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Jacobianfree NewtonKrylov methods: a survey of approaches and applications
 J. Comput. Phys
"... Jacobianfree NewtonKrylov (JFNK) methods are synergistic combinations of Newtontype 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 Jacobianvector product, which ..."
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Cited by 192 (6 self)
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Jacobianfree NewtonKrylov (JFNK) methods are synergistic combinations of Newtontype 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 Jacobianvector 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 dierential equations and integrodierential equations. In this survey article 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 xed point method (interpreted as a preconditioning process, potentially with signicant code reuse). The aim of this article 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 problems of physical interest and to provide sources of further practical information. A review paper solicited by the EditorinChief of the Journal of Computational
Nonlinearly preconditioned inexact Newton algorithms
 SIAM J. Sci. Comput
, 2000
"... Abstract. Inexact Newton algorithms are commonlyused for solving large sparse nonlinear system of equations F (u ∗ ) = 0 arising, for example, from the discretization of partial differential equations. Even with global strategies such as linesearch or trust region, the methods often stagnate at loc ..."
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Cited by 53 (18 self)
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Abstract. Inexact Newton algorithms are commonlyused for solving large sparse nonlinear system of equations F (u ∗ ) = 0 arising, for example, from the discretization of partial differential equations. Even with global strategies such as linesearch or trust region, the methods often stagnate at local minima of �F �, especiallyfor problems with unbalanced nonlinearities, because the methods do not have builtin machineryto deal with the unbalanced nonlinearities. To find the same solution u ∗ , one maywant to solve instead an equivalent nonlinearlypreconditioned system F(u ∗ ) = 0 whose nonlinearities are more balanced. In this paper, we propose and studya nonlinear additive Schwarzbased parallel nonlinear preconditioner and show numericallythat the new method converges well even for some difficult problems, such as high Reynolds number flows, where a traditional inexact Newton method fails. Key words. nonlinear preconditioning, inexact Newton methods, Krylov subspace methods, nonlinear additive Schwarz, domain decomposition, nonlinear equations, parallel computing, incompressible
Parallel NewtonKrylovSchwarz Algorithms For The Transonic Full Potential Equation
, 1998
"... We study parallel twolevel overlapping Schwarz algorithms for solving nonlinear finite element problems, in particular, for the full potential equation of aerodynamics discretized in two dimensions with bilinear elements. The overall algorithm, NewtonKrylovSchwarz (NKS), employs an inexact finite ..."
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Cited by 51 (32 self)
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We study parallel twolevel overlapping Schwarz algorithms for solving nonlinear finite element problems, in particular, for the full potential equation of aerodynamics discretized in two dimensions with bilinear elements. The overall algorithm, NewtonKrylovSchwarz (NKS), employs an inexact finitedifference Newton method and a Krylov space iterative method, with a twolevel overlapping Schwarz method as a preconditioner. We demonstrate that NKS, combined with a density upwinding continuation strategy for problems with weak shocks, is robust and economical for this class of mixed elliptichyperbolic nonlinear partial differential equations, with proper specification of several parameters. We study upwinding parameters, inner convergence tolerance, coarse grid density, subdomain overlap, and the level of fillin in the incomplete factorization, and report their effect on numerical convergence rate, overall execution time, and parallel efficiency on a distributedmemory parallel computer.
A quasiNewton algorithm based on a reduced model for fluidstructure interaction problems in blood flows
 M2AN Math. Model. Numer. Anal
"... Abstract. We propose a quasiNewton algorithm for solving fluidstructure interaction problems. The basic idea of the method is to build an approximate tangent operator which is cost effective and which takes into account the socalled added mass effect. Various test cases show that the method allow ..."
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Cited by 46 (6 self)
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Abstract. We propose a quasiNewton algorithm for solving fluidstructure interaction problems. The basic idea of the method is to build an approximate tangent operator which is cost effective and which takes into account the socalled added mass effect. Various test cases show that the method allows a significant reduction of the computational effort compared to relaxed fixed point algorithms. We present 2D and 3D fluidstructure simulations performed either with a simple 1D structure model or with shells in large displacements. Mathematics Subject Classification. 65M60, 74K25, 76D05, 76Z05. 1.
Analysis of Inexact TrustRegion SQP Algorithms
 RICE UNIVERSITY, DEPARTMENT OF
, 2000
"... In this paper we extend the design of a class of compositestep trustregion SQP methods and their global convergence analysis to allow inexact problem information. The inexact problem information can result from iterative linear systems solves within the trustregion SQP method or from approximatio ..."
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Cited by 26 (2 self)
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In this paper we extend the design of a class of compositestep trustregion SQP methods and their global convergence analysis to allow inexact problem information. The inexact problem information can result from iterative linear systems solves within the trustregion SQP method or from approximations of firstorder derivatives. Accuracy requirements in our trustregion SQP methods are adjusted based on feasibility and optimality of the iterates. Our accuracy requirements are stated in general terms, but we show how they can be enforced using information that is already available in matrixfree implementations of SQP methods. In the absence of inexactness our global convergence theory is equal to that of Dennis, ElAlem, Maciel (SIAM J. Optim., 7 (1997), pp. 177207). If all iterates are feasible, i.e., if all iterates satisfy the equality constraints, then our results are related to the known convergence analyses for trustregion methods with inexact gradient information fo...
Spectral residual method without gradient information for solving largescale nonlinear systems: Theory and experiments
, 2004
"... Abstract. A fully derivativefree spectral residual method for solving largescale nonlinear systems of equations is presented. It uses in a systematic way the residual vector as a search direction, a spectral steplength that produces a nonmonotone process and a globalization strategy that allows for ..."
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Cited by 23 (5 self)
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Abstract. A fully derivativefree spectral residual method for solving largescale nonlinear systems of equations is presented. It uses in a systematic way the residual vector as a search direction, a spectral steplength that produces a nonmonotone process and a globalization strategy that allows for this nonmonotone behavior. The global convergence analysis of the combined scheme is presented. An extensive set of numerical experiments that indicate that the new combination is competitive and frequently better than wellknown NewtonKrylov methods for largescale problems is also presented. 1.
Two Classes of Multisecant Methods for Nonlinear Acceleration ∗
, 2007
"... Many applications in science and engineering lead to models which require solving largescale fixed point problems, or equivalently, systems of nonlinear equations. Several successful techniques for handling such problems are based on quasiNewton methods that implicitly update the approximate Jacob ..."
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Cited by 21 (0 self)
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Many applications in science and engineering lead to models which require solving largescale fixed point problems, or equivalently, systems of nonlinear equations. Several successful techniques for handling such problems are based on quasiNewton methods that implicitly update the approximate Jacobian or inverse Jacobian to satisfy a certain secant condition. We present two classes of multisecant methods which allows to take into account a variable number of secant equations at each iteration. The first is the Broydenlike class, of which Broyden’s family is a subclass, and Anderson mixing is a particular member. The second class is that of the nonlinear EirolaNevanlinnatype methods. This work was motivated by a problem in electronic structure calculations, whereby a fixed point iteration, known as the selfconsistent field (SCF) iteration, is accelerated by various strategies termed ‘mixing’. 1
Globalization Techniques for NewtonKrylov Methods and Applications to the FullyCoupled SOLUTION OF THE NAVIERâSTOKES EQUATIONS
"... A NewtonâKrylov method is an implementation of Newton's method in which a Krylov subspace method is used to solve approximately the linear subproblems that determine Newton steps. To enhance robustness when good initial approximate solutions are not available, these methods are usually glob ..."
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Cited by 17 (5 self)
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A NewtonâKrylov method is an implementation of Newton's method in which a Krylov subspace method is used to solve approximately the linear subproblems that determine Newton steps. To enhance robustness when good initial approximate solutions are not available, these methods are usually globalized, i.e., augmented with auxiliary procedures (globalizations) that improve the likelihood of convergence from a starting point that is not near a solution. In recent years, globalized NewtonKrylov methods have been used increasingly for the fully coupled solution of largescale problems. In this paper, we review several representative globalizations, discuss their properties, and report on a numerical study aimed at evaluating their relative merits on largescale two and threedimensional problems involving the steadystate NavierStokes equations.
Consistent Initial Condition Calculation For DifferentialAlgebraic Systems
, 1995
"... In this paper we describe a new algorithm for the calculation of consistent initial conditions for a class of systems of differentialalgebraic equations which includes semiexplicit indexone systems. We consider initial condition problems of two typesone where the differential variables are speci ..."
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Cited by 16 (2 self)
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In this paper we describe a new algorithm for the calculation of consistent initial conditions for a class of systems of differentialalgebraic equations which includes semiexplicit indexone systems. We consider initial condition problems of two typesone where the differential variables are specified, and one where the derivative vector is specified. The algorithm requires a minimum of additional information from the user. We outline the implementation in a generalpurpose solver DASPK for differentialalgebraic equations, and present some numerical experiments which illustrate its effectiveness.
Practical quasiNewton methods for solving nonlinear systems
, 2000
"... Practical quasiNewton methods for solving nonlinear systems are surveyed. The definition of quasiNewton methods that includes Newton 's method as a particular case is adopted. However, especial emphasis is given to the methods that satisfy the secant equation at every iteration, which are ..."
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Cited by 14 (2 self)
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Practical quasiNewton methods for solving nonlinear systems are surveyed. The definition of quasiNewton methods that includes Newton 's method as a particular case is adopted. However, especial emphasis is given to the methods that satisfy the secant equation at every iteration, which are called here, as usually, secant methods. The leastchange secant update (LCSU) theory is revisited and convergence results of methods that do not belong to the LCSU family are discussed. The family of methods reviewed in this survey includes Broyden 's methods, structured quasiNewton methods, methods with direct updates of factorizations, rowscaling methods and columnupdating methods. Some implementation features are commented. The survey includes a discussion on global convergence tools and linearsystem implementations of Broyden's methods. In the final section, practical and theoretical perspectives of this area are discussed. 1 Introduction In this survey we consider nonlinear ...