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  • Articles  (21)
  • CR: G1.3  (15)
  • 65N20  (8)
  • Springer  (21)
  • American Meteorological Society
  • Blackwell Publishing Ltd
  • Copernicus
  • Hindawi
  • Institute of Electrical and Electronics Engineers
  • Molecular Diversity Preservation International
  • Springer Science + Business Media
  • 2020-2022
  • 2010-2014
  • 1985-1989  (21)
  • 1960-1964
  • 1989  (21)
  • Mathematics  (21)
  • Geography
  • Architecture, Civil Engineering, Surveying
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  • Articles  (21)
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  • Springer  (21)
  • American Meteorological Society
  • Blackwell Publishing Ltd
  • Copernicus
  • Hindawi
    • +
Years
  • 2020-2022
  • 2010-2014
  • 1985-1989  (21)
  • 1960-1964
Year
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  • Mathematics  (21)
  • Geography
  • Architecture, Civil Engineering, Surveying
  • Computer Science  (1)
  • 1
    Electronic Resource
    Electronic Resource
    On truncated incomplete decompositions (1989)
    Springer
    BIT 29 (1989), S. 719-740 
    Details
    ISSN: 1572-9125
    Keywords: 65N20 ; Incomplete decompositions ; preconditioning ; smoothing ; multi-grid method ; smoothing property ; robustness
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In the present paper we introduce truncated incomplete decompositions (TrILU) for constant coefficient matrices. This new ILU variant saves most of the memory and work usually needed to compute and store the factorization. Further it improves the smoothing and preconditioning properties of standard ILU-decompositions. Besides describing the algorithm, we give theoretical results concerning stability and convergence as well as the smoothing property and robustness for TrILU smoothing in a multi-grid method. Further, we add numerical results of TrILU as smoother in a multi-grid method and as preconditioner in a pcg-method fully confirming the theoretical results.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01932742
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  • 2
    Electronic Resource
    Electronic Resource
    Two-level hierarchically preconditioned conjugate gradient methods for solving linear elasticity finite element equations (1989)
    Springer
    BIT 29 (1989), S. 748-768 
    Details
    ISSN: 1572-9125
    Keywords: 65F10 ; 65N20
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper we study and compare some preconditioned conjugate gradient methods for solving large-scale higher-order finite element schemes approximating two- and three-dimensional linear elasticity boundary value problems. The preconditioners discussed in this paper are derived from hierarchical splitting of the finite element space first proposed by O. Axelsson and I. Gustafsson. We especially focus our attention to the implicit construction of preconditioning operators by means of some fixpoint iteration process including multigrid techniques. Many numerical experiments confirm the efficiency of these preconditioners in comparison with classical direct methods most frequently used in practice up to now.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01932744
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  • 3
    Electronic Resource
    Electronic Resource
    A survey of multilevel preconditioned iterative methods (1989)
    Springer
    BIT 29 (1989), S. 769-793 
    Details
    ISSN: 1572-9125
    Keywords: 65F10 ; 65N20 ; 65N30 ; two-level ; multilevel methods ; optimal preconditioners ; survey
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We survey multilevel iterative methods applied for solving large sparse systems with matrices, which depend on a level parameter, such as arise by the discretization of boundary value problems for partial differential equations when successive refinements of an initial discretization mesh is used to construct a sequence of nested difference or finite element meshes. We discuss various two-level (two-grid) preconditioning techniques, including some for nonsymmetric problems. The generalization of these techniques to the multilevel case is a nontrivial task. We emphasize several ways this can be done including classical multigrid methods and a recently proposed algebraic multilevel preconditioning method. Conditions for which the methods have an optimal order of computational complexity are presented.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01932745
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  • 4
    Electronic Resource
    Electronic Resource
    Block SSOR preconditionings for high order 3d fe systems (1989)
    Springer
    BIT 29 (1989), S. 805-823 
    Details
    ISSN: 1572-9125
    Keywords: 65F10 ; 65N20 ; 3D Navier equations ; high order hierarchical finite elements ; block SSOR preconditioned conjugate gradient method ; stripwise block two-color orderings
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract For solving 3D high order hierarchical FE systems the block SSOR preconditioned CG algorithms based on new stripwise block two-color orderings of degrees of freedom and providing for efficient concurrent/vector implementation are suggested. As demonstrated by numerical results for the 3D Navier equations approximated using hierarchical orderp, 2 ≤p ≤ 5, FE's the convergence rate of such BSSOR-CG algorithms is only slightly dependent onp and mesh nonunformity.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01932747
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  • 5
    Electronic Resource
    Electronic Resource
    Preconditioned conjugate gradient methods for large-scale fluid flow applications (1989)
    Springer
    BIT 29 (1989), S. 850-866 
    Details
    ISSN: 1572-9125
    Keywords: 65F10 ; 65N20
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The simulation of large-scale fluid flow applications often requires the efficient solution of extremely large nonsymmetric linear and nonlinear sparse systems of equations arising from the discretization of systems of partial differential equations. While preconditioned conjugate gradient methods work well for symmetric, positive-definite matrices, other methods are necessary to treat large, nonsymmetric matrices. The applications may also involve highly localized phenomena which can be addressed via local and adaptive grid refinement techniques. These local refinement methods usually cause non-standard grid connections which destroy the bandedness of the matrices and the associated ease of solution and vectorization of the algorithms. The use of preconditioned conjugate gradient or conjugate-gradient-like iterative methods in large-scale reservoir simulation applications is briefly surveyed. Then, some block preconditioning methods for adaptive grid refinement via domain decomposition techniques are presented and compared. These techniques are being used efficiently in existing large-scale simulation codes.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01932749
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  • 6
    Electronic Resource
    Electronic Resource
    Rehabilitation of the Gauss-Jordan algorithm (1989)
    Springer
    Numerische Mathematik 54 (1989), S. 591-599 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F05, 65G05, 15A06 ; CR: G1.3
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In this paper a Gauss-Jordan algorithm with column interchanges is presented and analysed. We show that, in contrast with Gaussian elimination, the Gauss-Jordan algorithm has essentially differing properties when using column interchanges instead of row interchanges for improving the numerical stability. For solutions obtained by Gauss-Jordan with column interchanges, a more satisfactory bound for the residual norm can be given. The analysis gives theoretical evidence that the algorithm yields numerical solutions as good as those obtained by Gaussian elimination and that, in most practical situations, the residuals are equally small. This is confirmed by numerical experiments. Moreover, timing experiments on a Cyber 205 vector computer show that the algorithm presented has good vectorisation properties.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396364
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  • 7
    Electronic Resource
    Electronic Resource
    Homotopy algorithm for symmetric eigenvalue problems (1989)
    Springer
    Numerische Mathematik 55 (1989), S. 265-280 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS) ; 65H10 ; 58C99 ; 55M25 ; CR: G1.3
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The homotopy method can be used to solve eigenvalue-eigenvector problems. The purpose of this paper is to report the numerical experience of the homotopy method of computing eigenpairs for real symmetric tridiagonal matrices together with a couple of new theoretical results. In practice, it is rerely of any interest to compute all the eigenvalues. The homotopy method, having the order preserving property, can provide any specific eigenvalue without calculating any other eigenvalues. Besides this advantage, we note that the homotopy algorithm is to a large degree a parallel algorithm. Numerical experimentation shows that the homotopy method can be very efficient especially for graded matrices.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01390054
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  • 8
    Electronic Resource
    Electronic Resource
    A note on a one-sided Jacobi algorithm (1989)
    Springer
    Numerische Mathematik 56 (1989), S. 627-633 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F15 ; CR: G1.3
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We propose a “one-sided” or “implicit” variant of the Jacobi diagonalization algorithm for positive definite matrices. The variant is based on a previous Cholesky decomposition and currently uses essentially one square array which, on output, contains the matrix of eigenvectors thus reaching the storage economy of the symmetric QL algorithm. The current array is accessed only columnwise which makes the algorithm attractive for various parallelized and/or vectorized implementations. Even on a serial computer our algorithm shows improved efficiency, in particular if the Cholesky step is made with diagonal pivoting. On matrices of ordern=25–200 our algorithm is about 2–3.5 times slower than QL thus being almost on the halfway between the standard Jacobi and QL algorithms. The previous Cholesky decomposition can be performed with higher precision without extra time or storage costs thus offering considerable gains in accuracy with highly conditioned input matrices.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396349
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  • 9
    Electronic Resource
    Electronic Resource
    Tchebychev acceleration technique for large scale nonsymmetric matrices (1989)
    Springer
    Numerische Mathematik 56 (1989), S. 721-734 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F10 ; CR: G1.3
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The acceleration by Tchebychev iteration for solving nonsymmetric eigenvalue problems is dicussed. A simple algorithm is derived to obtain the optimal ellipse which passes through two eigenvalues in a complex plane relative to a reference complex eigenvalue. New criteria are established to identify the optimal ellipse of the eigenspectrum. The algorithm is fast, reliable and does not require a search for all possible ellipses which enclose the spectrum. The procedure is applicable to nonsymmetric linear systems as well.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01405199
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  • 10
    Electronic Resource
    Electronic Resource
    An a posteriori error estimation of the Rayleigh-Ritz eigenvalues (1989)
    Springer
    Numerische Mathematik 54 (1989), S. 639-654 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 35P15, 47A55, 65M15 ; CR: G1.3
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary LetA, B be essentially self-adjoint and positive definite differential operators defined inL 2(G). Using Svirskij's construction of the base operator and some results from the analytic perturbation theory of linear operators a formula providing eigenvalue lower bounds of the problemAu=λBu is derived. In this formula a rough lower bound of some higher eigenvalue and the residual convergence of the Rayleigh-Ritz eigenfunction approximations are needed. Some numerical results are presented.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396487
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