P. N. Brown and Y. Saad. Hybrid Krylov methods for nonlinear systems of equations. SIAM J. Sci. Statistic. Comp., 11:450-481, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Newton iteration (see, e.g. 11] to Eq. 5) through the two step sequence of (approximately) solving the Newton correction equation ) 6) where I is the identity matrix, and then updating the iterate via u u . We employ matrix free Newton Krylov methods (see, e.g. [7]) in which we compute the action of the Jacobian on a vector v via directional differencing of the form f # (u)v# (u hv) f (u) h ,whereh is a differencing parameter. We precondition the Newton Krylov methods, whereby we increase the linear convergence rate at each nonlinear iteration by ....

Brown PN, Saad Y. Hybrid Krylov methods for nonlinear systems of equations. SIAM J Sci Statistical Comput 1990;11:450--481.

....and prompt turnaround forces consideration of parallelism, and, for cost effectiveness, particularly parallelism of the high latency, low bandwidth variety represented by workstation clusters. A Newton Krylov Schwarz (NKS) method combines a Newton Krylov (NK) method such as nonlinear GMRES [2], with a Krylov Schwarz (KS) method, such as additive Schwarz [8] The key linkage is provided by the Krylov method, of which the restarted form of GMRES [21] is perhaps the best known example for nonselfadjoint problems. From a computational point of view, the most important characteristic of a ....

....matrix vector evaluation routine with jjvjj 2 = 1. We therefore set h to be n Delta mach . When less is known about the scaling of u and v, a reasonable choice is mach Delta (u ; v) jjvjj , with guard code to set h = mach if jjvjj is too small. For a fuller discussion, see [2]. For numerical experiments demonstrating the importance of the relative scaling of h in the CFD context, see [17, 20] Preconditioning (5) by J low , for instance on the left, as in ( 7) shifts the inconsistency from the nonlinear to the linear aspects of the problem. This should ....

P. Brown and Y. Saad. Hybrid Krylov methods for nonlinear systems of equations. SIAM Journal of Scientific and Statistical Computing, 11:450--481, 1990.

....be achieved through the use of implicit boundary conditions as compared to explicit ones. However, the attractive Newton s method convergence property cannot be approached because of the mismatch of the right and left hand side operators. Therefore we propose to use Newton Krylov matrix free (see [3]) methods combined with mixed discretization in the implicitly defined Jacobian Preconditioner. Numerical experiments that show the performance of our approach are then presented. In the next section, we describe the Newton Krylov methodology together with mixed discretization. We present, in the ....

....the Euler solver. The description of the implicit boundary conditions is also given in the section 3. Numerical experiments are presented in the section 4. The last section is devoted to some remarks and extensions. 2 Newton Krylov Methods Newton Krylov methods first proposed by Brown and Saad [3], have been investigated for compressible Euler and Navier Stokes equations using unstructured grids in [16] 17] and [7] and for structured grids in [4] and [5] In [16] and [17] the author has applied the matrix free Newton Krylov methodology to both the transonic and supersonic ....

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P. N. Brown and Y. Saad, Hybrid Krylov Methods for Nonlinear Systems of Equations, SIAM J. Sci. Stat. Comp. 11(1990), 450--481.

....convergence in the vicinity of a root and the fact that the solution can be determined by solving a few linear systems with the Jacobian as matrix. Unfortunately Newton s method will only converge if the initial guess is suciently close to the root. A backtracking line search technique [2,5] is used to increase the radius of convergence. The underlying idea is to adaptively choose a damping factor such that there is a sucient decrease in the residual, and the directional derivative for the next iterate is not too small . This increases the radius of convergence while quadratic ....

....linear systems. The storage requirements are only a few vectors and they do not need the matrix itself but only its action on vectors . In the context of Newton iteration the action of the Jacobian can be approximated by a di erence quotient and the linear systems need not be solved exactly. In [2] it is shown how non linear Krylov subspace correction methods can be formulated, including a Newton GMRES algorithm based on backtracking line search techniques. Since the matrices considered here are non symmetric GMRES [12,14] has been chosen. Other possible choices include the Conjugate ....

Brown, P. N., Saad, Y.: Hybrid Krylov methods for nonlinear systems of equations, SIAM J. Stat. Comp ## (1990) 450-481

....solidi cation in realistic settings. The major diculties of direct simulation are associated with keeping track of the moving solid liquid interfaces so that the correct boundary conditions are prescribed. Front tracking methods t naturally in the framework of free boundary problems, see e.g. [2, 20]. The interface is represented byadiscrete set of marker points which are connected by interpolants to obtain the location of the free boundary. The time dependent problem is then solved byadvecting the marker points according to the velocity given by (2.1) and solving the heat equations in solid ....

....[35] is used to transfer data between a xed temperature grid and interfaces. Time stepping essentially reduces to solving a system of non linear equations for the normal velocities of the marker points guaranteeing that the Gibbs Thomson condition holds. Almgren considers a di erent algorithm in [2] where marker points are advected by minimizing an energy functional chosen such that the Gibbs Thomson condition is satis ed. Due to the complicated interfacial shapes which arise it will be necessary to add and delete points during simulation. Merging interfaces present a problem since it ....

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Brown, P. N., Saad, Y.: Hybrid Krylov methods for nonlinear systems of equations, SIAM J. Stat. Comp ## (1990) 450-481

.... a Krylov Schwarz domain decomposition algorithm for the finite element solution of the nonlinear full potential equation of aerodynamics, extending our model studies of linear convection diffusion problems in [5] and of linear aerodynamic design optimization problems in [33] Newton Krylov methods [2, 3, 14, 15, 39] are potentially well suited and increasingly popular for the implicit solution of nonlinear problems whenever it is expensive to compute or store a true Jacobian. We employ a combined algorithm, called Newton Krylov Schwarz, and focus on the interplay of the three nested components of the ....

P. N. Brown and Y. Saad, Hybrid Krylov methods for nonlinear systems of equations, SIAM J. Sci. Stat. Comput., 11 (1990), pp. 59--71.

....still be much less expensive than a full Monte Carlo sampling approach. 3. IMPLEMENTATION We have implemented a three dimensional variably saturated flow model based on Richards equation in the ParFlow software package [3] The Richards equation model uses the KINSOL inexact Newton Krylov [4] software package to solve the nonlinear systems at each time step [5] Each nonlinear Newton iteration is solved with GMRES [6] preconditioned with Schaffer s semi coarsening multigrid [7] method implemented in the hypre preconditioning library [8] Previous work has shown that this solution ....

P.N. Brown and Y. Saad. Hybrid krylov methods for nonlinear systems of equations. 8

....in local q linear convergence in the norm I1 I1. 2. The choice = 10 4 used by Cai, Gropp, Keyes, and Tidriri [3] which requires uniformly close approximations of Newton steps for all k and results in fst oc q ner convergence in the norm II 3. The choice k = 1 2 k l of Brown md Sttd [2]. This choice results in loctl q superlinetr convergence md dlows relatively inaccurate ppromtions of Newton steps for sm k, when my not be near , however, it incorpo rates no information bout F. 4. The choice = min(1 (k 2) llF(x)ll) of Dembo and Steihaug [5] This choice results in ....

....it may be appropriate to use forcing terms that are not asymptotically increasingly demanding, such as constant forcing terms that give adequately fast q hnear convergence. 10 whereg(zl,Z2) 1if z2 = i andg(zl,z2) 0if0 z2 1. This is awidely used test problem; see, e.g. Brown and Saad [2] or Glowinski, Keller, and Reinhart [8] The numerical problem becomes harder as the Reynolds number Re increases. Discretization was by piecewise linear finite elements on a uniform 63 x 63 grid 4, so that n = 3969. The discretized problem was preconditioned on the right using a fast biharmonic ....

P. N. BROWN AND Y. SAAD, Hybrid Krylov methods for nonlinear systems of equations, SIAM J. Sci. Stat. Comput., 11 (1990), pp. 450 481.

....always necessary to compute the exact Newton direction, and a sufficiently accurate approximation will do. It is possible to relate the accuracy needed in GMRES to the residual norm in the current Newton step, and formulate sufficient conditions for super linear or quadratic convergence [8] In [7] Brown and Saad use this to formulate non linear Krylov algorithms with stronger coupling between the linear and non linear solvers. Among the algorithms considered is a Newton GMRES algorithm based on backtracking 13 line search techniques. 2.3.3 Orthogonalization scheme in GMRES The inner ....

P. N. BROWN AND Y. SAAD, Hybrid Krylov methods for nonlinear systems of equations, SIAM J. Star. Comp., 11 (1990), pp. 450-481.

.... a Krylov Schwarz domain decomposition algorithm for the finite element solution of the nonlinear full potential equation of aerodynamics, extending our model studies of linear convection diffusion problems in [5] and of linear aerodynamic design optimization problems in [34] Newton Krylov methods [2, 3, 15, 16, 40] are potentially well suited and increasingly popular for the implicit solution of nonlinear problems whenever it is expensive to compute or store a true Jacobian. We employ a combined algorithm, called Newton KrylovSchwarz, and focus on the interplay of the three nested components of the ....

P. N. Brown and Y. Saad, Hybrid Krylov methods for nonlinear systems of equations, SIAM J. Sci. Stat. Comput., 11 (1990), pp. 59--71.

....(see also Section 8) Guided by the known approaches for the linear system (cf. 25, 29, 7] and the eigenproblem (cf. 28, 27] we will define accelerated Inexact Newton schemes for more general nonlinear systems. This leads to a combination of Krylov subspace methods for Inexact Newton (cf. [16, 4] and also [8] with acceleration techniques (as in [2] and offers us an overwhelming choice of techniques for further improving the efficiency of Newton type methods. As a side effect this leads to a surprisingly simple framework for the identification of many well known methods for linear , ....

....an approximation for the Jacobian (see, e.g. 8] Alg. 1 is an algorithmical representation of the resulting Inexact Newton scheme. If, for instance, Krylov subspace methods are used for the approximate solution of the correction equation (6) then only directional derivatives are required (cf. [8, 4]) there is no need to evaluate the Jacobian explicitly. If v is a given vector then the vector F (u)v can be approximated using the fact that F (u v) Gamma F (u) F (u)v for 0: The combination of a Krylov subspace method with directional derivatives combines the steps 2c and ....

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P. N. Brown and Y. Saad, Hybrid Krylov methods for nonlinear systems of equations, SIAM J. Sci. Statist. Comput., 11 (1990), pp. 450--481.

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P. N. Brown and Y. Saad, Hybrid Krylov methods for nonlinear systems of equations, SIAM J. Sci. Stat. Comp., 11 (1990), pp. 450-481.

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P. N. Brown and Y. Saad. Hybrid Krylov methods for nonlinear systems of equations. SIAM J. Sci. Stat. Comp., 11:450-481, 1990.

....the multiplication of J by an arbitrary vector x can be carried out, usually at low cost, with the help of the difference formula F(u ex,9) F(u, 9) 2) where e is some small and carefully chosen scalar. The approximation (2) has been useful in solving nonlinear systems of equations [3, 6, 12, 4, 44, 19] and to compute eigenvalues of various semi discrete operators [11] used in compressible fluid flow calculations. Here, an algorithm such as Arnoldi s method can be used but not the nonsymmetric Lanczos procedure since we do not know to compute the vector JTx for any vector x when the Jacobian ....

P. N. Brown and Y. Saad. Hybrid Krylov methods for nonlinear systems of equations. SIAM J. Sci. Stat. Comp., 11:450-481, 1990.

....method is mathematicay equivalent to Algorithm 2.2. Note that we can define zj in step 2 b without reference to any preconditioner, i.e. we can simply pick a given new vector z5. We would like to mention that the technique presented above can be viewed as an extension of a strategy presented in [1] in the context of using Krylov subspace methods for solving nonlinear equations. More recently, van der Vorst and Vuik developed a family of algorithms that have the same feature as FGMRES in that they also allow variations in the preconditioner [7] 2.2. Some basic properties. One notable ....

P. N. Brown and Y. Saad. Hybrid Krylov methods for nonlinear systems of equations. SIAM J. Sci. Star. Comp., 11:450 481, 1990.

....by the DOE Office of Energy Research, Applied Mathematical Sciences Research Program. lWork supported by Cooperative Agreement NCC 2 387 between the National Aeronautics and Space Adminis tration (NASA) and the Universities Space Research Association (USRA) 1 Introduction In a previous paper [5] we have proposed several basic methods based upon the idea of employing a Newton iteration in which the Jacobian equations are solved approximately by a Krylov subspace method. Several theoretical issues raised in [5] were left unanswered. The purpose of this paper is to fill this gap by laying ....

....Space Research Association (USRA) 1 Introduction In a previous paper [5] we have proposed several basic methods based upon the idea of employing a Newton iteration in which the Jacobian equations are solved approximately by a Krylov subspace method. Several theoretical issues raised in [5] were left unanswered. The purpose of this paper is to fill this gap by laying down the theoretical foundation of nonlinear Krylov subspace methods and by providing convergence results for them. In fact we will not limit ourselves to Krylov subspace methods. Rather, we discuss inexact Newton ....

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P. N. Brown and Y. Saad, Hybrid Krylov methods for nonlinear systems of equations, SIAM J. Stat. Scient. Cornput., to appear.

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P. N. Brown and Y. Saad. Hybrid Krylov methods for nonlinear systems of equations. SIAM J. Sci. Statistic. Comp., 11:450-481, 1990.

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Peter N. Brown and Yousef Saad. Hybrid Krylov methods for nonlinear systems of equations. SIAM J. Sci. Statist. Comput., 11:450--481, 1990.

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P. N. Brown and Y. Saad. Hybrid Krylov methods for nonlinear systems of equations. SIAM J. Sci. Stat. Comput., 11:450--481, 1990.

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P. N. Brown and Y. Saad, Hybrid Krylov methods for nonlinear systems of equations, SIAM J. Sci. Statist. Comput., 11 (1990), pp. 450--481.

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P. N. Brown and Y. Saad, Hybrid Krylov methods for nonlinear systems of equations, SIAM J. Sci. Stat. Comput., 11 (1990), pp. 450--481.

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P. N. Brown and Y. Saad, Hybrid Krylov Methods for Nonlinear Systems of Equations, SIAM J. Sci. Stat. Comp. 11(1990), 450--481.

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Peter N. Brown and Yousef Saad. Hybrid krylov methods for nonlinear systems of equations. SIAM J. Sci. Statist. Comput., 11:450-481, 1990.

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P. N. Brown and Y. Saad, Hybrid krylov methods for nonlinear systems of equations, SIAM J. Sci. Stat. Comput., 11 (1990), pp. 450--481.

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P.N. Brown and Y. Saad, "Hybrid Krylov Methods for Nonlinear Systems of Equations," SIAM J. Scientific and Statistical Computing, Vol. 11, 1990, pp. 450--481.

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