conjugate Gradient Methods with Variable Preconditioning](https://d3i71xaburhd42.cloudfront.net/d2a9d5bab02146a7fe3a244677432d21e33a2d98/11-Figure2-1.png)
Inexact Preconditioned Conjugate Gradient Method with Inner-Outer
![PDF] Steepest Descent and Conjugate Gradient Methods with Variable Preconditioning](https://media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-031-08720-2_3/MediaObjects/527858_1_En_3_Fig2_HTML.png)
Steepest Descent Methods
![PDF] Steepest Descent and Conjugate Gradient Methods with Variable Preconditioning](https://ars.els-cdn.com/content/image/1-s2.0-S0024379516304104-gr002.gif)
Preconditioned steepest descent-like methods for symmetric
![PDF] Steepest Descent and Conjugate Gradient Methods with Variable Preconditioning](https://i.ytimg.com/vi/9LT-i5LiK7A/maxresdefault.jpg)
Preconditioning: Gradient Descent, Conjugate Gradients and SESOP
![PDF] Steepest Descent and Conjugate Gradient Methods with Variable Preconditioning](https://i1.rgstatic.net/publication/341845895_The_steepest_descent_of_gradient-based_iterative_method_for_solving_rectangular_linear_systems_with_an_application_to_Poisson's_equation/links/5fc46cf6a6fdcc6cc6841481/largepreview.png)
PDF) The steepest descent of gradient-based iterative method for
![PDF] Steepest Descent and Conjugate Gradient Methods with Variable Preconditioning](https://media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-031-08720-2_3/MediaObjects/527858_1_En_3_Fig4_HTML.png)
Steepest Descent Methods
![PDF] Steepest Descent and Conjugate Gradient Methods with Variable Preconditioning](https://d3i71xaburhd42.cloudfront.net/6252990076647c9b14e3b1d4ff3ccdafb591a6ff/4-Figure3-1.png)
PDF] Nonsymmetric multigrid preconditioning for conjugate gradient
![PDF] Steepest Descent and Conjugate Gradient Methods with Variable Preconditioning](https://d3i71xaburhd42.cloudfront.net/58140dd819ce6d049571e35c6520d6232f5980e4/5-Figure2-1.png)
PDF] Comparison of steepest descent method and conjugate gradient
Hyperbolic Pairs in the Method of Conjugate Gradients
![PDF] Steepest Descent and Conjugate Gradient Methods with Variable Preconditioning](https://www.researchgate.net/publication/349914675/figure/fig3/AS:1012974680547328@1618523392105/Steepest-Descent-Method-e-Conjugate-Gradient-Method5-12-33-The-slope-drop-strategy.png)
Steepest Descent Method (e) Conjugate Gradient Method[5, 12, 33
![PDF] Steepest Descent and Conjugate Gradient Methods with Variable Preconditioning](https://d3i71xaburhd42.cloudfront.net/d2a9d5bab02146a7fe3a244677432d21e33a2d98/6-Figure1-1.png)
Inexact Preconditioned Conjugate Gradient Method with Inner-Outer
Conjugate gradient method - Wikipedia
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