ACCELERATED INEXACT NEWTON SCHEMES FOR LARGE SYSTEMS OF NONLINEAR EQUATIONS

Classical iteration methods for linear systems, such as Jacobi iterati on, can be accelerated considerably by Krylov subspace methods like GM RES. In this paper, we describe how inexact Newton methods for nonline ar problems can be accelerated in a similar way and how this leads to a general framework that includes many well-known techniques for solvi ng linear and nonlinear systems, as well as new ones. Inexact Newton m ethods are frequently used in practice to avoid the expensive exact so lution of the large linear system arising in the (possibly also inexac t) linearization step of Newton's process. Our framework includes acce leration techniques for the ''linear steps'' as well as for the ''nonl inear steps'' in Newton's process. The described class of methods, the accelerated inexact Newton (AIN) methods, contains methods like GMRES and GMRESR for linear systems, Arnoldi and Jacobi-Davidson for linear eigenproblems, and many variants of Newton's method, like damped Newt on, for general nonlinear problems. As numerical experiments suggest, the AIN approach may be useful for the construction of efficient schem es for solving nonlinear problems.