H. A. van der Vorst and C. Vuik. GMRESR: a family of nested GMRES methods. Num. Lin. Alg. Appl., pages 369-386, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....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 difference between FGMRES and the usual GMRES algorithm is that the action of AMj 1 on a vector v of the Krylov subspace is no longer in the span of Vm l. Instead, it is easy to show that the following equality takes place (1) AZ m Vm ] lra . This ....

H.A. van der Vorst and C. Vuik. GMRESR: a family of nested gmres methods. Technical Report 91-80, Delft University of Technology, Mathematics and Informatics, Delft, The Netherlands, 1991.

....of the form Ax = b: 1.1) In recent years, several authors studied Krylov subspace methods with variable (or exible) preconditioning, i.e. preconditioning with a di erent (possibly nonlinear) operator at each iteration of a Krylov subspace method. These include [1] 16] 26] 28] 35] [36], and [38] The usual (right) preconditioning consists of replacing (1.1) by y = b; with Mx = y; 1.2) for a suitable preconditioner M . One of the motivations for methods with variable preconditioners is the need to solve each preconditioning equation Mz = v; 1.3) only inexactly, as is ....

....Krylov method. One can also consider preconditioners which might improve using information from previous iterations; cf. 2] 12] 20] Experiments have been reported in the literature, where the preconditioner in (1.3) is itself a Krylov subspace method. For example, some versions of GMRESR [36] t this description. In [4] 28] one has GMRES for the preconditioner, or inner This version dated 4 February 2002 Dipartimento di Matematica, Universit a di Bologna, and Istituto di Analisi Numerica del CNR, Pavia, Italy. val dragon.ian.pv.cnr.it) Department of Mathematics, Temple ....

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H. A. Van der Vorst and C. Vuik. GMRESR: a family of nested GMRES methods. Numerical Linear Algebra with Applications, 1:369-386, 1994.

....results are not influenced by the choice on which continent the stream function is prescribed. Prescribing on the largest continent has the advantage that the linear system becomes better conditioned and hence easier to solve by an iterative method. The iterative solution method we used is GMRESR [12], combined with diagonal scaling. 7.2 Results The model we have used as our test problem is too simplified to expect very realistic results. e.g. bottom topography, nonlinear effects and three dimensional effects are all neglected. Our model mainly gives a balance between wind stress, Coriolis ....

Van der Vorst, H.A. and C. Vuik 1993 - GMRESR: A family of nested GMRES methods, Num. Lin. Alg. Appl. 1, 1-7

....to the inexact Krylov method and its variable preconditioner as the outer iteration and inner iteration respectively. Methods that can be used for the outer iteration are the so called flexible methods. These are methods that are specially designed for dealing with variable preconditioning, e.g. [11, 18, 23] which we combine with an approximate matrix vector product. We discuss a few choices for the outer iteration in the remainder of this section. For this purpose, we need the following notation: when an inexact Krylov subspace method is used for solving the linear system Az = y with a relative ....

....computed residual is that we do not have to estimate the residual reduction in the coming step. Notice that this is not precisely # j 1 in practice. Finally, the recursively computed residual is an essential ingredient of optimal methods like, for example, flexible GMRES [18] and GMRESR [23]. 4.2 The outer iteration: flexible GMRES The flexible GMRES method by Saad [18] is a variant of the GMRES method that can deal with variable preconditioning. It constructs an orthogonal basis V k for AZ k where z j = # j (v j ) which, with inexact matrix vector product, can be summarized by ....

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H. A. van der Vorst and C. Vuik, GMRESR: a family of nested GMRES methods, Numer. Linear Algebra Appl. 1 (1994), no. 4, 369--386. MR 95j:65034

....the result of a preconditioning step applied to the most recent basis vector vm 1 of the residual space Vm 1 , which may be viewed as an approximate solution of the equation Ac = vm 1 . A similar approach is taken in the GMRESR (which stands for GMRES Recursive) method of van der Vorst and Vuik [39]. In each step of GMRESR, the new correction vector cm 1 is chosen as the approximate solution of the equation Ac = rm obtained by a given number of GMRES steps, where rm is the residual of the MR approximation using the current correction space Cm . This method was improved upon by de Sturler ....

Henk A. van der Vorst and C. Vuik. GMRESR: a family of nested GMRES methods. Num. Lin. Alg. Appl., 1:369--386, 1994.

.... author was supported by the Netherlands organization for scientific research NWO, project 95MPR04 x Work of these authors was supported in part by the Netherlands organization for scientific research NWO, project 95MPR04 truncated and (or) nested (flexible) Krylov subspace iterative processes [11, 24, 18], use the same Krylov basis several times [11] reusing Krylov basis with sophisticated injecting technique [20] and application of Krylov like projections directly on the nonlinear level [22, 6, 8] are only some possibilities. These techniques are mainly aimed to capture the information, ....

H. A. van der Vorst and C. Vuik. GMRESR: a family of nested GMRES methods. Numer. Lin. Alg. Appl., 1:369--386, 1994.

....result of a preconditioning step applied to the most recent basis 23 vector vm 1 of the residual space Vm 1 , which may be viewed as an approximate solution of the equation Ac = vm 1 . A similar approach is taken in the GMRESR (which stands for GMRES Recursive) method of van der Vorst and Vuik [27]. In each step of GMRESR, the new correction vector cm 1 is chosen as the approximate solution of the equation Ac = rm obtained by a given number of GMRES steps, where rm is the residual of the MR approximation using the current correction space Cm . This method was improved upon by de Sturler ....

Henk A. van der Vorst and C. Vuik. GMRESR: a family of nested GMRES methods. Num. Lin. Alg. Appl., 1:369-386, 1994. 32

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H. A. van der Vorst and C. Vuik, GMRESR: a family of nested GMRES methods, Num. Lin. Alg. Appl., 1 (1994), pp. 369--386.

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H. A. van der Vorst and C. Vuik. GMRESR: a family of nested GMRES methods. Numerical Linear Algebra with Applications, 1(4):369--386, 1994.

....words. Nonlinear problems, Newton s method, Inexact Newton, Iterative methods. AMS subject classification. 65H10. 1. Introduction. Our goal in this paper is twofold. A number of iterative solvers for linear systems of equations, such as FOM [23] GMRES [26] GCR [31] Flexible GMRES [25] GMRESR [29] and GCRO [7] are in structure very similar to iterative methods for linear eigenproblems, like shift and invert Arnoldi [1, 24] Davidson [6, 24] and Jacobi Davidson [28] We will show that all these algorithms can be viewed as instances of an Accelerated Inexact Newton (AIN) scheme (cf. Alg. ....

....C Omega ) c 2 C( h; g 2 C (R) and f 2 L 2( Omega Gamma6 and u satisfies suitable boundary conditions. An example of (4) is, for instance ; 5) where Omega is some domain in R and u = 0 on (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 ....

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H. A. Van der Vorst and C. Vuik, GMRESR: A family of nested GMRES methods, Num. Lin. Alg. Appl., 1 (1994), pp. 369--386.

....if this gives a further reduction then GMRESR does not break down. This gives slightly more control over the method in comparison with FGMRES. In most cases though the results are about the same, but the efficient scheme for FGMRES has an advantage. We will briefly discuss the GMRESR method. In [101] it has been shown how the GMRES method, or more precisely, the GCR method, can be combined with other iterative schemes. The iteration steps of GMRES (or GCR) are called outer iteration steps, while the iteration steps of the preconditioning iterative method are referred to as inner iterations. ....

....outer iteration steps, while the iteration steps of the preconditioning iterative method are referred to as inner iterations. The combined method is called GMRES , where stands for any given iterative scheme; in the case of GMRES as the inner iteration method, the combined scheme is called GMRESR[101]. It was shown in [26, 25] that GMRESR can be implemented in a way that avoids about 30 of the overhead in the outerloop, which makes the method about as expensive per outer iteration step as FGMRES. The GMRES algorithm can be described as in Fig. 2. x 0 is an initial guess; r 0 = b Gamma Ax ....

[Article contains additional citation context not shown here]

H. A. Van der Vorst and C. Vuik. GMRESR: A family of nested GMRES methods. Num. Lin. Alg. with Appl., 1:369--386, 1994.

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H. A. van der Vorst and C. Vuik. GMRESR: a family of nested GMRES methods. Num. Lin. Alg. Appl., pages 369-386, 1994.

No context found.

H. A. Van der Vorst and C. Vuik. GMRESR: a family of nested GMRES methods. Numer. Lin. Alg. with Appl., 1:369--386, 1994.

No context found.

H. A. van der Vorst and C. Vuik. GMRESR: A family of nested GMRES methods. Num. Lin. Alg. Appl., 1:369--386, 1994.

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