Shanks and Anderson-type acceleration techniques for systems of nonlinear equations

Brezinski, Claude, Cipolla, Stefano, Redivo-Zaglia, Michela and Saad, Yousef (2021) Shanks and Anderson-type acceleration techniques for systems of nonlinear equations. IMA Journal of Numerical Analysis, 42 (4), 3058-3093. (doi:10.1093/imanum/drab061).

Record type: Article

Abstract

This paper examines a number of extrapolation and acceleration methods and introduces a few modifications of the standard Shanks transformation that deal with general sequences. One of the goals of the paper is to lay out a general framework that encompasses most of the known acceleration strategies. The paper also considers the Anderson Acceleration (AA) method under a new light and exploits a connection with quasi-Newton methods in order to establish local linear convergence results of a stabilized version of the AA method. The methods are tested on a number of problems, including a few that arise from nonlinear partial differential equations.

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Published date: 25 August 2021

Additional Information: Funding Information: Labex CEMPI (ANR-11-LABX-0007-01 to C.B.); GNCS–INdAM (project ‘Efficient methods for large scale problems with applications to data analysis and preconditioning’ to S.C.); Department of Computer Science & Engineering, University of Minnesota (project no. UMF0002384 to S.C.); University of Padua (‘Numerical linear algebra and extrapolation methods with applications’, project no. DOR 1903575/19 to M.R.-Z.); National Science Foundation grant (DMS-1912048 to Y.S.). Publisher Copyright: © The Author(s) 2021.

Keywords: Anderson acceleration, extrapolation methods, Krylov subspace methods, Navier–Stokes equation, nonlinear Poisson problems, quasi-Newton methods, regularization