Geodesic
Algebraic distance for anisotropic diffusion problems: multilevel results
Electronic transactions on numerical analysis, Tome 44 (2015), pp. 472-496
In this paper, we motivate, discuss the implementation, and present the resulting numerics for a new definition of strength of connection which uses the notion of algebraic distance as defined originally in the bootstrap algebraic multigrid framework (BAMG). We use this algebraic distance measure together with compatible relaxation and least-squares interpolation to derive an algorithm for choosing suitable coarse grids and accurate interpolation operators for algebraic multigrid algorithms. The main tool of the proposed strength measure is the least-squares functional defined by using a set of test vectors that in general is computed using the bootstrap process. The motivating application is the anisotropic diffusion problem, in particular, with non-grid aligned anisotropy. We demonstrate numerically that the measure yields a robust technique for determining strength of connectivity among variables for both two-grid and multigrid bootstrap algebraic multigrid methods. The proposed algebraic distance measure can also be used in any other coarsening procedure assuming that a rich enough set of near-kernel components of the matrix for the targeted system is known or is computed as in the bootstrap process.
Classification : 65N55, 65N22, 65F10
Keywords: bootstrap algebraic multigrid, least-squares interpolation, algebraic distances, strength of connection 
@article{ETNA_2015__44__a7,
     author = {Brandt,  Achi and Brannick,  James and Kahl,  Karsten and Livshits,  Irene},
     title = {Algebraic distance for anisotropic diffusion problems: multilevel results},
     journal = {Electronic transactions on numerical analysis},
     pages = {472--496},
     year = {2015},
     volume = {44},
     zbl = {1327.65261},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ETNA_2015__44__a7/}
}
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%A Livshits,  Irene
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%J Electronic transactions on numerical analysis
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Brandt,  Achi; Brannick,  James; Kahl,  Karsten; Livshits,  Irene. Algebraic distance for anisotropic diffusion problems: multilevel results. Electronic transactions on numerical analysis, Tome 44 (2015), pp. 472-496. http://geodesic.mathdoc.fr/item/ETNA_2015__44__a7/