PDF] Steepest Descent and Conjugate Gradient Methods with Variable Preconditioning
Por um escritor misterioso
Last updated 26 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
-
Steepest Descent Method Search Technique26 abril 2025
-
Steepest Descent and Newton's Method in Python, from Scratch: A Comparison, by Nicolo Cosimo Albanese26 abril 2025
-
Lecture 8: Gradient Descent (and Beyond)26 abril 2025
-
python - Steepest Descent Trace Behavior - Stack Overflow26 abril 2025
-
Curves of Steepest Descent for 3D Functions - Wolfram Demonstrations Project26 abril 2025
-
Using the Gradient Descent Algorithm in Machine Learning, by Manish Tongia26 abril 2025
-
Guide to Gradient Descent Algorithm: A Comprehensive implementation in Python - Machine Learning Space26 abril 2025
-
Chapter 4 Line Search Descent Methods Introduction to Mathematical Optimization26 abril 2025
-
nonlinear optimization - Do we need steepest descent methods, when minimizing quadratic functions? - Mathematics Stack Exchange26 abril 2025
-
PROBLEM ON STEEPEST DESCENT METHOD26 abril 2025
você pode gostar
-
CODES) all star tower defense TOP 5 hill unit26 abril 2025
-
Pokemon - Silver Tempest - Radiant Alakazam - Radiant Rare - 059/195 - NM/M26 abril 2025
-
Oxe Games on X: Avakin Life MOD APK v1.040.02(Ilimitado Dinheiro/Menu) #apk #avakin #dinheiromenu #life #mod #v104002ilimitado #jogos / X26 abril 2025
-
Super-Amigos Heróis de quadrinhos, Desenhos animados clássicos, Personagens icônicos26 abril 2025
-
Kiseijū: Sei no Kakuritsu - Anime de terror ganha trailer! - AnimeNew26 abril 2025
-
Anne Maria, Wiki26 abril 2025
-
Roblox mod apk 2023 - how to download roblox mod menu - get unlimited robux- free robux26 abril 2025
-
About: Bubble Shooter Original - Bubb (Google Play version26 abril 2025
-
Tabuleiro de xadrez - ícones de hobbies e tempo livre grátis26 abril 2025
-
Prime Video's The Lord of the Rings The Rings of Power to have 5 seasons - India Today26 abril 2025