@TechReport{ AmDuL'RoRoUç2010.1,

author = {Amestoy, Patrick and Duff, Iain and L'Excellent, Jean-Yves and Robert, Yves and Rouet, François-Henry and Uçar, Bora},

title = "{On computing inverse entries of a sparse matrix in an out-of-core environment}",

year = {2010},

month = {juin},

type = {Rapport de recherche},

number = {RT-APO-10-06},

institution = {Institut National Polytechnique de Toulouse},

address = {Toulouse, France},

language = {anglais},

URL = {http://mumps.enseeiht.fr/doc_n7/RT_APO_10_06.pdf},

keywords = {Sparse matrices, direct methods for linear systems and matrix inversion, multifrontal method, graphs and hypergraphs.},

note = {modified version accepted to SIAM journal on Scientific Computing},

abstract = {The inverse of an irreducible sparse matrix is structurally full,so that it is impractical to think of computing or storing it.
However, there are several applications where a subset of the entries of the inverse is required. Given a factorization of the sparse matrix held in out-of-core storage, we show how to compute such a subset efficiently, by accessing only parts of the factors.
When there are many inverse entries to compute, we need to
guarantee that the overall computation scheme
has reasonable memory requirements, while minimizing the cost of loading the factors. This leads to
a partitioning problem that we prove is NP-complete. We also show that we cannot get a close approximation to the optimal solution in polynomial time. We thus need to develop heuristic algorithms, and we propose: (i) a lower bound on the cost of an optimum solution;
(ii) an exact algorithm for a particular case; (iii) two other heuristics for a more general case; and (iv) hypergraph partitioning models for the most general setting.
We illustrate the performance of our algorithms in practice using the MUMPS softwarepackage on a set of real-life problems as well as some standard test matrices.
We show that our techniques can improve the execution time by a factor of 50.}

}