This talk focuses on developing a generalized bootstrap algebraic
multigrid algorithm for solving sparse matrix equations. As a
motivation of the proposed generalization, we consider an optimal form
of classical algebraic multigrid interpolation that has as columns
eigenvectors with small eigenvalues of the generalized eigen-problem
involving the system matrix and its symmetrized smoother. We use this
optimal form to design an algorithm for choosing and analyzing the
suitability of the coarse grid. In addition, it provides insights into
the design of the bootstrap algebraic multigrid setup algorithm that we
propose, which uses as a main tool a multilevel eigensolver to compute
approximations to these eigenvectors. A notable feature of the approach
is that it allows for general block smoothers and, as such, is well
suited for systems of partial differential equations. In addition, we
combine the GAMG setup algorithm with a least-angle regression
coarsening scheme that uses local regression to improve the choice of
the coarse variables. These new algorithms and their performance are
illustrated numerically for scalar diffusion problems with highly
varying (discontinuous) diffusion coefficient and for the linear
elasticity system of partial differential equations.`