Accelerated Inexact Newton Schemes for Large Systems of Nonlinear Equations

Abstract

Classical iteration methods for systems, such as Jacobi iteration, can be accelerated considerably by Krylov subspace methods like GMRES @. In this paper, we describe how inexact Newton methods for problems can be accelerated in a similar way and how this leads to a general framework that includes many well-known techniques for solving and systems, as well as new ones. Inexact Newton methods are frequently used in practice to avoid the expensive exact solution of the large system arising in the (possibly also inexact) linearization step of Newton's process. Our framework includes acceleration techniques for the linear steps as well as for the nonlinear steps in Newton's process. The described class of methods, the accelerated inexact Newton (AIN) methods, contains methods like GMRES and GMRESR for systems, Arnoldi and JacDav{} for eigenproblems, and many variants of Newton's method, like damped Newton, for general problems. As numerical experiments suggest, the AIN{} approach may be useful for the construction of efficient schemes for solving problems.