International audienceIn this paper, we discuss the usage of overlapping techniques for improving the convergence of preconditioners based on incomplete factorizations. To enable parallelism, these preconditioners are usually applied after the input matrix is permuted into a nested arrow form using k-way nested dissection. This graph partitioning technique uses k-way partitionning by vertex separator to recursively partition the graph of the input matrix into k subgraphs using a subset of its vertices called a separator. The overlapping technique is then based on algebraically extending the associated subdomains of these subgraphs and their corresponding separators obtained from k-way nested dissection by their direct neighbours. A similar approach is known to accelerate the convergence of domain decomposition methods, where the input matrix is partitioned into a number of independent subdomains using k-way vertex partitioning of a graph by edge separators, a different graph decomposition technique. We discuss the effect of the overlapping technique on the convergence of two classes of preconditioners, on the basis of nested factorization and block incomplete LDU factorization

We present the parallel design and performance of the nested filtering factorization preconditioner (NFF), which can be used for solving linear systems arising from the discretization of a system of PDEs on unstructured grids. NFF has limited memory requirements, and it is based on a two level recursive decomposition that exploits a nested block arrow structure of the input matrix, obtained priorly by using graph partitioning techniques. It also allows to preserve several directions of interest of the input matrix to alleviate the effect of low frequency modes on the convergence of iterative methods. For a boundary value problem with highly heterogeneous coefficients, discretized on three-dimensional grids with 64 millions unknowns and 447 millions nonzero entries, we show experimentally that NFF scales up to 2048 cores of Genci's Bull system (Curie), and it is up to 2.6 times faster than the domain decomposition preconditioner Restricted Additive Schwarz implemented in PETSc.