## Theory of inexact Krylov subspace methods and applications to scientific computing (2002)

Citations: | 48 - 6 self |

### BibTeX

@TECHREPORT{Simoncini02theoryof,

author = {Valeria Simoncini and Daniel and B. Szyld},

title = {Theory of inexact Krylov subspace methods and applications to scientific computing},

institution = {},

year = {2002}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. We provide a general frameworkfor the understanding of inexact Krylov subspace methods for the solution of symmetric and nonsymmetric linear systems of equations, as well as for certain eigenvalue calculations. This frameworkallows us to explain the empirical results reported in a series of CERFACS technical reports by Bouras, Frayssé, and Giraud in 2000. Furthermore, assuming exact arithmetic, our analysis can be used to produce computable criteria to bound the inexactness of the matrix-vector multiplication in such a way as to maintain the convergence of the Krylov subspace method. The theory developed is applied to several problems including the solution of Schur complement systems, linear systems which depend on a parameter, and eigenvalue problems. Numerical experiments for some of these scientific applications are reported.

### Citations

1518 |
Iterative methods for sparse linear systems
- Saad
- 2003
(Show Context)
Citation Context ...therto proposed. We present a general framework for inexact Krylov subspace methods, including FOM (or CG), Lanczos, MINRES, GMRES, and QMR. For general description of these methods, see, e.g., [18], =-=[28]-=-. The main results of this paper weresrst presented in [34] and were developed independently of similar results in [35], where another point of view is adopted. In the next section, we begin with the ... |

764 |
The Symmetric Eigenvalue Problem
- Parlett
- 1980
(Show Context)
Citation Context ...hat j (i) k j is small. The components of the eigenvectors 22 V. Simoncini and D. B. Szyld corresponding to the extreme eigenvalues do have a decreasing pattern, at least in the symmetric case [17], [=-=23-=-], and this allows us to eectively relax the accuracy in the computation of the matrix{vector products with A [4]. We note that this fact was already noticed and exploited in the symmetric case in [17... |

538 |
Direct Methods for Sparse Matrices
- DUFF, ERISMAN, et al.
- 1986
(Show Context)
Citation Context ...he magnitude ofsm = kr m ~ r m k; note that for FOM, kV T m r m k sm . 7. Schur complement systems. Block structured linear systems can be solved by using the associated Schur complement; see, e.g., [8], [36]. For instance, if the system stems from a saddle point problem, then the algebraic equation has the following form, S B B T 0 w x = f 0 ; (7.1) with S symmetric. The corresponding (... |

461 |
Domain decomposition: parallel multilevel methods for elliptic partial differential equations
- Bjørstad
- 1996
(Show Context)
Citation Context .... Simoncini and D. B. Szyld There are many scientic applications where the inexact matrix{vector product (1.2) appears naturally. For example, when using approximately a Schur complement [19], [20], [=-=36]-=-, or other situations where the operator in question implies a solution of a linear system, such as in certain eigenvalue algorithms [17], [31]. Other examples include cases when the matrix is very la... |

427 |
Numerical Methods for Large Eigenvalue Problems
- Saad
- 1992
(Show Context)
Citation Context ...also in the eigenvalue context. The results here though are less general than in the linear system setting. If the exact Arnoldi method is employed to approximate the eigenpairs of A (see, e.g., [1], =-=[26-=-]), then starting with a unit norm vector v 1 , the Krylov subspace Km (A; v 1 ) is constructed and ( ~ i ; Vm y i ), i = 1; : : : ; m, are approximate (Ritz) eigenpairs to some of the (exact) eigenp... |

324 |
Iterative Methods for Solving Linear Systems
- Greenbaum
- 1997
(Show Context)
Citation Context ...nes hitherto proposed. We present a general framework for inexact Krylov subspace methods, including FOM (or CG), Lanczos, MINRES, GMRES, and QMR. For general description of these methods, see, e.g., =-=[18]-=-, [28]. The main results of this paper weresrst presented in [34] and were developed independently of similar results in [35], where another point of view is adopted. In the next section, we begin wit... |

285 | A Flexible Inner-Outer Preconditioned GMRES Algorithm
- SAAD
- 1993
(Show Context)
Citation Context ... More precisely, at each iteration the operation (9.1) A˜zk, ˜zk ≈P −1 vk, is carried out, where ˜zk can be thought of as some approximation to the solution zk of the linear system Pz = vk, k =1,...,m=-=[28]-=-, [34], [39]. The operation in (9.1) thus replaces the exact preconditioning product AP −1 vk. In the variable preconditioning case, the Arnoldi relation AP −1 Vm = Vm+1Hm is transformed into (9.2) A[... |

234 |
Inexact Newton methods
- Dembo, Eisentat, et al.
- 1982
(Show Context)
Citation Context ...uitive situation is in contrast to other inexact or two-stage methods, where the inner tolerance has to stay at least constant [13], or it needs to decrease as the iterates get closer to the solution =-=[-=-7], [29]. This version is dated 24 April 2002. y Dipartimento di Matematica, Universita di Bologna, and Istituto di Matematica Applicata e Tecnologie Informatiche del CNR, Pavia, Italy (val@dragon.ia... |

114 | Pseudospectra of linear operators
- Trefethen
- 1997
(Show Context)
Citation Context ...m of the matrices as driving the iterative methods can be applied in this context [18], [28]. The large norm of the perturbations considered may also limit the applicability of pseudospectra analysis =-=[39]-=-. 3. Properties of the computed approximate solution. General Krylov subspace methods produce their approximation to the solution of (1.1) by either a minimization (or in some cases quasi-minimization... |

93 |
Vorst. Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide
- Bai, Demmel, et al.
- 2000
(Show Context)
Citation Context ...genvalue computation: when using certain formulations of the inexact shift-and-invert Arnoldi method, a system of the form (10.2) (M + ωK)K −1 x = b needs to be solved at each iteration of the method =-=[1]-=-. If K is nonsingular, the system (10.1) can be transformed into (10.3) (A + ωI)˜x = b, A = MK −1 , ˜x = Kx, yielding a shifted system with coefficient matrix A + ωI that can be solved efficiently wit... |

80 |
der Vorst. Approximate solutions and eigenvalue bounds from Krylov subspaces. Numerical Linear Algebra with Applications
- Paige, Parlett, et al.
- 1995
(Show Context)
Citation Context ... may considerably vary with m, showing that FOM may be very sensitive to the use of an inexact operator A; see, e.g., [6, section 6], for an example where Hm is singular at every other step, and also =-=[22]. In -=-both GMRES and FOM methods, we have determined a condition on kE k k of the type kE k ksm 1 k~r k 1 k " (5.12) which guarantees overall convergence below the given tolerance. In [3] a similar con... |

56 | A theoretical comparison of the Arnoldi and GMRES algorithms - Brown - 1991 |

51 | Inexact preconditioned conjugate gradient method with inner-outer iterations
- GOLUB, YE
- 1999
(Show Context)
Citation Context ...ion of the operator is required. Alternatively, we can think of (1.2) as y = Av + p(v); (1.3) again with p changing from step to step. In the experiments reported in the mentioned papers, and also in =-=[15]-=-, [31], the norm kEk (or kpk) is allowed to grow as the Krylov iteration progresses, without apparent degradation of the convergence of the iterative method. This counter-intuitive situation is in con... |

50 | The Test Matrix Toolbox for Matlab, version 3.0, Numerical Analysis
- Higham
- 1995
(Show Context)
Citation Context ...erent values of ℓm⋆ . Example 5.6. We consider the Grcar matrix of size n = 100, i.e., A is a Toeplitz matrix with minus ones on the subdiagonal, ones on the diagonal, and five superdiagonals of ones =-=[18]-=-. This matrix is known to have sensitive eigenvalues. Note that ‖A‖ ≈4.9985 and σmin(A) ≈ 0.7898. The right-hand side is b = e1. We compare solving with the exact GMRES method and with the inexact sch... |

42 | Flexible conjugate gradients
- Notay
(Show Context)
Citation Context ...n the case of Vm nonorthogonal. In this section, we relate our results to the cases when the columns of Vm in (2.2) are not orthogonal. This is the case, e.g., in inexact or flexible CG methods [14], =-=[22]-=-. In exact CG, the columns of Vm are (implicitly) generated by short recurrences. In inexact CG, short recurrences are used, but the global orthogonality is lost. The assumption V T m Vm = I is used i... |

40 | A multigrid method enhanced by Krylov subspace iteration for discrete Helmholtz equations
- Elman, Ernst, et al.
(Show Context)
Citation Context ... preconditioning is employed, the original system is transformed into AP 1 x = b; x = P 1 x: At each iteration of the solver the application of P 1 is thus required. In several cases, e.g., as in [1=-=0-=-], neither P nor P 1 can be applied exactly, but only through an operator, yielding a variable preconditioning procedure in which a dierent preconditioning operator is applied at each iteration. Often... |

40 |
The convergence of inexact Chebyshev and Richardson iterative methods for solving linear systems
- Golub, Overton
- 1988
(Show Context)
Citation Context ...trix{vector multiplication in iterative methods, some times in the context of small perturbations, and in some other instances allowing for large tolerances (though not letting them grow); see, e.g., =-=[14]-=-, [15], [16], [37]. The problem we consider in this paper is the case in which kEk (or kpk) can be monitored, usually through an additional (inner) tolerance. We are interested in evaluating how large... |

33 |
Block-diagonal and indefinite symmetric preconditioners for mixed finite element formulations
- Perugia, Simoncini
(Show Context)
Citation Context ...se if the Arnoldi algorithm is explicitly used to construct the Krylov subspace. If ‖v‖ ̸= 1, then the bound in (8.3) requires the additional factor ‖v‖ −1 .468 VALERIA SIMONCINI AND DANIEL B. SZYLD =-=[25]-=- for additional experiments. In the data we employ, S has size 1272×1272 while B has dimension 1272 × 816. The Schur complement system is (8.2), with ‖B T S −1 ‖≈ 3.5792·10 3 and σmin(B T S −1 B) ≈ 3.... |

32 |
Inexact Rayleigh quotient-type methods for eigenvalue computations
- Simoncini, Eldén
- 2002
(Show Context)
Citation Context ... the operator is required. Alternatively, we can think of (1.2) as y = Av + p(v); (1.3) again with p changing from step to step. In the experiments reported in the mentioned papers, and also in [15], =-=[31]-=-, the norm kEk (or kpk) is allowed to grow as the Krylov iteration progresses, without apparent degradation of the convergence of the iterative method. This counter-intuitive situation is in contrast ... |

30 |
Fast Iterative Solution of Stabilized Stokes Systems
- Silvester, Wathen
- 1994
(Show Context)
Citation Context ... and we consider a now well-established technique which consists of preconditioning the original structured problem (7.1) with P = D 0 0 B T B ; (8.5) where D is an approximation to S; see, e.g., [3=-=0]-=- and the references given therein. Here we use D = I as in [24]. At each iteration, the application of the exact preconditioner P requires solving a system with coecient matrix B T B, with sparse B, w... |

29 | Geometric aspects of the theory of Krylov subspace methods
- EIERMANN, ERNST
(Show Context)
Citation Context ...ce methods produce their approximation to the solution of (1.1) by either a minimization (or in some cases quasi-minimization) or a projection procedure over a subspace of the form AKm (A; v 1 ); cf. =-=[9]-=-. We show in this section that in the inexact case, as described in section 2, the appropriate subspace is R(Wm ), where Wm = Vm+1Hm , cf. (2.2) or (2.3). Given r 0 = b (A + E 0 )x 0 , the inexact Kry... |

29 |
H-splittings and two-stage iterative methods
- Frommer, Szyld
- 1992
(Show Context)
Citation Context ... degradation of the convergence of the iterative method. This counter-intuitive situation is in contrast to other inexact or two-stage methods, where the inner tolerance has to stay at least constant =-=[1-=-3], or it needs to decrease as the iterates get closer to the solution [7], [29]. This version is dated 24 April 2002. y Dipartimento di Matematica, Universita di Bologna, and Istituto di Matematica ... |

24 | Inexact inverse iteration for generalized eigenvalue problems - Golub, Ye |

22 |
A relaxation strategy for inexact matrix-vector products for Krylov methods
- Bouras, Frayssé
- 2000
(Show Context)
Citation Context ...der the iterative solution of large sparse (symmetric or) nonsymmetric n n linear systems of the form Ax = b (1.1) with a Krylov subspace method. In a series of papers, Bouras, Fraysse, and Giraud [3=-=]-=-, [4], [5], reported experiments in which the matrix{vector multiplication with A (at each step of the Krylov subspace method) is not performed exactly. In other words, given a vector v, the vector y ... |

22 |
A relaxation strategy for inner-outer linear solvers in domain decomposition methods
- Bouras, Frayssé, et al.
- 2000
(Show Context)
Citation Context ...erative solution of large sparse (symmetric or) nonsymmetric n n linear systems of the form Ax = b (1.1) with a Krylov subspace method. In a series of papers, Bouras, Fraysse, and Giraud [3], [4], [5=-=]-=-, reported experiments in which the matrix{vector multiplication with A (at each step of the Krylov subspace method) is not performed exactly. In other words, given a vector v, the vector y ex = Av is... |

21 | Flexible inner-outer Krylov subspace methods
- Simoncini, Szyld
(Show Context)
Citation Context ...ubtle distinction between inexactly preconditioned methods andsexible Krylov subspace methods, where the matrix{vector multiplication is exact, but the preconditioner is allowed to change; see, e.g., =-=[33]-=- and the references given therein. These two ideas can be combined and in fact this is used in [40]. We expand on these concepts in section 8. Matlab notation is used throughout the paper. Given a vec... |

20 |
der Vorst (editors). Templates for the solution of Algebraic Eigenvalue Problems: A Practical Guide
- Bai, Demmel, et al.
- 2000
(Show Context)
Citation Context ...igenvalue computation: when using certain formulations of the inexact shift-and-invert Arnoldi method, a system of the form (M + !K)K 1 x = b; (9.2) needs to be solved at each iteration of the method =-=[1]-=-. If K is nonsingular, the system (9.1) can be transformed into (A + !I)~x = b; A = MK 1 ; ~ x = Kx; (9.3) yielding a shifted system with coecient matrix A+!I that can be solved eciently with a Krylov... |

20 |
Large sparse symmetric eigenvalue problems with homogeneous linear constraints: the Lanczos process with inner–outer iterations
- Golub, Zhang, et al.
- 2000
(Show Context)
Citation Context ...ample, when using approximately a Schur complement [19], [20], [36], or other situations where the operator in question implies a solution of a linear system, such as in certain eigenvalue algorithms =-=[17-=-], [31]. Other examples include cases when the matrix is very large (and/or dense), and a reasonable approximation can be used [2], [11], [12]. Several authors have studied dierent aspects of the use ... |

19 |
Linear algebra methods in a mixed approximation of magnetostatic problems
- PERUGIA, SIMONCINI, et al.
- 1999
(Show Context)
Citation Context ...t methods on small problems. On large and denser problems (e.g., in 3D applications), direct solution of systems with B T B is very time consuming, and approximations to P need to be considered [24], =-=[25]-=-. Here we consider solving with B T B by an iterative method, relaxing the accuracy of the solution as the outer process converges. It is important to notice that for this structured problem, the cond... |

15 |
der Vorst. Numerical methods for the QCD overlap operator: I. signfunction and error bounds
- Lippert, van
(Show Context)
Citation Context ...ion of a linear system, such as in certain eigenvalue algorithms [17], [31]. Other examples include cases when the matrix is very large (and/or dense), and a reasonable approximation can be used [2], =-=[11-=-], [12]. Several authors have studied dierent aspects of the use of inexact matrix{vector multiplication in iterative methods, some times in the context of small perturbations, and in some other insta... |

14 |
Steihaug T.: Inexact Newton methods
- Dembo, Eisenstat
- 1982
(Show Context)
Citation Context ...uitive situation is in contrast to other inexact or two-stage methods, where the inner tolerance has to stay at least constant [12], or it needs to decrease as the iterates get closer to the solution =-=[7]-=-, [30]. There are many scientific applications where the inexact matrix-vector product (1.2) appears naturally. For example, when operating with A implies a solution of a linear system, as is the case... |

12 | Preconditioning a mixed discontinuous finite element method for radiation diffusion. Numerical Linear Algebra with Applications
- Warsa, Benzi, et al.
- 2004
(Show Context)
Citation Context ...ner) tolerance. We are interested in evaluating how large ‖E‖ can be at each step while still achieving convergence of the Krylov subspace method to the sought-after solution x. In [3], [4], [5], and =-=[41]-=-, ad hoc criteria were used to determine an appropriate inner tolerance. In this paper, we address essentially three questions: 1. What are the variational properties of the computed approximate solut... |

10 |
On the numerical solution of (λ 2 A + λB + C)x = b and application to structural dynamics
- Simoncini, Perotti
(Show Context)
Citation Context ...) becomes ‖Ek‖ ≤‖p (k) j ‖/‖vk‖. 10. Linear systems with a parameter. Consider the linear system (10.1) (M + ωK)x = b, which needs to be solved for several values of the parameter ω; see, e.g., [21], =-=[33]-=-. A similar problem arises in the context of eigenvalue computation: when using certain formulations of the inexact shift-and-invert Arnoldi method, a system of the form (10.2) (M + ωK)K −1 x = b need... |

9 | A relaxation strategy for the Arnoldi method in eigenproblems
- Bouras, Frayssé
- 2000
(Show Context)
Citation Context ...ence history and other norms. Left: Inner stopping criterion with ` = 1. Right: Inner stopping criterion with ` = 1=m? . 10. Application to eigenvalue computations. Following the empirical results in =-=[4]-=-, in this section we show that inexactness of the computed matrix{vector multiplication can be monitored also in the eigenvalue context. The results here though are less general than in the linear sys... |

8 |
A parallel SSOR preconditioner for lattice QCD
- Lippert, Schilling
- 1996
(Show Context)
Citation Context ... a linear system, such as in certain eigenvalue algorithms [17], [31]. Other examples include cases when the matrix is very large (and/or dense), and a reasonable approximation can be used [2], [11], =-=[12-=-]. Several authors have studied dierent aspects of the use of inexact matrix{vector multiplication in iterative methods, some times in the context of small perturbations, and in some other instances a... |

6 | On Newton-iterative methods for the solution of systems of nonlinear equations - SHERMAN - 1978 |

5 | On block diagonal and Schur complement preconditioning
- Mandel
- 1990
(Show Context)
Citation Context ....edu). 1 2 V. Simoncini and D. B. Szyld There are many scientic applications where the inexact matrix{vector product (1.2) appears naturally. For example, when using approximately a Schur complement [=-=19]-=-, [20], [36], or other situations where the operator in question implies a solution of a linear system, such as in certain eigenvalue algorithms [17], [31]. Other examples include cases when the matri... |

5 |
A flexible inner–outer preconditioned GMRES
- Saad
- 1993
(Show Context)
Citation Context ...each iteration the operation y = A~z k ; ~ z k P 1 v k (8.1) is carried out, where ~ z k can be thought of as some approximation to the solution z k of the linear system Pz = v k , k = 1; : : : ; m [=-=27]-=-, [33], [38]. The operation in (8.1) thus replaces the exact preconditioning product y = AP 1 v k . In the variable preconditioning case, the Arnoldi relation AP 1 Vm = Vm+1Hm is transformed into A[~z... |

5 |
Flexible inner-outer Krylov methods (and inexact Krylov methods). presentation
- Simoncini, Szyld
- 2002
(Show Context)
Citation Context ... Krylov subspace methods, including FOM (or CG), Lanczos, MINRES, GMRES, and QMR. For general description of these methods, see, e.g., [18], [28]. The main results of this paper weresrst presented in =-=[34]-=- and were developed independently of similar results in [35], where another point of view is adopted. In the next section, we begin with the analysis of the inexact Krylov subspace methods, setting th... |

5 |
A flexible quasi-minimal residual method with inexact preconditioning
- Szyld, Vogel
- 2001
(Show Context)
Citation Context ...on the operation y = A~z k ; ~ z k P 1 v k (8.1) is carried out, where ~ z k can be thought of as some approximation to the solution z k of the linear system Pz = v k , k = 1; : : : ; m [27], [33], [=-=-=-38]. The operation in (8.1) thus replaces the exact preconditioning product y = AP 1 v k . In the variable preconditioning case, the Arnoldi relation AP 1 Vm = Vm+1Hm is transformed into A[~z 1 ; ;... |

5 |
The solution of parametrized symmetric linear systems
- Meerbergen
(Show Context)
Citation Context ..., (9.8) becomes ‖Ek‖ ≤‖p (k) j ‖/‖vk‖. 10. Linear systems with a parameter. Consider the linear system (10.1) (M + ωK)x = b, which needs to be solved for several values of the parameter ω; see, e.g., =-=[21]-=-, [33]. A similar problem arises in the context of eigenvalue computation: when using certain formulations of the inexact shift-and-invert Arnoldi method, a system of the form (10.2) (M + ωK)K −1 x = ... |

4 | Schur complement systems in the mixedhybrid nite element approximation of the potential problem
- Maryska, Rozloznk, et al.
- 2000
(Show Context)
Citation Context ... 1 2 V. Simoncini and D. B. Szyld There are many scientic applications where the inexact matrix{vector product (1.2) appears naturally. For example, when using approximately a Schur complement [19], [=-=20]-=-, [36], or other situations where the operator in question implies a solution of a linear system, such as in certain eigenvalue algorithms [17], [31]. Other examples include cases when the matrix is v... |

4 |
Block-diagonal and inde symmetric preconditioners for mixed element formulations
- Perugia, Simoncini
- 2000
(Show Context)
Citation Context ... the basis vectors, so as to work with a banded b Hm . Example 7.2. We consider a problem of the form (7.1) derived from a 2D saddle point magnetostatic problem described in [25, section 3]; see also =-=-=-[24] for additional experiments. In the data we employ, S has size 1272 1272 while B has dimension 1272 816. The Schur complement system is B T S 1 Bx = B T S 1 f , with kB T S 1 k 3:5792 10 3 and... |

4 |
den Eshof. Inexact Krylov subspace methods for linear systems
- Sleijpen, van
- 2005
(Show Context)
Citation Context ...NRES, GMRES, and QMR. For general description of these methods, see, e.g., [18], [28]. The main results of this paper weresrst presented in [34] and were developed independently of similar results in =-=[35]-=-, where another point of view is adopted. In the next section, we begin with the analysis of the inexact Krylov subspace methods, setting the stage for our answers to the questions stated above. In se... |

4 | Backward error bounds for approximate Krylov subspaces, Numerical Linear Algebra With Applications 340
- Stewart
- 2002
(Show Context)
Citation Context ...lication in iterative methods, some times in the context of small perturbations, and in some other instances allowing for large tolerances (though not letting them grow); see, e.g., [14], [15], [16], =-=[37]-=-. The problem we consider in this paper is the case in which kEk (or kpk) can be monitored, usually through an additional (inner) tolerance. We are interested in evaluating how large kEk can be while ... |

3 |
Schur complement systems in the mixed-hybrid finite element approximation of the potential fluid flow problem
- Maryˇska, Rozloˇzník, et al.
(Show Context)
Citation Context ... the inexact matrix-vector product (1.2) appears naturally. For example, when operating with A implies a solution of a linear system, as is the case in Schur complement computations (see, e.g., [19], =-=[20]-=-, [37]), and in certain eigenvalue algorithms [16], [32], or when the matrix is very large ∗Received by the editors April 25, 2002; accepted for publication (in revised form) February 9, 2003; publish... |

1 |
Preconditioning of improved and `perfect' actions. Computer Phys. Comm
- Lippert, Schilling, et al.
- 1999
(Show Context)
Citation Context ...solution of a linear system, such as in certain eigenvalue algorithms [17], [31]. Other examples include cases when the matrix is very large (and/or dense), and a reasonable approximation can be used =-=[2-=-], [11], [12]. Several authors have studied dierent aspects of the use of inexact matrix{vector multiplication in iterative methods, some times in the context of small perturbations, and in some other... |

1 |
The solution of Parametrized Linear Systems from Mechanical Systems. Part I: Linear parameters
- Meerbergen
- 2000
(Show Context)
Citation Context ... k k. Inexact Krylov Subspace Methods 19 9. Linear systems with a parameter. Consider the linear system (M + !K)x = b; (9.1) which needs to be solved for several values of the parameter !; see, e.g., =-=[21]-=-, [32]. A similar problem arises in the context of eigenvalue computation: when using certain formulations of the inexact shift-and-invert Arnoldi method, a system of the form (M + !K)K 1 x = b; (9.2)... |

1 |
On the numerical solution of ( A+ B + C)x = b and application to structural dynamics
- Simoncini, Perotti
- 2002
(Show Context)
Citation Context ...Inexact Krylov Subspace Methods 19 9. Linear systems with a parameter. Consider the linear system (M + !K)x = b; (9.1) which needs to be solved for several values of the parameter !; see, e.g., [21], =-=[32]-=-. A similar problem arises in the context of eigenvalue computation: when using certain formulations of the inexact shift-and-invert Arnoldi method, a system of the form (M + !K)K 1 x = b; (9.2) needs... |

1 |
Preconditioning a mixed discontinuous element method for radiation diusion
- Warsa, Benzi, et al.
- 2001
(Show Context)
Citation Context ...tional (inner) tolerance. We are interested in evaluating how large kEk can be while still achieving convergence of the Krylov subspace method to the sought after solution x. In [3], [4], [5], and in =-=[40]-=-, ad-hoc criteria were used to determine an appropriate inner tolerance. In these papers, and also in [15], [16], the Krylov subspace method is implemented as if the matrix{vector multiplication were ... |