Optimal Algorithms for Well-Conditioned Nonlinear Systems of Equations
IEEE Transactions on Computers
Inexact Newton Methods and Mixed Nonlinear Complementary Problems
NAA '00 Revised Papers from the Second International Conference on Numerical Analysis and Its Applications
Nonlinear Krylov and moving nodes in the method of lines
Journal of Computational and Applied Mathematics - Special issue on the method of lines: Dedicated to Keith Miller
A choice of forcing terms in inexact Newton method
Journal of Computational and Applied Mathematics
BiCGStab, VPAStab and an adaptation to mildly nonlinear systems
Journal of Computational and Applied Mathematics
A globally convergent Newton-GMRES method for large sparse systems of nonlinear equations
Applied Numerical Mathematics
Journal of Computational Physics
Nonlinear Krylov and moving nodes in the method of lines
Journal of Computational and Applied Mathematics - Special issue on the method of lines: Dedicated to Keith Miller
Accelerating an inexact Newton/GMRES scheme by subspace decomposition
Applied Numerical Mathematics
Low-rank update of preconditioners for the inexact Newton method with SPD Jacobian
Mathematical and Computer Modelling: An International Journal
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Classical iteration methods for linear 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 nonlinear problems can be accelerated in a similar way and how this leads to a general framework that includes many well-known techniques for solving linear and nonlinear systems, as well as new ones. Inexact Newton methods are frequently used in practice to avoid the expensive exact solution of the large linear 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 linear systems, Arnoldi and JacDav{} for linear eigenproblems, and many variants of Newton's method, like damped Newton, for general nonlinear problems. As numerical experiments suggest, the AIN{} approach may be useful for the construction of efficient schemes for solving nonlinear problems.