Parallel Newton-Chebyshev Polynomial Preconditioners for the Conjugate Gradient method

TL;DR

This paper introduces a parallel polynomial preconditioning technique combining Newton methods and Chebyshev polynomials to efficiently solve large symmetric positive definite systems, demonstrating significant scalability and convergence improvements.

Contribution

It presents a novel parallel preconditioner based on Newton-Chebyshev polynomials that enhances convergence speed for large-scale systems.

Findings

Effective for matrices with up to 8 billion unknowns

Avoids eigenvalue clustering to speed convergence

Demonstrates efficiency in parallel computing environments

Abstract

In this note we exploit polynomial preconditioners for the Conjugate Gradient method to solve large symmetric positive definite linear systems in a parallel environment. We put in connection a specialized Newton method to solve the matrix equation X^{-1} = A and the Chebyshev polynomials for preconditioning. We propose a simple modification of one parameter which avoids clustering of extremal eigenvalues in order to speed-up convergence. We provide results on very large matrices (up to 8 billion unknowns) in a parallel environment showing the efficiency of the proposed class of preconditioners.