Using approximate inverses in algebraic multilevel methods

This paper deals with the iterative solution of large sparse symmetric positive definite systems. We investigate preconditioning techniques of the two-level type that are based on a block factorization of the system matrix. Whereas the basic scheme assumes an exact inversion of the submatrix related to the first block of unknowns, we analyze the effect of using an approximate inverse instead. We derive condition number estimates that are valid for any type of approximation of the Schur complement and that do not assume the use of the hierarchical basis. They show that the two-level methods are stable when using approximate inverses based on modified ILU techniques, or explicit inverses that meet some row-sum criterion. On the other hand, we bring to the light that the use of standard approximate inverses based on convergent splittings can have a dramatic effect on the convergence rate. These conclusions are numerically illustrated on some examples.