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Testing for effects of asymmetry and instability on preconditioned iterations of conjugate gradient type
Authors:STMPSON   R. B.
Affiliation:Department of Computer Science, University of Waterloo Ontario Canada N2L 3G1
Abstract:
We develop a parametrized family of matrices and use them totest the performance of some preconditioned iterative methodsas we vary the asymmetry and stability of the test matrices.The test matrices are based on a simple discretization of adynamic, two-species, contant coefficient, reaction-diffusionsystem of partial differential equations. The reaction coefficientsprovide natural parameters for varying the properties of thetest matrices, which are typical of modelling applications.These matrices are reducible via a red-black ordering, and itis shown that the reduced matrices are M-matrices for a largerrange of parameters than the ‘unreduced’ test matrices. The iterative methods tested are of conjugate gradient type,using incomplete factorization preconditioning. The componentsof the methods tested are: the acceleration technique (conjugategradient squared, stabilized biconjugate gradi ent, orthomin),the level of fill-in of the incomplete factorization preconditioner,the use of the reduced system, and the effect of time-step sizereduction (for dynamic simulations). The tests are carried Outby extensive sampling in regions of the parameter space. The results appear to confirm observations of other studiesusing diffusion-convection based tests, and, in particular,show that in these instances the performance of the methodsis essentially unaffected by asymmetry, but is strongly affectedby instability.
Keywords:
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