Energy optimization of algebraic multigrid bases

We propose a fast iterative method to optimize coarse basis functions in algebraic multigrid by minimizing the sum of their energies, subject to the condition that linear combinations of the basis functions equal to given zero energy modes, and subject to restrictions on the supports of the coarse basis functions. For a particular selection of the supports, the first iteration gives exactly the same basis functions as our earlier method using smoothed aggregation. The convergence rate of the minimization algorithm is bounded independently of the mesh size under usual assumptions on finite elements. The construction is presented for scalar problems as well as for linear elasticity. Computational results on difficult industrial problems demonstrate that the use of energy minimal basis functions improves algebraic multigrid performance and yields a more robust multigrid algorithm than smoothed aggregation.