Inverse-Based Algebraic Multigrid

We discuss two approaches of algebraic multigrid for the solution of linear systems. The common objective of the both approaches is to detect a submatrix of the original matrix such that its inverse is approximately sparse. The first approach is based on onstructing an adapted sparse approximate inverse of the associated leading submatrix, while the second approach directly constructs an incomplete LU decomposition. Different strategies are discussed for finding a submatrix with sparse approximate inverse.

Either case leads to an algebraic multilevel hierarchy. Numerical examples will be shown to illustrate the effectiveness of this approach.