The Multifrontal Method for Sparse Matrix Solution: Theory and Practice

Liu, Joseph W. H.

doi:10.1137/1034004

Cited by 364 publications

(232 citation statements)

References 31 publications

Supporting

Mentioning

224

Contrasting

Unclassified

Order By: Relevance

“…The work in this paper is based on the solver MUMPS, a MUltifrontal Massively Parallel Solver [1]. For an overview of the multifrontal method we refer to [7,8,16]. The work presented in [12] has shown how to use memory-based dynamic scheduling to improve the memory management of a parallel multifrontal approach.…”

Section: Introductionmentioning

confidence: 99%

Hybrid scheduling for the parallel solution of linear systems

et al. 2006

View full text Add to dashboard Cite

In this paper, we consider the problem of designing a dynamic scheduling strategy that takes into account both workload and memory information in the context of the parallel multifrontal factorization. The originality of our approach is that we base our estimations (work and memory) on a static optimistic scenario during the analysis phase. This scenario is then used during the factorization phase to constrain the dynamic decisions. The task scheduler has been redesigned to take into account these new features. Moreover performance have been improved because the new constraints allow the new scheduler to make optimal decisions that were forbidden or too dangerous in unconstrained formulations. Performance analysis show that the memory estimation becomes much closer to the memory effectively used and that even in a constrained memory environment we decrease the factorization time with respect to the initial approach.Key-words: sparse matrices, parallel multifrontal method, dynamic scheduling, memory.This text is also available as a research report of the Laboratoire de l'Informatique du Parallélisme http://www.ens-lyon.fr/LIP and as a technical report from ENSEEIHT-IRIT. Stratégies d'ordonnancement hybrides pour la résolution parallèle de systèmes linéairesRésumé : Nous proposons des stratégies d'ordonnancement bi-critères, qui s'inté-ressentà la foisà la performance età la consommation mémoire d'un algorithme parallèle de factorisation de matrices creuses, basé sur la méthode multifrontale. L'originalité de notre approche est que nous basons nos estimations mémoire sur un scénario optimiste (simulation lors de la phase d'analyse), qui est ensuite utilisé lors de la factorisation pour contraindre les décisions dynamiques d'ordonnancement. Un nouvel ordonnanceur aété implanté, qui prend en compte ces nouvelles contraintes. De plus, la performance aété améliorée parce que notre nouvelle approche permet a l'ordonnanceur de prendre des décisions meilleures, quiétaient interdites ou trop dangereuses auparavant. Une analyse de performance montre que les estimations mémoire sont beaucoup plus proches de la mémoire effectivement utilisée, et que le temps de factorisation est amélioré de façon significative par rapportà l'approche initiale.

show abstract

Section: Introductionmentioning

confidence: 99%

Hybrid scheduling for the parallel solution of linear systems

et al. 2006

View full text Add to dashboard Cite

show abstract

“…We have chosen the ultraweak setting because it is the most actively researched variational setting for DPG methods, at this time. For the normal equation, we solved the system with MUMPS 5.0.1 [52,1]; and for the overdetermined system, we used qr mumps 1.2 [15]. Our results are reported in Figure 5.6.…”

Section: 2mentioning

confidence: 99%

Discrete least-squares finite element methods

Keith

Petrides

Fuentes

et al. 2017

Computer Methods in Applied Mechanics and Engineering

View full text Add to dashboard Cite

ABSTRACT. A finite element methodology for large classes of variational boundary value problems is defined which involves discretizing two linear operators: (1) the differential operator defining the spatial boundary value problem; and (2) a Riesz map on the test space. The resulting linear system is overdetermined. Two different approaches for solving the system are suggested (although others are discussed): (1) solving the associated normal equation with linear solvers for symmetric positive-definite systems (e.g. Cholesky factorization); and (2) solving the overdetermined system with orthogonalization algorithms (e.g. QR factorization). The finite element assembly algorithm for each of these approaches is described in detail. The normal equation approach is usually faster for direct solvers and requires less storage. The second approach reduces the condition number of the system by a power of two and is less sensitive to round-off error. The rectangular stiffness matrix of second approach is demonstrated to have condition number O(h −1 ) for a variety of formulations of Poisson's equation. The stiffness matrix from the normal equation approach is demonstrated to be related to the monolithic stiffness matrices of least-squares finite element methods and it is proved that the two are identical in some cases. An example with Poisson's equation indicates that the solutions of these two different linear systems can be nearly indistinguishable (if round-off error is not an issue) and rapidly converge to each other. The orthogonalization approach is suggested to be beneficial for problems which induce poorly conditioned linear systems. Experiments with Poisson's equation in single-precision arithmetic as well as the linear acoustics problem near resonance in double-precision arithmetic verify this conclusion. The methodology described here was developed as an outgrowth of the discontinuous Petrov-Galerkin (DPG) methodology of Demkowicz and Gopalakrishnan [29]. The strength of DPG is highlighted throughout, however, the majority of theory presented is general. Extensions to constrained minimization principles are also considered throughout but are not analyzed in experiments.

show abstract

“…Indirect element ordering with default MC63 parameters is used prior to calling the frontal solver factorization routines. In general, memory consumption for a multifrontal solver (for a full explanation on multifrontal solvers, see for example [20]), such as the ones used in this work, is linearly related to the máximum wavefront (see, for instance, [19]), which in turn is strongly dependent on the order in which the elements are assembled. For this reason, ordering algorithms have an enormous impact on the final memory consumption, thus making it very difficult to supply general rules regarding precise memory requirements.…”

Section: Monitorizationmentioning

confidence: 99%

Parallelizing a hybrid finite element-boundary integral method for the analysis of scattering and radiation of electromagnetic waves

Díaz

Rico

Garcı́a-Castillo

et al. 2010

Finite Elements in Analysis and Design

View full text Add to dashboard Cite

ARTICLE INFO ABSTRACT Keywords:Electromagnetics Finite element method Boundary integral Parallel computing MPI Sparse direct solvers This paper presents the practical experience of parallelizing a simulator of general scattering and radiation electromagnetic problems. The simulator stems from an existing sequential simulator in the frequency domain, which is based on a finite element analysis. After the analysis of a test case, two steps were carried out: first, a "hand-crafted" code parallelization of a convolution-type operation was developed within the kernel of the simulator. Second, the sequential HSL library, used in the existing simulator, was replaced by the parallel MUMPS (MUltifrontal Massively Parallel sparse direct Solver) library in order to solve the associated linear algebra problem in parallel. Such a library allows for the distribution of the factorized matrix and some of the computational load among the available processors. A test problem and three realistic (in terms of the number of unknowns) cases have been run using the parallelized versión of the code, and the results are presented and discussed focusing on the memory usage and achieved speed-up.

show abstract

The Multifrontal Method for Sparse Matrix Solution: Theory and Practice

Cited by 364 publications

References 31 publications

Hybrid scheduling for the parallel solution of linear systems

Hybrid scheduling for the parallel solution of linear systems

Discrete least-squares finite element methods

Parallelizing a hybrid finite element-boundary integral method for the analysis of scattering and radiation of electromagnetic waves

Contact Info

Product

Resources

About