In this paper we introduce an algebraic recursive multilevel incomplete factorization preconditioner, based on a distributed Schur complement formulation, for solving general linear systems.The novelty of the proposed method is to combine factorization techniques of both implicit and explicit type, recursive combinatorial algorithms, multilevel mechanisms and overlapping strategies to maximize sparsity in the inverse factors and consequently reduce the factorization costs. Numerical experiments demonstrate the good potential of the proposed solver to precondition effectively general linear systems, also against other state-ofthe-art iterative solvers of both implicit and explicit form.

We introduce an algebraic recursive multilevel approximate inverse-based preconditioner, based on a distributed Schur complement formulation. The proposed preconditioner combines recursive combinatorial algorithms and multilevel mechanisms to maximize sparsity during the facorization. Experiments on selected sparse linear systems from electromagnetics applications demonstrate the potential of the proposed method, also against other state-of-the-art solvers.

We propose an adaptive reordered method to deal with the PageRank problem. It has been shown that one can reorder the hyperlink matrix of PageRank problem to calculate a reduced system and get the full PageRank vector through forward substitutions. This method can provide a speedup for calculating the PageRank vector. We observe that in the existing reordered method, the cost of the recursively reordering procedure could offset the computational reduction brought by minimizing the dimension of linear system. With this observation, we introduce an adaptive reordered method to accelerate the total calculation, in which we terminate the reordering procedure appropriately instead of reordering to the end. Numerical experiments show the effectiveness of this adaptive reordered method.

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