## A New Adaptive Gmres Algorithm For Achieving High Accuracy (0)

Venue: | Numer. Linear Algebra Appl |

Citations: | 5 - 3 self |

### BibTeX

@ARTICLE{Sosonkina_anew,

author = {Maria Sosonkina and Layne T. Watson and Rakesh K. Kapania},

title = {A New Adaptive Gmres Algorithm For Achieving High Accuracy},

journal = {Numer. Linear Algebra Appl},

year = {},

volume = {5},

pages = {275--297}

}

### Years of Citing Articles

### OpenURL

### Abstract

. GMRES(k) is widely used for solving nonsymmetric linear systems. However, it is inadequate either when it converges only for k close to the problem size or when numerical error in the modified Gram-Schmidt process used in the GMRES orthogonalization phase dramatically affects the algorithm performance. An adaptive version of GMRES(k) which tunes the restart value k based on criteria estimating the GMRES convergence rate for the given problem is proposed here. This adaptive GMRES(k) procedure outperforms standard GMRES(k), several other GMRES-like methods, and QMR on actual large scale sparse structural mechanics postbuckling and analog circuit simulation problems. There are some applications, such as homotopy methods for high Reynolds number viscous flows, solid mechanics postbuckling analysis, and analog circuit simulation, where very high accuracy in the linear system solutions is essential. In this context, the modified GramSchmidt process in GMRES can fail causing the entire GMR...

### Citations

337 | AMR: A quasi-minimal residual method for non-hermitian linear systems
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Citation Context ...k): the standard implementation of GMRES(k) [16], the adaptive GMRES(k) with modified GramSchmidt orthogonalization, GMRES(k) with GMRES-ILU preconditioning [15], and the popular iterative method QMR =-=[6]-=-. A brief description of the GMRES(k) with GMRES-ILU preconditioning and QMR algorithms follows. The GMRES(k) with GMRES-ILU preconditioning method is a variation of flexible GMRES developed in [15]. ... |

285 | A Flexible Inner-Outer Preconditioned GMRES Algorithm
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Citation Context ...(k) considers some other versions of GMRES(k): the standard implementation of GMRES(k) [16], the adaptive GMRES(k) with modified GramSchmidt orthogonalization, GMRES(k) with GMRES-ILU preconditioning =-=[15]-=-, and the popular iterative method QMR [6]. A brief description of the GMRES(k) with GMRES-ILU preconditioning and QMR algorithms follows. The GMRES(k) with GMRES-ILU preconditioning method is a varia... |

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Citation Context ...); kEHR k 2su; where u is the machine unit roundoff. Clearly the orthogonalization with Householder reflections is more robust. An implementation of GMRES(k) using Householder reflections is given in =-=[20]-=-. In theory, the implementation of GMRES using Householder reflections is about twice as expensive as when modified Gram-Schmidt is used [21]. However, the Householder reflection method produces a mor... |

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26 |
personal communication
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Citation Context ...tion of GMRES(k) using Householder reflections is given in [20]. In theory, the implementation of GMRES using Householder reflections is about twice as expensive as when modified Gram-Schmidt is used =-=[21]-=-. However, the Householder reflection method produces a more accurate orthogonalization of the Krylov subspace basis when the basis vectors are nearly linearly dependent and the modified Gram-Schmidt ... |

24 | der Vorst and C. Vuik, GMRESR: a family of nested GMRES methods - van - 1994 |

18 |
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Citation Context ...(bgv) of the remaining number of steps allowed (itmax is a bound on the number of steps permitted). Slow progress of GMRES(k) which indicates that an increase in the restart value k may be beneficial =-=[18]-=- can be detected with a similar test. The near-stagnation test uses a different, smaller multiples(smv) of the remaining allowed number of steps. If near-stagnation occurs, the restart value k is incr... |

16 |
type methods for unconstrained and linearly constrained optimization
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Citation Context ...for (i; j) 2 Z; Q ij = A ij ; for (i; j) = 2 Z; i 6= j; Q ii = A ii ; whenever possible. The factors of the preconditioning matrix Q preserve the sparsity pattern of A. The Gill-Murray preconditioner =-=[7]-=- for a matrix A with a skyline sparsity pattern is a positive definite approximation Q = GG T to A. The lower triangular matrix G has the same skyline structure as A, and is chosen such that G is alwa... |

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Citation Context ...gnificant "team players" in a sophisticated problem solving process, the overall success of which largely depends on linear system solutions. In the context of homotopy zero curve tracking a=-=lgorithms [22]-=-, for example, to locate the difficult parts of the curve (sharp turning points) a linear system solution in the correction phase of the stepping algorithm along the curve has to be very accurate. In ... |

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Preconditioned iterative methods for sparse linear algebra problems arising in circuit simulation
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Citation Context ...i fi fi fijjr 0 jje 1 \Gamma H (k) e z fi fi fi fi fi fi ; where H (k) e is a k \Theta k block tridiagonal matrix augmented with a row of the form aee T k . Based on the conclusions of [4], [10], and =-=[12]-=- incomplete LU factorization with zero fill in (ILU(0)) and Gill-Murray preconditioning are considered for nonsymmetric unstructured and symmetric skyline structured matrices, respectively. Let Z ae f... |

5 | Sframe: An efficient system for detailed DC simulation of bipolar analog integrated circuits using continuation methods. Analog Integrated Circuits Signal Process - MELVILLE, MOINIAN, et al. - 1993 |

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Citation Context ...izer of fi fi fi fi fi fijjr 0 jje 1 \Gamma H (k) e z fi fi fi fi fi fi ; where H (k) e is a k \Theta k block tridiagonal matrix augmented with a row of the form aee T k . Based on the conclusions of =-=[4]-=-, [10], and [12] incomplete LU factorization with zero fill in (ILU(0)) and Gill-Murray preconditioning are considered for nonsymmetric unstructured and symmetric skyline structured matrices, respecti... |

3 |
Stability of lattice structures under combined loads
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Citation Context ...outines within the homotopy curve tracking software HOMPACK [24]. 3.2. Space truss stability analysis. The stability characteristics of a space truss under multiple independent loads are described in =-=[9]-=-. A brief derivation of the nonlinear equilibrium equations, whose solution by a homotopy method generates the linear systems to be solved, is given here. Finite element model. The axial deformation o... |

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Citation Context ...igh accuracy solutions is investigated. High accuracy computation is known to be especially important for some circuit design and simulation problems, which involve nonsymmetric unstructured matrices =-=[14]-=-. These problems have proven to be very difficult for any iterative method, and presently, only specially tailored direct methods are used to solve them in a production environment. The impact of nume... |

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orck, Solving linear least squares problems by Gram-Schmidt orthogonalization
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Citation Context ...trix M , and define the error matrix E = Q T Q \Gamma I . The error matrices EMGS , EHR using the modified Gram-Schmidt and Householder reflection methods, respectively, to construct Q from M satisfy =-=[2]-=- kEMGS k 2su cond(M); kEHR k 2su; where u is the machine unit roundoff. Clearly the orthogonalization with Householder reflections is more robust. An implementation of GMRES(k) using Householder refle... |

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Citation Context ...of fi fi fi fi fi fijjr 0 jje 1 \Gamma H (k) e z fi fi fi fi fi fi ; where H (k) e is a k \Theta k block tridiagonal matrix augmented with a row of the form aee T k . Based on the conclusions of [4], =-=[10]-=-, and [12] incomplete LU factorization with zero fill in (ILU(0)) and Gill-Murray preconditioning are considered for nonsymmetric unstructured and symmetric skyline structured matrices, respectively. ... |

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Citation Context ... phenomena, with horizontal and vertical tangents. A finite element model based on a curved quadrilateral 48-degree-of-freedom thin-shell element (Fig. 6) is used to form the equations of equilibrium =-=[11]-=-. The element has four corner nodes and each node has 12 degrees of freedom. Figure 6 shows the undeformed middle surface of the shell element embedded in a fixed Cartesian coordinate system x i (i = ... |

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Tracking nonlinear equilibrium paths by a homotopy method. Nonlinear Anal
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Citation Context ... i=1 K i ; K i = @ 2 i @q k @q l : K i is the tangent stiffness matrix of the element i expressed relative to the generalized displacements of the assemblage. It is computed by the code number method =-=[23]-=- from the global stiffness matrix of element i. Structure 1. Figure 4 shows a 21-degree-of-freedom lamella dome. The joints lie on the surface of a spherical cap with a radius of 157.25 inches. The su... |

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Citation Context ... different, with different buckling loads. 3.3. Shallow shell stability analysis. Another test problem is the finite element analysis of a shallow cylindrical shell (Fig. 5) under a concentrated load =-=[25]-=-. The shell shown in Fig. 5 is hinged at the longitudinal edges and free along the curved boundaries. = 0.1 Rad P 508 mm 6.35 R = 2540 q q q A Fig. 5. Cylindrical shell geometry; load applied at A. Th... |