  In this paper we motivate, discuss the implementation and present the
resulting numerics for a new definition of strength of connection which is
based on the notion of algebraic distance. This algebraic distance measure,
combined with compatible relaxation, is used to choose suitable coarse grids
and accurate interpolation operators for algebraic multigrid algorithms. The
main tool of the proposed measure is the least squares functional defined using
a set of relaxed test vectors. The motivating application is the anisotropic
diffusion problem, in particular problems 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 solvers. %We illustrate the use of the measure to construct, in
addition, an adaptive aggregation form of interpolation for the targeted
anisotropic problems. %Our approach is not a two-level approach -- we provide
preliminary results that show its extendability to multigrid. The proposed
algebraic distance measure can also be used in any other coarsening procedure,
assuming a rich enough set of the near-kernel components of the matrix for the
targeted system is known or computed.
