On Computing Inverse Entries of a Sparse Matrix in an Out-of-Core Environment

Abstract

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 eciently, 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.Key words. Sparse matrices, direct methods for linear systems and matrix inversion, multifrontal method, graphs and hypergraphs.

Application of the perfectly matched layer in 3-D marine controlled-source electromagnetic modelling

“…The formed linear equations in (4) given by the discretization of eq. (3) are solved by a multifrontal direct solver MUMPS 5.0.2 parallelized by OpenMP (Amestoy et al 2001(Amestoy et al , 2012, which could avoid uncertainties in pre-conditioning and convergence for iterative solutions, especially for low frequencies (Farquharson & Miensopust 2011;Oldenburg et al 2013). In this section, we will focus on the implementation of CFS-PML in details.…”

“…The numerical test is performed under the Dell Precision Tower 3620 (3.50 GHz CPU Intel@Xeon E3-1240 v5 family including 2 processors with up to 4 cores per processor, memory up to 16 G), which is suitable for OpenMP parallel programming. Note the MUMPS 5.0.2 used for factorizing the complex sparse matrix formed by SFD discretization is parallelized by OpenMP in an "out-of-core" environment (Amestoy et al 2001(Amestoy et al , 2012. When the "out-of-core" phase is activated and the complete matrix of factors is written to disk and will be read each time a solution phase is requested, therefore the memory requirement can be significantly reduced while not increasing much of the factorization time on a reasonably small number of processors.…”

Scheduling for Large-Scale Systems

“…We assume that the cache size is adapted to the application, therefore ensuring that the execution time is linearly related to the frequency [38]: Exe(w i , f ) = w i f . When a task is scheduled to be re-executed at two different speeds f (1) and f (2) , we always account for both executions, even when the first execution is successful, and hence Exe(…”

“…Since the reliability increases with speed, we must have f ≥ f rel to match the reliability constraint. If task T i is re-executed (speeds f (1) and f (2) ), then the execution of T i is successful if and only if one of the attempts do not fail, so that the reliability of T i is (2) )), and this quantity should be at least equal to R i (f rel ).…”

“…Otherwise, if task T i is re-executed at speeds f (1) and f (2) , it is natural to add up the energy consumed during both executions, just as we add up both execution times when enforcing the makespan deadline. Again, this corresponds to the worst-case execution scenario.…”