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21st GAMM-Seminar Leipzig on
Robust Fast Solvers

Max-Planck-Institute for Mathematics in the Sciences
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January, 26th-28th, 2005
 
     
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  Abstract Oliver Bröker, Wed, 17.30-18.00 Previous Contents Next  
  Multigrid Smoothing with Explicit Approximate Inverse Stencils
Oliver Bröker (ETH Zürich)

Approximate inverses based on Frobenius-norm minimization [4] have been shown to yield good smoothers in (algebraic) multigrid [1,2,3]. This approach offers some degrees of freedom, mainly the sparsity pattern of the inverse, and thus potential improvement. The essential advantage of the approach is the smoothing efficiency, coupled with high parallelism.

We investigate this smoother class for stencil based problems, including the Laplacian, the Mehrstellen-Operator, convection-diffusion equations, mixed-derivatives, the Helmholtz-operator and the biharmonic-operator. Exploiting regular grid structures enables us to compute explicit formulas and may render the initialization and storage of an approximate inverse unnecessary.

Critical values in the construction of an efficient multigrid scheme are the smoothing-, the two-grid- and the three-grid-factor [5]. Analyzing these factors for the various stencils shows that in some cases Frobenius-norm minimization yields a near optimal smoother, while for others not even convergence is obtained. We show that in most cases approximate inverse stencil entries with better smoothing-factor and overall convergence properties can be found and propose several strategies to construct such values.

We conclude the talk with numerical results on large problems.

[1] O. Broeker Parallel Algebraic Multigrid with Sparse Approximate Inverses Dissertation No. 15129, ETH Zurich, 2003/May

[2] O. Broeker and M.J. Grote Sparse Approximate Inverse Smoothers for Geometric and Algebraic Multigrid Applied Numerical Mathematics, Vol. 41, Number 1, pp. 61-80, 2002/Mar

[3] O. Broeker, M.J. Grote, C. Mayer, A. Reusken Robust Parallel Smoothing for Multigrid Via Sparse Approximate Inverses SIAM Journal on Scientific Computing, Vol. 23, Number 4, pp. 1396-1417, 2001/Oct

[4] M.J. Grote and T. Huckle Parallel Preconditioning with Sparse Approximate Inverses SIAM J. of Scientific Computing, Vol. 18, Number 3, pp. 838--853, 1997/May

[5] R. Wienands and W. Joppich Practical Fourier Analysis for Multigrid Methods CRC Press, 2004/Nov


 

 
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