CERFACS CERFACS Scientific Activity Report Jan. 2010 – Dec. 2011

4 Dense and Sparse Matrix Computations
4.1 Analysis of the solution phase of a parallel multifrontal
approach.
P. Amestoy : ENSEEIHT-IRIT, UNIVERSITY OF TOULOUSE, France ; I. S. Duff : RUTHERFORD
APPLETON LABORATORY AND CERFACS, UK and France ; A. Guermouche : LABRI, UNIV.
BORDEAUX 1 / INRIA BORDEAUX SUD-OUEST, France ; Tz. Slavova : CERFACS, France
In [ALG9], we study the forward and backward substitution phases of a sparse multifrontal factorization.
These phases are often neglected in papers on sparse direct factorization but, in many applications, they can
be the bottleneck so it is crucial to implement them efficiently. In this work, we assume that the factors have
been written on disk during the factorization phase, and we discuss the design of an efficient solution phase.
We will look at the issues involved when we are solving the sparse systems on parallel computers and will
consider in particular their solution in a limited memory environment when out-of-core working is required.
Two different approaches are presented to read data from the disk, with a discussion on the advantages and
the drawbacks of each. We present some experiments on realistic test problems using an out-of-core version
of a sparse multifrontal code called MUMPS (MUltifrontal Massively Parallel Solver).
4.2 On computing inverse entries of a sparse matrix in an out-of-core
environment.
P. Amestoy : INPT(ENSEEIHT)-IRIT, UNIVERSITY OF TOULOUSE, France ; I. S. Duff : RUTHERFORD
APPLETON LABORATORY AND CERFACS, UK and France ; Y. Robert : CERFACS, France ; F.-
H. Rouet : INPT(ENSEEIHT)-IRIT, France ; B. Uçar : ENS LYON AND CNRS, France
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 software package 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. For more details, see [ALG33].
CERFACS ACTIVITY REPORT 11