Parallel Newton-Chebyshev Polynomial Preconditioners for the Conjugate\n Gradient method

2020/08/04 by Luca Bergamaschi, Bergamaschi, Luca, Ángeles Martínez +1
Computer Science · Physics and Astronomy · Mathematics · #Matrix Theory and Algorithms #Electromagnetic Scattering and Analysis #Advanced Optimization Algorithms Research

paper · pdf · doi:10.48550/arxiv.2008.01440

Abstract

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

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