Library

X
feed icon rss

Your email was sent successfully. Check your inbox.

An error occurred while sending the email. Please try again.

Proceed reservation?

Export
  • 1
    Electronic Resource
    Electronic Resource
    On the convergence of line iterative methods for cyclically reduced non-symmetrizable linear systems (1994)
    Springer
    Numerische Mathematik 67 (1994), S. 177-190 
    Details
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991): 65F10
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary. We derive analytic bounds on the convergence factors associated with block relaxation methods for solving the discrete two-dimensional convection-diffusion equation. The analysis applies to the reduced systems derived when one step of block Gaussian elimination is performed on red-black ordered two-cyclic discretizations. We consider the case where centered finite difference discretization is used and one cell Reynolds number is less than one in absolute value and the other is greater than one. It is shown that line ordered relaxation exhibits very fast rates of convergence.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002110050023
    Library Location Call Number Volume/Issue/Year Availability
    BibTip Others were also interested in ...
    Paper (German National Licenses)
    Fulltext
  • 2
    Electronic Resource
    Electronic Resource
    A hybrid Arnoldi-Faber iterative method for nonsymmetric systems of linear equations (1993)
    Springer
    Numerische Mathematik 64 (1993), S. 213-240 
    Details
    ISSN: 0945-3245
    Keywords: 65F10
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We present here a new hybrid method for the iterative solution of large sparse nonsymmetric systems of linear equations, say of the formAx=b, whereA ∈ ℝ N, N , withA nonsingular, andb ∈ ℝ N are given. This hybrid method begins with a limited number of steps of the Arnoldi method to obtain some information on the location of the spectrum ofA, and then switches to a Richardson iterative method based on Faber polynomials. For a polygonal domain, the Faber polynomials can be constructed recursively from the parameters in the Schwarz-Christoffel mapping function. In four specific numerical examples of non-normal matrices, we show that this hybrid algorithm converges quite well and is approximately as fast or faster than the hybrid GMRES or restarted versions of the GMRES algorithm. It is, however, sensitive (as other hybrid methods also are) to the amount of information on the spectrum ofA acquired during the first (Arnoldi) phase of this procedure.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01388688
    Library Location Call Number Volume/Issue/Year Availability
    BibTip Others were also interested in ...
    Paper (German National Licenses)
    Fulltext
  • 3
    Electronic Resource
    Electronic Resource
    On hybrid iterative methods for nonsymmetric systems of linear equations (1996)
    Springer
    Numerische Mathematik 73 (1996), S. 489-506 
    Details
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991):65F10
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary. Hybrid methods for the solution of systems of linear equations consist of a first phase where some information about the associated coefficient matrix is acquired, and a second phase in which a polynomial iteration designed with respect to this information is used. Most of the hybrid algorithms proposed recently for the solution of nonsymmetric systems rely on the direct use of eigenvalue estimates constructed by the Arnoldi process in Phase I. We will show the limitations of this approach and propose an alternative, also based on the Arnoldi process, which approximates the field of values of the coefficient matrix and of its inverse in the Krylov subspace. We also report on numerical experiments comparing the resulting new method with other hybrid algorithms.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002110050202
    Library Location Call Number Volume/Issue/Year Availability
    BibTip Others were also interested in ...
    Paper (German National Licenses)
    Fulltext
  • 4
    Electronic Resource
    Electronic Resource
    Block triangular preconditioners for nonsymmetric saddle point problems: field-of-values analysis (1999)
    Springer
    Numerische Mathematik 81 (1999), S. 577-594 
    Details
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991):65F10, 65N30
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract. A preconditioned minimal residual method for nonsymmetric saddle point problems is analyzed. The proposed preconditioner is of block triangular form. The aim of this article is to show that a rigorous convergence analysis can be performed by using the field of values of the preconditioned linear system. As an example, a saddle point problem obtained from a mixed finite element discretization of the Oseen equations is considered. The convergence estimates obtained by using a field–of–values analysis are independent of the discretization parameter h. Several computational experiments supplement the theoretical results and illustrate the performance of the method.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002110050405
    Library Location Call Number Volume/Issue/Year Availability
    BibTip Others were also interested in ...
    Paper (German National Licenses)
    Fulltext
  • 5
    Electronic Resource
    Electronic Resource
    Field-of-values analysis of preconditioned iterative methods for nonsymmetric elliptic problems (1997)
    Springer
    Numerische Mathematik 78 (1997), S. 103-117 
    Details
    ISSN: 0945-3245
    Keywords: Key words: Krylov subspace methods, GMRES, FOM, field of values, hierarchical basis, multilevel preconditioning, nonsymmetric elliptic problems ; Mathematics Subject Classification (1991): 65F10, 65N30, 65N55
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary. The convergence rate of Krylov subspace methods for the solution of nonsymmetric systems of linear equations, such as GMRES or FOM, is studied. Bounds on the convergence rate are presented which are based on the smallest real part of the field of values of the coefficient matrix and of its inverse. Estimates for these quantities are available during the iteration from the underlying Arnoldi process. It is shown how these bounds can be used to study the convergence properties, in particular, the dependence on the mesh-size and on the size of the skew-symmetric part, for preconditioners for finite element discretizations of nonsymmetric elliptic boundary value problems. This is illustrated for the hierarchical basis and multilevel preconditioners which constitute popular preconditioning strategies for such problems.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002110050306
    Library Location Call Number Volume/Issue/Year Availability
    BibTip Others were also interested in ...
    Paper (German National Licenses)
    Fulltext
  • 6
    Electronic Resource
    Electronic Resource
    Gauss–Newton Multilevel Methods for Least-Squares Finite Element Computations of Variably Saturated Subsurface Flow (2000)
    Springer
    Computing 64 (2000), S. 323-338 
    Details
    ISSN: 1436-5057
    Keywords: AMS Subject Classifications: 65M55, 65M60, 76S05. ; Key Words: Variably saturated flow, nonlinear elliptic problems, least-squares finite element method, inexact Gauss–Newton method, Raviart–Thomas spaces, multilevel method.
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Notes: Abstract We apply the least-squares mixed finite element framework to the nonlinear elliptic problems arising in each time-step of an implicit Euler discretization for variably saturated flow. This approach allows the combination of standard piecewise linear H 1-conforming finite elements for the hydraulic potential with the H(div)-conforming Raviart–Thomas spaces for the flux. It also provides an a posteriori error estimator which may be used in an adaptive mesh refinement strategy. The resulting nonlinear algebraic least-squares problems are solved by an inexact Gauss–Newton method using a stopping criterion for the inner iteration which is based on the change of the linearized least-squares functional relative to the nonlinear least-squares functional. The inner iteration is carried out using an adaptive multilevel method with a block Gauss–Seidel smoothing iteration. For a realistic water table recharge problem, the results of computational experiments are presented.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s006070070028
    Library Location Call Number Volume/Issue/Year Availability
    BibTip Others were also interested in ...
    Paper (German National Licenses)
    Fulltext
  • 7
    Electronic Resource
    Electronic Resource
    The near-best solution of a polynomial minimization problem by the Carathéodory-Fejér method (1990)
    Springer
    Constructive approximation 6 (1990), S. 303-319 
    Details
    ISSN: 1432-0940
    Keywords: Polynomial Chebyshev approximation problems with interpolatory constraints ; Nevanlinna-Pick interpolation ; Faber-Carathéodory-Fejér approximation method
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract For a compact set Ω⊑C, 1∉Ω, we consider the Chebyshev problem $$\min \{ ||p_m ||_\Omega :p \in \Pi _m :p(1) = 1;p^{(j)} (1) = 0,j = 1,2,...,n\} ,$$ , wheren is a fixed nonnegative integer and II m denotes the space of all complex polynomials of degreem. This problem is of importance for the construction of semi-iterative methods for singular systems of linear algebraic equations. In the case when Ω is a Jordan region whose boundary is sufficiently smooth, we determine the asymptotic behavior of ‖p m * ‖Ω, whereP m * , denotes the solution of the above Chebyshev problem. Moreover, using the Carathéodory-Fejér method, we construct a “near minimal” solution $$\hat p_m $$ of this problem if CΩ is simply connected.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01890413
    Library Location Call Number Volume/Issue/Year Availability
    BibTip Others were also interested in ...
    Paper (German National Licenses)
    Fulltext
  • 8
    Electronic Resource
    Electronic Resource
    Multilevel Minimal Residual Methods for Nonsymmetric Elliptic Problems (1996)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 3 (1996), S. 351-367 
    Details
    ISSN: 1070-5325
    Keywords: multilevel preconditioning ; Krylov subspace methods ; GMRES, nonsymmetric elliptic problems ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The subject of this paper is to study the performance of multilevel preconditioning for nonsymmetric elliptic boundary value problems. In particular, a minimal residual method with respect to an appropriately scaled norm, measuring the size of the residual projections on all levels, is studied. This norm, induced by the multilevel splitting, is also the basis for a proper stopping criterion. Our analysis shows that the convergence rate of this minimal residual method using the multilevel preconditioner by Bramble, pasciak and Xu is bounded independently of the mesh-size. However, the convergence rate deteriorates with increasing size of the skew-symmetric part. Our numerical results show that by incorporating this into a multilevel cycle starting on the coarsest level, one can save fine-level-iterations and, therefore, computational work.
    Additional Material: 2 Ill.
    Type of Medium: Electronic Resource
    Library Location Call Number Volume/Issue/Year Availability
    BibTip Others were also interested in ...
    Paper (German National Licenses)
Close ⊗
This website uses cookies and the analysis tool Matomo. More information can be found here...