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Jacobian-free Newton–Krylov (JFNK) methods are synergistic combinations of Newton-type methods for superlinearly convergent solution of nonlinear equations and Krylov subspace methods for solving the Newton correction equations. The link between the two methods is the Jacobian-vector product, which may be probed approximately without forming and storing the elements of the true Jacobian, through a variety of means. Various approximations to the Jacobian matrix may still be required for preconditioning the resulting Krylov iteration. As with Krylov methods for linear problems, successful application of the \JFNK\ method to any given problem is dependent on adequate preconditioning. \JFNK\ has potential for application throughout problems governed by nonlinear partial differential equations and integro-differential equations. In this survey paper, we place \JFNK\ in context with other nonlinear solution algorithms for both boundary value problems (BVPs) and initial value problems (IVPs). We provide an overview of the mechanics of \JFNK\ and attempt to illustrate the wide variety of preconditioning options available. It is emphasized that \JFNK\ can be wrapped (as an accelerator) around another nonlinear fixed point method (interpreted as a preconditioning process, potentially with significant code reuse). The aim of this paper is not to trace fully the evolution of JFNK, nor to provide proofs of accuracy or optimal convergence for all of the constituent methods, but rather to present the reader with a perspective on how \JFNK\ may be applicable to applications of interest and to provide sources of further practical information.

@Article{ Knoll_2004aa, abstract = {Jacobian-free Newton–Krylov (JFNK) methods are synergistic combinations of Newton-type methods for superlinearly convergent solution of nonlinear equations and Krylov subspace methods for solving the Newton correction equations. The link between the two methods is the Jacobian-vector product, which may be probed approximately without forming and storing the elements of the true Jacobian, through a variety of means. Various approximations to the Jacobian matrix may still be required for preconditioning the resulting Krylov iteration. As with Krylov methods for linear problems, successful application of the \{JFNK\} method to any given problem is dependent on adequate preconditioning. \{JFNK\} has potential for application throughout problems governed by nonlinear partial differential equations and integro-differential equations. In this survey paper, we place \{JFNK\} in context with other nonlinear solution algorithms for both boundary value problems (BVPs) and initial value problems (IVPs). We provide an overview of the mechanics of \{JFNK\} and attempt to illustrate the wide variety of preconditioning options available. It is emphasized that \{JFNK\} can be wrapped (as an accelerator) around another nonlinear fixed point method (interpreted as a preconditioning process, potentially with significant code reuse). The aim of this paper is not to trace fully the evolution of JFNK, nor to provide proofs of accuracy or optimal convergence for all of the constituent methods, but rather to present the reader with a perspective on how \{JFNK\} may be applicable to applications of interest and to provide sources of further practical information. }, author = {Knoll, D.A. and Keyes, David E.}, doi = {10.1016/j.jcp.2003.08.010}, file = {Knoll_2004aa.pdf}, issn = {0021-9991}, journal = {Journal of Computational Physics}, keywords = {newton,pcg,iterative,linear,linear-systems,nonlinear}, langid = {english}, number = {2}, pages = {357--397}, title = {Jacobian-free {Newton}–{Krylov} methods: a survey of approaches and applications}, volume = {193}, year = {2004}, shortjournal = {J. Comput. Phys.} }

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