An Algebraic Multilevel Preconditioner for Symmetric Positive Definite and Indefinite Problems

Abstract

We present a preconditioning method for the iterative solution of large sparse systems of equations. The preconditioner is based on ideas both from ILU preconditioning and from multigrid. The resulting preconditioning technique requires the matrix only. A multilevel structure is obtained by constructing a maximal independent set of the graph of a reduced matrix. The computation of a Schur complement approximation is based on a Galerkin approach with a matrix dependent prolongation and restriction. The resulting preconditioner has a transparant modular structure similar to the algorithmic structure of a multigrid V-cycle. The method is applied to symmetric positive definite and indefinite Helmholtz problems. The multilevel preconditioner is compared with standard ILU preconditioning methods.KeywordsHelmholtz EquationMultigrid MethodTridiagonal MatrixDecomposition PhaseCyclic ReductionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.