## Recycling Krylov Subspaces for Sequences of Linear Systems (2004)

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Venue: | SIAM J. Sci. Comput |

Citations: | 50 - 3 self |

### BibTeX

@TECHREPORT{Parks04recyclingkrylov,

author = {Michael L. Parks and Eric De Sturler and Greg Mackey and Duane D. Johnson and Spandan Maiti},

title = {Recycling Krylov Subspaces for Sequences of Linear Systems},

institution = {SIAM J. Sci. Comput},

year = {2004}

}

### Years of Citing Articles

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### Abstract

Many problems in engineering and physics require the solution of a large sequence of linear systems. We can reduce the cost of solving subsequent systems in the sequence by recycling information from previous systems. We consider two dierent approaches. For several model problems, we demonstrate that we can reduce the iteration count required to solve a linear system by a factor of two. We consider both Hermitian and non-Hermitian problems, and present numerical experiments to illustrate the eects of subspace recycling.

### Citations

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Citation Context ...ng the dominant invariant subspace by retaining all previously generated complete Krylov spaces [25, 26]. Moreover, both approaches use full recurrences, so the CG iteration is really a FOM iteration =-=[27]-=-. Clearly, both in memory and floating point operations, these methods are very expensive. This drawback is somewhat alleviated as the methods are presented in the context of the finite element tearin... |

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Citation Context ...n 4, we give the experimental results, which show that recycling can be very benecial. Conclusions and future work are given in section 5. 2. Truncated and Augmented Krylov Methods. Restarting GMRES [=-=28]-=- may lead to poor convergence and even stagnation. Therefore, recent research has focused on truncated methods that improve convergence by retaining a carefully selected subspace between cycles. A tax... |

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Citation Context ...an be signicant. Solving a sequence of linear systems where the matrix is invariant is a special case of (1.1). When all right hand sides are available simultaneously, block methods such as block CG [=-=23]-=-, block GMRES [34], and the family of block EN-like methods [35] are often suitable. However, block methods do not generalize to the case (1.1). If only one right hand side is available at a time, the... |

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Citation Context ...this section, we discuss those choices and solvers implementing them. We then investigate how those solvers might be modied to recycle subspaces between linear systems. Morgan's GMRES-DR and GMRES-E [=-=20]-=- retain an approximately invariant subspace between cycles. In particular, both methods focus on removing the eigenvalues of smallest magnitude, and retain a subspace spanned by approximate eigenvecto... |

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Citation Context ... case (1.1). If only one right hand side is available at a time, the method of Fischer [12], the de ated conjugate gradient method (de ated CG) [29], or the hybrid method of Simoncini and Gallopoulos =-=[30]-=- may be employed. Fischer's methodsrst looks for a solution in the space spanned by the previous solution vectors in the sequence, which is only helpful if the solution vectors are correlated. In de a... |

56 |
Matrix Algorithms, Volume II: Eigensystems
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Citation Context ... satisfied. 3. Convergence Analysis for Deflation-Based Krylov Subspace Recycling. Recent work on the convergence of GMRES [31] together with the theory on invariant subspaces and their perturbations =-=[34]-=- provides a good framework to analyze the GCRO-DR method. Unfortunately, a similar convergence theory for the GCROT method is still lacking. However, in section 4 we show by numerical experiment that,... |

54 | Adaptively preconditioned GMRES algorithms
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Citation Context ... over the recycled subspace, and then maintain orthogonality with the image of this space in the Arnoldi recurrence. In a preconditioning approach, we construct preconditioners that shift eigenvalues =-=[1, 10]-=-. When using exactly invariant subspaces, an augmentation approach is superior to a preconditioning approach [8]. Hence, we consider only the augmentation and orthogonalization approaches. In secton 2... |

48 | Restarted GMRES preconditioned by deflation
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Citation Context ... over the recycled subspace, and then maintain orthogonality with the image of this space in the Arnoldi recurrence. In a preconditioning approach, we construct preconditioners that shift eigenvalues =-=[1, 10]-=-. When using exactly invariant subspaces, an augmentation approach is superior to a preconditioning approach [8]. Hence, we consider only the augmentation and orthogonalization approaches. In secton 2... |

45 |
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Citation Context ...Solving a sequence of linear systems where the matrix is invariant is a special case of (1.1). When all right hand sides are available simultaneously, block methods such as block CG [23], block GMRES =-=[34]-=-, and the family of block EN-like methods [35] are often suitable. However, block methods do not generalize to the case (1.1). If only one right hand side is available at a time, the method of Fischer... |

44 | Dynamic Thick Restarting of the Davidson, and the Implicitly Restarted Arnoldi Methods
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Citation Context ...parameterized medium in a tomography application to fit measured data. Finally, we note that GCRO-DR uses more or less the Arnoldi method with dense restarting for approximating an invariant subspace =-=[33, 38]-=-. This method generally offers fast convergence for the exterior components of the spectrum. This fast convergence and Theorems 3.1 and 3.2 together indicate that GCRO-DR satisfies the three important... |

44 | GMRES with Deflated Restarting
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Citation Context ... advancement with respect to related work in this area. We consider two approaches for the solution of (1.1) which are related to two existing truncated and restarted solvers. These solvers, GMRES-DR =-=[21]-=- and GCROT [7], were developed for solving single linear systems; both recycle a judiciously selected subspace between restarts to maintain good convergence. In the following, we define a truncation i... |

42 | A deflated version of the conjugate gradient algorithm
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(Show Context)
Citation Context ... [39]. However, block methods do not generalize to the case (1.1). If only one right-hand side is available at a time, the method of Fischer [11], the deflated conjugate gradient method (deflated CG) =-=[29]-=-, or the hybrid method of Simoncini and Gallopoulos [30] may be employed. Fischer’s method looks for a starting vector in the space spanned by the previous solution vectors in the sequence, which is h... |

42 | Truncation strategies for optimal Krylov subspace methods
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Citation Context ...th respect to related work in this area. We consider two approaches for the solution of (1.1) which are related to two existing truncated and restarted solvers. These solvers, GMRES-DR [21] and GCROT =-=[7]-=-, were developed for solving single linear systems; both recycle a judiciously selected subspace between restarts to maintain good convergence. In the following, we define a truncation in the sense us... |

38 |
H.: Thick-restart Lanczos method for large symmetric eigenvalue problems
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Citation Context ...parameterized medium in a tomography application to fit measured data. Finally, we note that GCRO-DR uses more or less the Arnoldi method with dense restarting for approximating an invariant subspace =-=[33, 38]-=-. This method generally offers fast convergence for the exterior components of the spectrum. This fast convergence and Theorems 3.1 and 3.2 together indicate that GCRO-DR satisfies the three important... |

36 | O.: Analysis of acceleration strategies for restarted minimal residual methods
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Citation Context ... In a preconditioning approach, we construct preconditioners that shift eigenvalues [1, 10]. When using exactly invariant subspaces, an augmentation approach is superior to a preconditioning approach =-=[8]-=-. Hence, we consider only the augmentation and orthogonalization approaches. In secton 2, we discuss several truncated or restarted linear solvers that use the ideas above to reduce the total number o... |

34 | Implicitly restarted GMRES and Arnoldi methods for nonsymmetric linear systems of equations
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Citation Context ...t the end of the previous cycle. For the first cycle, the harmonic Ritz vectors can be computed from Hm in (2.1). It can be shown that these harmonic Ritz vectors fit naturally into a Krylov subspace =-=[20]-=-. In each cycle, GMRES-DR proceeds by first orthogonalizing � Yk to give � Υk. GMRES-DR then carries out the Arnoldi recurrence for m − k iterations while maintaining orthogonality to � Υk. This gives... |

29 |
Analysis of augmented krylov subspace methods
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Citation Context ...ogonalization, preconditioning. In an augmentation approach, we append additional vectors at the end of the Arnoldi recurrence, in the manner of FGMRES, such that an Arnoldi-like relation is formed [=-=27]-=-. In an orthogonalization approach, wesrst minimize the residual over the recycled subspace, and then maintain orthogonality with the image of this space in the Arnoldi recurrence. In a preconditionin... |

26 | Convergence of restarted Krylov subspaces to invariant subspaces
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Citation Context ...tor onto Q. Finally, we define the one-sided distance from the subspace Q to the subspace C as δ (Q, C) ≡�(I − ΠC)ΠQ�2, (3.1) which is equal to the sine of the largest principal angle between Q and C =-=[1]-=-. This means that any unit vector in Q has a component of at most length δ orthogonal to C. Theorem 3.1. Given a space C, letV = range � � Vm−k+1H m−k be the (m − k)dimensional Krylov subspace generat... |

24 |
Projection techniques for iterative solution of Ax b with successive right-hand sides
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Citation Context ... and the family of block EN-like methods [35] are often suitable. However, block methods do not generalize to the case (1.1). If only one right hand side is available at a time, the method of Fischer =-=[12]-=-, the de ated conjugate gradient method (de ated CG) [29], or the hybrid method of Simoncini and Gallopoulos [30] may be employed. Fischer's methodsrst looks for a solution in the space spanned by the... |

22 |
Solution of the Schrödinger Equation in Periodic Lattices with an Application to Metallic Lithium,” Phys
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Citation Context ...-structure calculations based on the Schrodinger equation are used to predict key physical properties of materials systems with a large number of atoms. We consider systems arising in the KKR method [=-=17, 16-=-]. For an electron that is not scattered going from atom i to atom j, the Green's function solution is the structural Green's function G 0 (r i ; r j ; E) = e i p Ejr i r j j 4jr i r j j ; where r i a... |

22 |
The superlinear convergence behaviour of
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Citation Context ... On the right sides, the first term represents the convergence of a deflated problem where all components in the subspace Q have been removed, which typically leads to an improved rate of convergence =-=[21, 31, 36]-=-. The second term in the right sides represents a constant times the residual of m − k iterations of GCRO-DR solving for r1. If the recycle space C contains an invariant subspace Q, then δ = γ = 0 for... |

20 | On the occurrence of superlinear convergence of exact and inexact Krylov subspace methods
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Citation Context ...; the cost savings in GCRO-DR(m, k) arise because (2.5) and (2.6) are already satisfied. 3. Convergence analysis for deflation-based Krylov subspace recycling. Recent work on the convergence of GMRES =-=[31]-=- together with the theory on invariant subspaces and their perturbations [34] provides a good framework for analyzing the GCRO-DR method. Unfortunately, a similar convergence theory for the GCROT meth... |

19 |
der Vorst, Numerical methods for the QCD overlap operator: I. Signfunction and error bounds
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Citation Context ...undamental theory describing the strong interaction between quarks and gluons. Numerical simulations of QCD on a four-dimensional space-time lattice are considered the only way to solve QCD ab initio =-=[4,-=- 33]. As the problem has a 12 12 block structure, we are often interested in solving for 12 right hand sides related to a single lattice site. The linear system to be solved is (I D)x = b with 0 sc ... |

19 |
der Vorst. Numerical methods for the qcd overlap operator. i: Signfunction and error bounds
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Citation Context ...undamental theory describing the strong interaction between quarks and gluons. Numerical simulations of QCD on a four-dimensional space-time lattice are considered the only way to solve QCD ab initio =-=[4, 35]-=-. As the problem has a 12 × 12 block structure, we are often interested in solving for 12 right-hand sides related to a single lattice site. The linear system to be solved is (I − κD)x = b with 0 ≤ κ<... |

15 | Galerkin projection methods for solving multiple linear systems
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Citation Context ... of a drawback, because the interface problem is small relative to the overall problem, and it is common to use a full recurrence in FETI. The two Galerkin projection methods developed by Chan and Ng =-=[3]-=- could also be used. These methods require all systems to be available simultaneously, or at least the right hand sides. Moreover, they focus on situations where all the matrices are very close. Howev... |

15 | Sturler, Recycling subspace information for diffuse optical tomography
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Citation Context ...The proposed methods can be tuned to recycle a variety of subspaces based on computed information from the matrix, background knowledge of the application, and other information; this is discussed in =-=[15]-=-. However, for a first evaluation,sKRYLOV SUBSPACE RECYCLING 3 it is reasonable to analyze the effectiveness of these existing methods appropriately modified for solving (1.1). The application of thes... |

14 |
Nested Krylov methods based on GCR
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Citation Context ...pace recycling, even when the matrix does not change. We discuss GMRES-E and GMRES-DR further in section 2.4. Because GMRES-DR cannot be used for Krylov subspace recycling, we combine ideas from GCRO =-=[5]-=- and GMRES-DR to produce a new linear solver, GCRO-DR. GCRO-DR is suitable for the solution of individual linear systems as well as sequences of them, and is moresexible than GMRES-DR. We discuss GCRO... |

13 |
On the calculation of the energy of a Bloch wave in a metal
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Citation Context ...-structure calculations based on the Schrodinger equation are used to predict key physical properties of materials systems with a large number of atoms. We consider systems arising in the KKR method [=-=17, 16-=-]. For an electron that is not scattered going from atom i to atom j, the Green's function solution is the structural Green's function G 0 (r i ; r j ; E) = e i p Ejr i r j j 4jr i r j j ; where r i a... |

13 |
Nested Krylov methods based on
- Sturler
- 1996
(Show Context)
Citation Context ...alues. Note that GMRES-DR must use only harmonic Ritz vectors, and that it cannot be modified for Krylov subspace recycling even when the matrix does not change. Therefore, we combine ideas from GCRO =-=[6]-=- and GMRES-DR to produce a new linear solver, GCRO-DR, which is suitable for the solution of individual linear systems as well as sequences of them, and is more flexible than GMRES-DR. We discuss GCRO... |

12 |
Iterative accelerating algorithms with krylov subspaces for the solution to large-scale nonlinear problems
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Citation Context ...imate invariant subspace and use it for deflation, following the GMRES-DR method. Since GMRES-DR cannot be adapted for recycling, we propose a more general method, GCRO-DR. The work by Rey and Risler =-=[25, 26]-=- has a similar motivation as GMRES-DR; see below. However, our implementation is cheaper, more effective, and more adaptive. An alternative idea is to recycle a subspace that minimizes the loss of ort... |

10 |
A de version of the Conjugate Gradient algorithm
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Citation Context ...uitable. However, block methods do not generalize to the case (1.1). If only one right hand side is available at a time, the method of Fischer [12], the de ated conjugate gradient method (de ated CG) =-=[29]-=-, or the hybrid method of Simoncini and Gallopoulos [30] may be employed. Fischer's methodsrst looks for a solution in the space spanned by the previous solution vectors in the sequence, which is only... |

7 |
Theory and convergence properties of the screened Korringa-Kohn-Rostoker method, Phys
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Citation Context ...bove can be inverted very rapidly. The second requires the inversion of a sparse, complex, non-Hermitian matrix, where the relative number of nonzeros in the matrix decreases with the number of atoms =-=[15, 36, 32]-=-. We give results in Section 4.2, using a model problem provided by Duane Johnson (Materials Science, UIUC) and Andrei Smirnov (Oak Ridge National Laboratory). Only the block-diagonal elements (corres... |

7 |
Inner-outer methods with deflation for linear systems with multiple right-hand sides
- Sturler
(Show Context)
Citation Context ...hat significant convergence improvements are obtained using recycled subspaces of a small dimension. The application of an early version of GCROT to multiple right-hand sides was briefly discussed in =-=[5]-=-. The tuning of subspace recycling for diffuse optical tomography, leading to further convergence improvements, is discussed and relevant theory presented in [15]. We discuss the basic derivation of o... |

6 |
A Rayleigh-Ritz preconditioner for the iterative solution to large scale nonlinear problems
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- 1998
(Show Context)
Citation Context ...ace recycling. For the Hermitian positive denite case, Rey and Risler have proposed to reduce the eective condition number by retaining all converged Ritz vectors arising in a previous CG iteration [2=-=4, 25, 26-=-]. In general, this requires signicant storage. Moreover, memory-wise, they lose the advantage of a short recurrence, as they keep the full recurrence during the solution of a single system. Since the... |

6 |
On the reuse of Ritz vectors for the solution to nonlinear elasticity problems by domain decomposition methods
- Risler, Rey
- 1998
(Show Context)
Citation Context ...ace recycling. For the Hermitian positive denite case, Rey and Risler have proposed to reduce the eective condition number by retaining all converged Ritz vectors arising in a previous CG iteration [2=-=4, 25, 26-=-]. In general, this requires signicant storage. Moreover, memory-wise, they lose the advantage of a short recurrence, as they keep the full recurrence during the solution of a single system. Since the... |

6 |
with deflated restarting
- GMRES
(Show Context)
Citation Context ...t advancement with respect to related work in this area. We consider two approaches for the solution of (1.1) that are related to two existing truncated and restarted solvers. These solvers, GMRES-DR =-=[21]-=- and GCROT [7], were developed for solving single linear systems; both recycle a judiciously selected subspace between restarts to maintain good convergence. In the following, we define a truncation i... |

5 |
Energy and pressure calculations for random substitutional alloys
- Johnson, Nicholson, et al.
- 1990
(Show Context)
Citation Context ...bove can be inverted very rapidly. The second requires the inversion of a sparse, complex, non-Hermitian matrix, where the relative number of nonzeros in the matrix decreases with the number of atoms =-=[15, 36, 32]-=-. We give results in Section 4.2, using a model problem provided by Duane Johnson (Materials Science, UIUC) and Andrei Smirnov (Oak Ridge National Laboratory). Only the block-diagonal elements (corres... |

5 |
restarted GMRES and Arnoldi methods for nonsymmetric systems of equations
- Implicitly
(Show Context)
Citation Context ...ation, rather than orthogonality constraints. Removing or de ating certain eigenvalues can greatly improve convergence. Based on this idea, Morgan has proposed three linear solvers (GMRES-E, GMRES-IR =-=[21]-=-, and GMRES-DR) that aim to de ate the eigenvalues of smallest magnitude. However, these solvers can be changed to de ate other eigenvalues. We consider only GMRES-E and GMRES-DR. 6 PARKS, DE STURLER,... |

5 | On the superlinear convergence of exact and inexact krylov subspace methods
- Simoncini, Szyld
- 2003
(Show Context)
Citation Context ...riant Subspaces. When recycling nearly invariant subspaces, we show a residual bound demonstrating improved convergence under certain assumptions. The following theorem is adapted to our purpose from =-=[31-=-], which was in turn inspired by [27]. Theorem 2.1. Let range(Q k ) be a k-dimensional invariant subspace of A 2 C nn . Let PQ be the spectral projector onto range(Q k ). Let range(Y k ) be a kdimensi... |

5 |
Accuracy and limitations of localized Green’s function methods for materials science applications
- Smirnov, Johnson
(Show Context)
Citation Context ...bove can be inverted very rapidly. The second requires the inversion of a sparse, complex, non-Hermitian matrix, where the relative number of nonzeros in the matrix decreases with the number of atoms =-=[15, 36, 32]-=-. We give results in Section 4.2, using a model problem provided by Duane Johnson (Materials Science, UIUC) and Andrei Smirnov (Oak Ridge National Laboratory). Only the block-diagonal elements (corres... |

4 |
MPI-based implementation of a PCG solver using an EBE architecture and preconditioner for implicit, 3-D finite element analyses
- Gullerud, Dodds
(Show Context)
Citation Context ...solving nonlinear equations. They also occur in modeling fatigue and fracture viasnite element analysis. These analyses use dynamic loading, requiring many loading steps, and rely on implicit solvers =-=[14]-=-. Generally, several thousand loading increments are required to resolve the fracture progression. The matrix and right hand side, at each loading step, depend on the previous solution, so that only o... |

3 |
Restarted GMRES preconditioned by de
- Erhel, Burrage, et al.
- 1996
(Show Context)
Citation Context ... over the recycled subspace, and then maintain orthogonality with the image of this space in the Arnoldi recurrence. In a preconditioning approach, we construct preconditioners that shift eigenvalues =-=[1, 10]-=-. When using exactly invariant subspaces, an augmentation approach is superior to a preconditioning approach [8]. Hence, we consider only the augmentation and orthogonalization approaches. In secton 2... |

3 |
Reusing Krylov subspaces for sequences of linear systems
- Mackey
- 2003
(Show Context)
Citation Context ...led optimal truncation. We discuss the idea of optimal truncation in the context of restarted GMRES, although it can be described in more general terms, and independently of any specic linear solver [=-=6, 18]-=-. Consider solving Ax = b with initial residual r 0 . The idea is to determine, after each cycle, a subspace to retain for the next cycle in order to maintain good convergence after the restart. At th... |

3 |
Set QCD: Quantum Chromodynamics. Description of matrix set
- Medeke
(Show Context)
Citation Context ...e linear system to be solved is (I D)x = b with 0 sc , where D is a sparse, complex, non-Hermitian matrix representing periodic nearest neighbor coupling on the fourdimensional space-time lattice [19]. For = c the system becomes singular. The physically interesting case is for close to c ; c depends on D. We present results in Section 4.3. 3.4. Convection Diusion. We consider thesnite d... |

3 | A new family of block methods
- Yang, Gallivan
- 1999
(Show Context)
Citation Context ... matrix is invariant is a special case of (1.1). When all right hand sides are available simultaneously, block methods such as block CG [23], block GMRES [34], and the family of block EN-like methods =-=[35]-=- are often suitable. However, block methods do not generalize to the case (1.1). If only one right hand side is available at a time, the method of Fischer [12], the de ated conjugate gradient method (... |

2 | The Iterative Solution of a Sequence of Linear Systems Arising from Nonlinear Finite Element Analysis
- Parks
- 2005
(Show Context)
Citation Context ... of the present paper, instead we discuss two main theoretical results and their implications and demonstrate these numerically in section 4. For more details on these theoretical results we refer to =-=[15, 23, 24]-=-. The first result concerns the convergence of GCRO-DR; see [23, 24]. We show that the recycle space need not approximate an invariant subspace accurately to improve the rate of convergence significan... |