Overlapping for preconditioners based on incomplete factorizations and nested arrow form

Abstract

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

A hybrid recursive multilevel incomplete factorization preconditioner for solving general linear systems

“…We display a simple example from [20] to describe shortly how the overlapping procedure works in practice. We consider a 5 × 5 matrix having the structure shown in Figure 2(a).…”

“…From Eqn. (20), we can construct the edges fromΩ 1 toS. These are the nonzero entries of the overlapped interface blockF 1 , adopting the same notation as in (6).…”

“…The following property, established in [20], ensures an equivalence between the equations of the overlapped system and those of the original system.…”

Parallel design and performance of nested filtering factorization preconditioner

“…This leads to a preconditioner with a very cheap construction cost, since it requires only the factorization of diagonal blocks. In our recent work [13], we have shown that by using overlapping techniques similar to the ones from domain decomposition methods, the convergence of NSSOR is greatly improved. Without going further into the details of the overlapping techniques, we note that the experiments from section 4 present the convergence of NSSOR with overlapping, in which the variables at the interface between the different subdomains are duplicated.…”

“…The number of innermost diagonal blocks that can be computed in parallel corresponds to P , the number of processors used in the parallel computation, however the recursion depth is only two. While NFF remains an efficient preconditioner with this new reordering, the other preconditioners presented in [10] require the usage of overlapping techniques as explained in [13].…”