Coarsening by Compatible Relaxation

Oren E. Livne

Computer Science Department, Gates 2B, Stanford University, Stanford, CA 94305-9025


Algebraic Multigrid (AMG) solvers of large sparse linear system of equations are based on multigrid principles but no do explicitly use the geometry of the grids. The emphasis in AMG is on black on procedures for coarsening the set of equations, relying on its algebraic relations. Although AMG is widely employed, e.g. for solving second-order elliptic discretized PDEs with disordered coefficients on unstructured grids, its scope is rather limited to these cases. However, AMG may be remedied, by systemtically understanding and improving its ingredients: relaxation, coarse levels and inter-level transfers. We present a novel approach for selecting the coarse variables for AMG. Classical AMG coarse set construction relies on the strength of local algebraic connections between variables, that are often misleading and inaccurate. Alternatively, our coarse set quality measure is the rate of convergence of compatible relaxation (CR) [Brandt, ETNA, 2000]. It also leads to a construction algorithm of the coarse set. We will present numerical examples of our algorithm for simple model problems, e.g., anistropic diffusion and the biharmonic operator, for which classical AMG often fails. This work makes it possible to assess and construct a coarse set of variables for a given system, prior to its actual use by an AMG solver. Undergoing research is focused on attaining CR rates (that play a similar role to the smoothing rate in geometric multigrid, predicting the ideal multigrid efficiency). This relates to self-correcting AMG (a joint work with Prof. A. Brandt, Weizmann Institute) that iteratively derives its own interpolation operators.