H. A. van der Vorst and C. Vuik, GMRESR: a family of nested GMRES methods, Numer. Linear Algebra Appl., 1 (1994), pp. 369--386.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....arises in the matrix formulation of the Krylov subspace method. Where possible we point out the di#erences with techniques used in literature and discuss implications for rounding error analysis. Another related problem is when a variable preconditioner is used in the Krylov subspace method. See [9, 22, 28, 8, 11] for some results. The outline of this paper is as follows. In Sections 2 and 3 we setup the framework that we need in the rest of this paper. We give an expression for the residual gap for a general Krylov subspace method in Section 3. This general expression is exploited in the remainder of ....

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

....of the linear solver. These effects may compensate for the more expensive steps. For precisely these reasons, a strategy is followed in [6, 4] where in (2) is replaced by the matrix of all Ritz vectors that could be associated with eigenvalues in a cluster near the target eigenvalue. GMRESR [21] and GCRO [3] are nested methods for solving linear systems Ax = b iteratively. They both use GCR in the outer loop to update the approximate solution and GMRES in the inner loop to compute a new search direction from a correction equation. As argued in [7] Jacobi Davidson with (2) can be ....

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

....iteration. This means that the generated vectors generally do not span a Krylov space. However, if we precondition from the right we nevertheless get the image of some (approximate Krylov) subspace under multiplication by the linear operator, and we can minimize the residual over this subspace [22, 3, 15]. We use the Flexible GMRES (FGMRES) implementation outlined in [17] Note that we have approximated A l in (15) by A . Hence the result of [14] does not guarantee convergence of GMRES for P l M l or M l P l in three steps; we no longer have exactly three distinct eigenvalues. A complete ....

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

....with speci c choices of linear solvers for the saddle point problem and the convection di usion and pressure Poisson subproblems. To solve the saddle point linear problem associated with each Oseen iteration, we use GMRESR. GMRESR is a variation on GMRES proposed by van der Vorst and Vuik [55] allowing the preconditioner to vary at each iteration. For the pressure Poisson problem, A p , we use CG preconditioned with algebraic multigrid, and for the convection di usion problem, F , we use GMRES preconditioned with algebraic multigrid. For transient and pseudotransient problems, we use ....

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

....accurate solutions to the dual equations (1.2) in each of the neighborhoods surrounding the interpolation points oe k . The impact of various DS preconditioners (various choices for the oe ) is considered in Chapter 6. The use of multiple varying preconditioners has appeared in the literature [70, 71] for solving fixed systems of linear equations. Similarities can be seen between these algorithms and the model reduction algorithms developed in Chapter 4. The modelreduction problem is frequency dependent; adapting multiple preconditioners to cover a range of frequencies is novel. One may ....

....in standard fashion without underlining. 142 When an iterative solver is utilized, Phi m represents a nonlinear operation which is no longer associated with a fixed matrix P k . The use of iterative solvers to implicitly perform DS preconditioning is common in the linear solver literature [70, 71, 97]. Methods of this type are known as inner outer iterations. The outer iteration constructs a search subspace for the solution of the original problem. In our case, the outer iteration constructs the projection matrices V and Z. During each outer step, an entire loop of inner iterations is executed ....

H. A. van der Vorst and C. Vuik, "GMRESR: A family of nested GMRES methods," Numer. Linear Algebr. Appl., vol. 1, pp. 369--386, 1994.

....one can of course restart the GMRES process after a certain number of iterations. However it is more profitable to use a recursive version of GCR in which the inner GMRES loop generates a better preconditioner and the outer GCR loop solves a well preconditioned system. In VLUGR2[3] we used GMRESR[23] and we reported some stagnations. In [12] a remedy is given against these stagnations. The idea is that the new Krylov basis vectors in the innerloop should not only be orthogonal to the previous ones but also to the corrections on the approximate solution already computed in the outerloop. This ....

H.A. van der Vorst and C. Vuik. GMRESR: A family of nested GMRES methods. Report 91-80, Faculty of Technical Mathematics and Informatics, TU Delft, the Netherlands, 1991.

....arises in the matrix formulation of the Krylov subspace method. Where possible we point out the di#erences with techniques used in literature and discuss implications for rounding error analysis. Another related problem is when a variable preconditioner is used in the Krylov subspace method. See [9, 22, 28, 8, 11] for some results. The outline of this paper is as follows. In Sections 2 and 3 we setup the framework that we need in the rest of this paper. We give an expression for the residual gap for a general Krylov subspace method in Section 3. This general expression is exploited in the remainder of ....

H. A. van der Vorst and C. Vuik, GMRESR: a family of nested GMRES methods, Numer.

....iteration. This means that the generated vectors generally do not span a Krylov space. However, if we precondition from the right we nevertheless get the image of some (approximate Krylov) subspace under multiplication by the linear operator, and we can minimize the residual over this subspace [22, 3, 15]. We use the FGMRES implementation outlined in [17] Note that we have approximated A Gamma1 l in (15) by A Gamma1 . Hence the result of [14] does not guarantee convergence of GMRES for P l M l or M l P l in three steps; we no longer have exactly three distinct eigenvalues. A complete ....

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

....Q H m #r 0 # 2 e 1 . 1.17) Comparing the GCR and the GMRES algorithm we see that Cm = Wm 1 Qm , Um = WmR 1 m . We would like to have a method that converges like full GMRES, but uses only limited resources. Several methods derived from GMRES or GCR have been proposed recently: GMRESR [25], FGMRES [20] and GCRO [9, 7, 6] These methods strike a balance between optimality and cost by keeping a limited number of vectors from previous search spaces, hence doing some form of truncation. However, GCRO is the only method that computes the optimal solution over the subspace spanned by ....

H. A. Van der Vorst and C. Vuik, GMRESR: A family of nested GMRES methods, Numer. Linear Algebra with Appl., 1 (1994), pp. 369--386.

.... estimations are usually done by the power method or by the Arnoldi technique but they can also be computed from modified moments [2] Other hybrid solutions do not rely on eigenvalue estimations but use directly a polynomial generated by GMRES itself [10] An alternative approach discussed in [17, 14] is to build a preconditioner based on the application of GMRES. In this paper the eigenvalue technique is not used, but rather an invariant subspace approach. This idea has been developed in [5, 15] for the solution of nonlinear parameter dependent systems of equations, in which a Newton method ....

H.A. van der Vorst and C. Vuik. GMRESR : a family of nested GMRES methods. Technical Report, June 1992.

....# Richard Stockton College, Pomona, NJ 08240 (judith.vogel stockton.edu) 363 364 DANIEL B. SZYLD AND JUDITH A. VOGEL In recent years, several authors worked on the idea of preconditioning with a di#erent matrix at each outer iteration of a Krylov subspace method [1] 16] 20] 21] [25]; see also [6] 14] 15] for other instances of inner outer iterations. Preconditioning of this form is referred to as flexible preconditioning, also known as variable or inexact preconditioning. Our approach to a flexible version of QMR, which we call FQMR, is similar to that of Saad for FGMRES ....

H. A. Van der Vorst and C. Vuik, GMRESR: a family of nested GMRES methods, Numer. Linear Algebra Appl., 1 (1994), pp. 369--386.

....corresponding to the pressure correction equation these numbers are 9 and 19 respectively. Various solution methods have been studied, cf. 16] 17] 19] 18] 33] 32] 31] 34] 39] 42] 43] 41] 44] We use GCR [3] preconditioned with multigrid for 12 the pressure and GMRESR [28] (a combination of GCR and GMRES [22] for the velocity. In three dimensions this requires much storage. Per finite volume cell, we store 5 reals for geometric quantities, 7 for ; ae; k; and a right hand side, 8 for two solution fields, 3 Theta 51 = 153 for the velocity matrix, 6 for the ....

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

....(inner) iterative method. This is the case, e.g. when the preconditioner used is some version of multigrid, such as in [10] In recent years, several authors worked on the idea of preconditioning with a di erent matrix at each outer iteration of a Krylov subspace method [1] 16] 20] 21] [25]; see also [6] 14] 15] for other instances of inner outer iterations. Preconditioning of this form is referred to as exible preconditioning, also known as variable or inexact preconditioning. Our approach to a exible version of QMR, which we call FQMR, is similar to that of Saad for FGMRES ....

H.A. Van der Vorst and C. Vuik. GMRESR: a family of nested GMRES methods. Numerical Linear Algebra with Applications. 1:369-386, 1994. 16

....v j can be discarded immediately after it is used in the above formula. In fact, we can simply overwrite it onto the space used for p j and modify step 4 of algorithm 4.2 accordingly. We note that DQGMRES is similar in nature to the GMRESR family of algorithms introduced by Van der Vorst and Vuik [9]. Finally, we would like to mention that convergence results identical to those of the QMR algorithm [2, 3] hold. For example, it is easy to prove that the norms of the residuals r Q m and r G m obtained after m steps of the DQGMRES and GMRES algorithms respectively are related by kr Q m k ....

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. 25

.... that some researchers formalized the inexact version of this algorithm [14, 28] In same fashion, intensive work has been devoted to extending current nonsymmetric iterative solvers to be able to accommodate the inexactness or variability of the preconditioner from iteration to iteration; e.g. [3, 43, 49, 50]. In this work, we use right preconditioning. It is well known that this form is preferable over left preconditioning for comparing different preconditioners since it makes the relative residual norms measured within the iterative solver invariant. This norm size invariance simplifies the ....

H. van der Vorst and C. Vuik, GMRESR: A Family of Nested GMRES Methods, Tech. Rep. TR91--80, Technological University of Delft, 1991.

....usage in large simulation codes. For parallel linear algebra solvers, it can also be though to use these asynchronous relaxation iterations as smoother in multigrid or to consider few steps of asynchronous relaxations as a preconditioner of FGMRES [140] or III.1 Asynchronous iterations 89 GMRESR [153], that are Krylov solvers in which it is allowed to take a di erent preconditioner in each step. 90 Parallel performance Chapter 2 Parallel performance of two level non overlapping domain decomposition methods 2.1 Introduction The full exploitation of the new computer architecture with large ....

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.

....6 which is equivalent to Q T N Gamma1 AQv = Q T N Gamma1 f: 19) In this way, accurate solution of subdomain problems finally leads to a system involving only the interface equations. Accelerated domain decomposition in [11] amounts to solving the interface equations (19) using GMRESR [46]. In the present paper, we use GMRES: the required matrix vector product can be computed by doing one domain decomposition iteration, see [11] for details. 3.3 Inaccurate subdomain solution Domain decomposition iteration (10) is typically implemented as Nu m 1 = N Gamma A)u m f ; 20) ....

.... = kv k 1 k 2 v k 1 = v k 1 =fi s k 1 = s k 1 =fi # end Gram Schmidt # update x and r fl = f; v k 1 ) x k 1 = x k 1 fls k 1 r k 1 = r k 1 Gamma fls k 1 k = k 1 end while Figure 4: The GCR algorithm with general search directions without restart and with a relative stopping criterion [46] For the special case of the search direction s k 1 = r k , we obtain the classical GCR algorithm, which is equivalent to GMRES [41] For this choice of search direction, the space S k is called the Krylov space. The difference between GCR and GMRES is that, with the benefit of allowing more ....

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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), 1994.

.... Johnson, Micchelli and Paul [126] Saad [183] and Nachtigal, Reichel and Trefethen [159] In more sophisticated approaches, the polynomial preconditioner is adapted to the iterations (Saad [187] or the preconditioner may even be some other (iterative) method of choice (Van der Vorst and Vuik [209], Axelsson and Vassilevski [24] Stagnation is prevented in the GMRESR method (Van der Vorst and Vuik [209] by including LSQR steps in some phases of the process. In De Sturler and Fokkema [64] part of the optimality of GMRES is maintained in the hybrid method GCRO, in which the iterations of ....

.... approaches, the polynomial preconditioner is adapted to the iterations (Saad [187] or the preconditioner may even be some other (iterative) method of choice (Van der Vorst and Vuik [209] Axelsson and Vassilevski [24] Stagnation is prevented in the GMRESR method (Van der Vorst and Vuik [209]) by including LSQR steps in some phases of the process. In De Sturler and Fokkema [64] part of the optimality of GMRES is maintained in the hybrid method GCRO, in which the iterations of the preconditioning method are kept orthogonal to the iterations of the underlying GCR method. All these ....

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

....multigrid terminology, Z is the projection or interpolation operator, and Y T is the restriction operator. 12 and form Q x, or solve y from PAK Gamma1 y = P b; 5.3) and form QK Gamma1 y. Both systems can be solved by one s favorite Krylov subspace solver, such as: GMRES [20] GCR [6, 23], Bi CGSTAB [22] etc. The question remains how to choose Y . We consider two possibilities: 1. Suppose Z consists of eigenvectors of A. Choose Y as the corresponding eigenvectors of A T . 2. Choose Y = Z. For both choices we can prove some results about the spectrum of PA. Assumption 5.1 We ....

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

....as GMRES. However GCR does not take advantage of the recursion (4) Rather, GCR requires the storage of an extra set of orthogonal residual search vectors. Three advantages of this method are: 1) the preconditioner K need not remain constant (nor even be a linear operator; the implementation in [40] uses GMRES(m) as a preconditioner) 2) one is free to employ truncation strategies such as in [41] and (3) the method will not break down if the LSQR switch is employed [40] As described in Figure 1, the method is unrestricted in the number of iterations, and therefore in the number of vectors ....

.... this method are: 1) the preconditioner K need not remain constant (nor even be a linear operator; the implementation in [40] uses GMRES(m) as a preconditioner) 2) one is free to employ truncation strategies such as in [41] and (3) the method will not break down if the LSQR switch is employed [40]. As described in Figure 1, the method is unrestricted in the number of iterations, and therefore in the number of vectors v k and q k which must be stored. Since most modern computers are equipped with only finite memory, it is necessary either to restart the iteration periodically, discarding ....

[Article contains additional citation context not shown here]

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

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