PDF] Steepest Descent and Conjugate Gradient Methods with Variable Preconditioning
Por um escritor misterioso
Last updated 04 abril 2025
![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://miro.medium.com/v2/resize:fit:1400/1*-XzvdqP1LApHMRypbxhazA.png)
Descent carefully on a gradient!. In machine learning, gradient
![PDF] Steepest Descent and Conjugate Gradient Methods with Variable Preconditioning](https://d3i71xaburhd42.cloudfront.net/58140dd819ce6d049571e35c6520d6232f5980e4/3-Figure1-1.png)
PDF] Comparison of steepest descent method and conjugate gradient
![PDF] Steepest Descent and Conjugate Gradient Methods with Variable Preconditioning](https://www.researchgate.net/publication/2854011/figure/fig1/AS:669564601917445@1536648049571/The-preconditioned-conjugate-gradient-method.png)
1: The preconditioned conjugate gradient method
![PDF] Steepest Descent and Conjugate Gradient Methods with Variable Preconditioning](https://indrag49.github.io/Numerical-Optimization/bookdownproj_files/figure-html/unnamed-chunk-10-1.png)
Chapter 5 Conjugate Gradient Methods
![PDF] Steepest Descent and Conjugate Gradient Methods with Variable Preconditioning](https://ars.els-cdn.com/content/image/1-s2.0-S0024379516304104-gr001.jpg)
Preconditioned steepest descent-like methods for symmetric
![PDF] Steepest Descent and Conjugate Gradient Methods with Variable Preconditioning](https://static.hindawi.com/articles/mpe/volume-2021/6219062/figures/6219062.fig.006.jpg)
A Descent Four-Term Conjugate Gradient Method with Global
![PDF] Steepest Descent and Conjugate Gradient Methods with Variable Preconditioning](https://media.springernature.com/m685/springer-static/image/art%3A10.1186%2Fs13660-019-2192-6/MediaObjects/13660_2019_2192_Fig3_HTML.png)
A conjugate gradient algorithm and its application in large-scale
![PDF] Steepest Descent and Conjugate Gradient Methods with Variable Preconditioning](https://media.springernature.com/m685/springer-static/image/art%3A10.1007%2Fs11590-022-01867-9/MediaObjects/11590_2022_1867_Figa_HTML.png)
A link between the steepest descent method and fixed-point
![PDF] Steepest Descent and Conjugate Gradient Methods with Variable Preconditioning](https://i1.rgstatic.net/publication/294118573_Conjugate_gradient_and_steepest_descent_approach_on_quasi-Newton_search_direction/links/56f6920d08ae38d710a1bc14/largepreview.png)
PDF) Conjugate gradient and steepest descent approach on quasi
![PDF] Steepest Descent and Conjugate Gradient Methods with Variable Preconditioning](https://0.academia-photos.com/attachment_thumbnails/59852465/mini_magick20190624-19559-1t5imq9.png?1561401861)
PDF) A New Extended PR Conjugate Gradient Method for Solving
Recomendado para você
-
Descent method — Steepest descent and conjugate gradient, by Sophia Yang, Ph.D.04 abril 2025
-
Steepest Descent Method04 abril 2025
-
Descent method — Steepest descent and conjugate gradient in Python, by Sophia Yang, Ph.D.04 abril 2025
-
The Steepest-Descent Method - ppt video online download04 abril 2025
-
Conjugate gradient methods - Cornell University Computational Optimization Open Textbook - Optimization Wiki04 abril 2025
-
Gradient Descent in Machine Learning: Optimized Algorithm04 abril 2025
-
example of steepest Descent (left) and Conjugate Gradient (right)04 abril 2025
-
Solved The steepest descent method for minimize f(x) is the04 abril 2025
-
Steepest Descent - an overview04 abril 2025
-
Visualizing the gradient descent method04 abril 2025
você pode gostar
-
Is Honkai Star Rail coming to PS4?04 abril 2025
-
Mini-P, Five Nights at Freddy's Wiki04 abril 2025
-
Brenna Collins - Content Creation Manager - Playeasy.com04 abril 2025
-
Dancing Queens (2021) - IMDb04 abril 2025
-
How to change your shift in king of legacy|TikTok Search04 abril 2025
-
Elijah Wood and WoTC react to LoTR x Magic: The Gathering set and interpretation - Xfire04 abril 2025
-
Mortal Kombat 11 Review • Codec Moments04 abril 2025
-
Esposa do motorista de app morto com capitão-tenente italiano soube do acidente ao rastrear moto - Segurança - Diário do Nordeste04 abril 2025
-
How to Get Made in Heaven (Obtaining mih + Showcase)04 abril 2025
-
Seirei Gensouki: Spirit Chronicles Fated Reunion - Watch on04 abril 2025