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  • 1
    Electronic Resource
    Electronic Resource
    An analysis of the composite step biconjugate gradient method (1993)
    Springer
    Numerische Mathematik 66 (1993), S. 295-319 
    Details
    ISSN: 0945-3245
    Keywords: 65N20 ; 65F10
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The composite step biconjugate gradient method (CSBCG) is a simple modification of the standard biconjugate gradient algorithm (BCG) which smooths the sometimes erratic convergence of BCG by computing only a subset of the iterates. We show that 2×2 composite steps can cure breakdowns in the biconjugate gradient method caused by (near) singularity of principal submatrices of the tridiagonal matrix generated by the underlying Lanczos process. We also prove a “best approximation” result for the method. Some numerical illustrations showing the effect of roundoff error are given.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01385699
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    A composite step bi-conjugate gradient algorithm for nonsymmetric linear systems (1994)
    Springer
    Numerical algorithms 7 (1994), S. 1-16 
    Details
    ISSN: 1572-9265
    Keywords: Biconjugate gradients ; nonsymmetric linear systems ; 65N20 ; 65F10
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract The Bi-Conjugate Gradient (BCG) algorithm is the simplest and most natural generalization of the classical conjugate gradient method for solving nonsymmetric linear systems. It is well-known that the method suffers from two kinds of breakdowns. The first is due to the breakdown of the underlying Lanczos process and the second is due to the fact that some iterates are not well-defined by the Galerkin condition on the associated Krylov subspaces. In this paper, we derive a simple modification of the BCG algorithm, the Composite Step BCG (CSBCG) algorithm, which is able to compute all the well-defined BCG iterates stably, assuming that the underlying Lanczos process does not break down. The main idea is to skip over a step for which the BCG iterate is not defined.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02141258
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    A composite step conjugate gradients squared algorithm for solving nonsymmetric linear systems (1994)
    Springer
    Numerical algorithms 7 (1994), S. 17-32 
    Details
    ISSN: 1572-9265
    Keywords: Lanczos method ; conjugate gradients squared algorithm ; breakdowns ; composite step ; 65F10 ; 65F25
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We propose a new and more stable variant of the CGS method [27] for solving nonsymmetric linear systems. The method is based on squaring the Composite Step BCG method, introduced recently by Bank and Chan [1,2], which itself is a stabilized variant of BCG in that it skips over steps for which the BCG iterate is not defined and causes one kind of breakdown in BCG. By doing this, we obtain a method (Composite Step CGS or CSCGS) which not only handles the breakdowns described above, but does so with the advantages of CGS, namely, no multiplications by the transpose matrix and a faster convergence rate than BCG. Our strategy for deciding whether to skip a step does not involve any machine dependent parameters and is designed to skip near breakdowns as well as produce smoother iterates. Numerical experiments show that the new method does produce improved performance over CGS on practical problems.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02141259
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    Paper (German National Licenses)
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  • 4
    Electronic Resource
    Electronic Resource
    Additive Schwarz domain decomposition methods for elliptic problems on unstructured meshes (1994)
    Springer
    Numerical algorithms 8 (1994), S. 329-346 
    Details
    ISSN: 1572-9265
    Keywords: Unstructured meshes ; non-nested coarse meshes ; additive Schwarz algorithm ; optimal convergence rate ; 65N30 ; 65F10
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We give several additive Schwarz domain decomposition methods for solving finite element problems which arise from the discretizations of elliptic problems on general unstructured meshes in two and three dimensions. Our theory requires no assumption (for the main results) on the substructures which constitute the whole domain, so each substructure can be of arbitrary shape and of different size. The global coarse mesh is allowed to be non-nested to the fine grid on which the discrete problem is to be solved and both the coarse meshes and the fine meshes need not be quasi-uniform. In this general setting, our algorithms have the same optimal convergence rate of the usual domain decomposition methods on structured meshes. The condition numbers of the preconditioned systems depend only on the (possibly small) overlap of the substructures and the size of the coares grid, but is independent of the sizes of the subdomains.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02142697
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    Paper (German National Licenses)
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