PDF] Steepest Descent and Conjugate Gradient Methods with Variable Preconditioning

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.

Steepest Descent Methods

Hyperbolic Pairs in the Method of Conjugate Gradients

PDF) Preconditioned steepest descent-like methods for symmetric

Conjugate Gradient Method (CG)

Steepest Descent Methods

Conjugate gradient method - Wikipedia

Inexact Preconditioned Conjugate Gradient Method with Inner-Outer

PDF) Conditional Gradient (Frank-Wolfe) Method

A Descent Four-Term Conjugate Gradient Method with Global

Search paths of the steepest descent and the conjugate gradient

Gradient descent - Wikipedia

matrices - How is the preconditioned conjugate gradient algorithm

PDF] Nonsymmetric multigrid preconditioning for conjugate gradient

Nonlinear Conjugate Gradient Methods for Unconstrained Optimization