Jacobian-free Newton-Krylov methods: a survey of approaches and applications

By Dana A. Knoll, David E. Keyes. 2004

2,081 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/1ac8bf57669da9afce7a19d9a09dca0d6d4a9784

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Citations in the corpus

Fast derivatives of likelihood functionals for ODE based models using adjoint-state method (2016, methodology): “Then, the preconditioning of Newton solver step during the CVODES integration of (15) is an interesting possibility to speed up the ASM (Knoll and Keyes, 2004).”
High‐order semi‐implicit time‐integrators for a triangular discontinuous Galerkin oceanic shallow water model (2009, methodology): “The reason for choosing the SI method over the fully implicit method is that in doing so we only need to contend with a linear matrix problem; for the fully implicit method, we would have to solve a nonlinear matrix problem, that while not impossible, requires many more iterations for convergence (o”
A Jacobian-free Newton-GMRES(m) method with adaptive preconditioner and its application for power flow calculations (2006, methodology): “marized in [1], these methods are synergistic combinations of”
Joint three-dimensional inversion of coupled groundwater flow and heat transfer based on automatic differentiation: sensitivity calculation, verification, and synthetic examples (2006, methodology): “This non-linear system of eqs is solved by a simple alternating fixed-point iteration (Huyakorn & Pinder 1983), or a Jacobian-free Newton‐Krylov solver ( Knoll & Keyes 2004; Kelley 2003).”
Inexact Schwarz‐algebraic multigrid preconditioners for crack problems modeled by extended finite element methods (2012, methodology): “In addition, these techniques have been found to give fast convergence when employed as preconditioners to Krylov-type solvers [34].”
The Magnetic Connection between the Convection Zone and Corona in the Quiet Sun (2007, methodology): “We follow the method outlined in Knoll & Keyes (2004) and construct a first-order Taylor expansion of J(q)v, which leads to the approximation J(q)v F(qþ v) F(q) ; ð37Þ where is a small quantity.”
The deal.II Library, Version 9.5 (2023, methodology): “The default configuration is set up to use a Jacobian-free Newton-Krylov (JFNK) approach [46], using solve_with_jacobian as linear solver.”
Recent computational developments in Krylov subspace methods for linear systems (2007, background): “2; for KKT systems or saddle point problems [33], [100]; or for integral operators [203]; considerations for parallel computers [82], [104], [178]; methods for singular matrices [134], [198], [262]; or for regularization of ill-posed problems [58], [179]; multiple right-hand side and block methods [”
A numerical method for solving the 3D unsteady incompressible Navier-Stokes equations in curvilinear domains with complex immersed boundaries (2007, methodology): “In what follows, we will briefly review the basic concepts of such methods in the context of the present problem–see [24] for a recent comprehensive review of the current state of development and application of matrix-free Newton-Krylov methods.”
MOOSE: Enabling Massively Parallel Multiphysics Simulation (2019, methodology): “Eﬃcient solution of nonlinear systems of equations using Newton’s method requires accurate computation of the Jacobian matrix or its action on a vector [5].”
What Color Is Your Jacobian? Graph Coloring for Computing Derivatives (2005, methodology): “A recent survey article by Knoll and Keyes [79] discusses various applications where the “Jacobian-free Newton–Krylov” method is used.”
Fourierization of the Legendre--Galerkin method and a new space--time spectral method (2007, methodology): “9) is a coupled nonlinear system of size (M − 1)N and should be solved by using a suitable Newton-type iterative scheme such as the preconditioned Jacobian-free Newton– Krylov method (see [11] for a review on this subject).”
Fixed-point fluid–structure interaction solvers with dynamic relaxation (2008, methodology): “However, as the interface Jacobian (37) is not easily available, a matrix free Krylov subspace solver [23] has to be applied inside Newton’s method.”
Parallel Newton-Krylov Solver for the Euler Equations Discretized Using Simultaneous-Approximation Terms (2008, background): “Approximate-Newton Phase The approximate-Newton phase uses a form of pseudotransient continuation to find the initial iterate [24,32,34].”
Finite difference methods for ordinary and partial differential equations - steady-state and time-dependent problems (2007, methodology): “These methods are also called Jacobian–free Newton–Krylov methods (JFNK), and a good survey of these methods and their history and applicability is given in the review paper of Knoll and Keyes [57].”