Signalons que pour la mise au point de certains de algorithmes, des tests on ´et´e effectu´es sur d’autres machines telles qu’une grappe de Sun Enterprise de la f´ed´eration lyonnaise de calcul haute-performance ou une grappe de bi-processeurs Itanium II (projet i-cluster 2, Grenoble).
C.4 Probl` emes de test
Les probl`emes de test ont ´et´e extraits de plusieurs collections de matrices creuse : la collection Rutherford-Boeing [26], la collection de l’universit´e de Floride5 et la collec-tion PARASOL6. Signalons que les probl`emes MCHLNF, ULTRASOUND3 et ULTRA-SOUND80 [8, 9] nous ont ´et´e fournis par M. Sosonkina, qui travaille sur la r´esolution de syst`emes lin´eaires creux par des m´ethodes r´ecursives multi-niveaux (PARMS [50]). La matrice CONV3D64 a quant `a elle ´et´e fournie par le CEA-CESTA. Les tableauxC.1,C.2 donnent une description des diff´erentes matrices que nous avons utilis´ees.
Matrice Ordre NZ Description
3DTUBE 45330 1629474 3-D pressure tube
AUDIKW 1 943695 39297771 Automotive crankshaft model with over 900,000 TETRA elements (MSC-CRANKSHAFT-900K)
BCSSTK34 588 11003 NASTRAN buckling problem stiffness matrix BCSSTK38 8032 181746 Stiffness matrix, airplane engine component
BMWCRA 1 148770 5396386 Automotive crankshaft model with nearly 150,000 TETRA ele-ments (MSC-CRANKSHAFT-150K)
CFD2 123440 1605669 CFD, symmetric pressure matrix, from Ed Rothberg, Silicon Gra-phics, Inc.
CRANKSG2 63838 7106348 Linear static analysis of a crankshaft detail (MSC-CRANKSHAFT-SEGMENT-2)
GUPTA1 31802 1098006 Linear programming matrix (A*A’), Anshul Gupta GUPTA2 62064 2155175 Linear programming matrix (A*A’), Anshul Gupta GUPTA3 16783 4670105 Linear programming matrix (A*A’), Anshul Gupta
M T1 97578 5328 Tubular joint MSDOOR 415863 10328399 Medium size door
NASA1824 1824 20516 Structure from NASA langley, 1824 degrees of freedom NASA2910 2910 88603 Structure from NASA langley, 2910 degrees of freedom NASA4704 4704 54730 Structure from NASA langley, 4704 degrees of freedom
OILPAN 73752 1835470 Oilpan
SHIP 003 121728 4103881 Ship structure from production run
STRUCT4 4350 121074 Finite element matrix, from Ed Rothberg, Silicon Graphics, Inc.
S3DKQ4M2 90449 2455670 cyl shell R/t=1000 unif 100 x 150 quad mesh DK-el with drill rotat S3DKT3M2 90449 1921955 cyl shell 100 x 150 unif triang mesh R/t = 1000
THREAD 29736 2249892 Threaded connector/contact problem
VIBROBOX 12328 177578 Vibroacoustic problem (flexible box, structure only) Andre Cote
Tab. C.1 – Description des probl`emes de test sym´etriques.
5http://www.cise.ufl.edu/~davis/sparse/
6http://www.parallab.uib.no/parasol
Matrice Ordre NZ Description
AF23560 23560 484256 1 NACA 0012 airfoil M=0.8, eigenvalue calculation
BIG 13209 91465 1Structuresymmetric Matrix big K. Gaertner ETH Zurich Oct 1996.
CIRCUIT 4 80209 307604 Circuit DAE with BDF method & Newton. W. Bomhof, Univ.
Utrecht
CONV3D64 836550 12548250 provided by CEA-CESTA ; generated using AQUILON (http ://www.enscpb.fr/master/aquilon)
EPB3 84617 463625 Plate-fin heat exchanger (large case), (DAE) Da-vid.Averous@ensigct.fr
GARON02 13535 390607 2D FEM, Navier-Stokes, CFD. Square inlet and outlet on opp. sides GRAHAM1 9035 335504 Galerkin FE disc. of Nav.Stokes 2phase fluid flow. D Graham, U
Illinois
GRID48 47045 3705625 Unsymmetric assembled finite element problem. Nine node, five va-riable ELT48
LI 22695 1350309 3D Finite element matrix, magnetohydrodynamic problem with 5 variables
MCHLNF 49800 4136484 3D tire design, provided by John Melson (Michelin Research and Development Corporation).
MIXING TANK 29957 1995041 Mixing tank. Unsymmetric matrix. Modified version of MIXTANK to avoid underflow errors on some machines
ONETONE1 36057 341088 AT&T,harmonic balance method, one-tone. Approx Jacobian for SSOR precon
PRE2 659033 5959282 AT&T,harmonic balance method, large example
RMA10 46835 2374001 3D CFD model, Charleston harbor. Steve Bova, US Army Eng., WES
SAYLR1 238 1128 1UNSYMMETRIC MATRIX OF PAUL SAYLOR - 14 BY 17 2D GRID MAY, 1983
THERMAL 3456 66528 Noel.Brunetiere@lms.univ-poitiers.fr FEM, thermal problem TWOTONE 120750 1224224 AT&T,harmonic balance method, two-tone. More off-diag nz than
onetone
VENKAT50 62424 1717792 Unstructured 2D EULER solver, V. Venkatakrishnan NASA, time step = 50
WANG1 2903 19093 Discretized electron continuity, 3d diode, nonuniform 14-14-16 mesh WANG3 26064 177168 Discretized electron continuity, 3d diode, uniform 30-30-30 mesh XENON2 157464 3866688 Complex zeolite,sodalite crystals. D Ronis
ro-nis@onsager.chem.mcgill.ca
ULTRASOUND3 185193 11390625 Propagation of 3D ultrasound waves generated by X. Cai (Simula Research Laboratory, Norway) using Diffpack.
ULTRASOUND80 531441 33076161 Propagation of 3D ultrasound waves, provided by M. Sosonkina.
Larger than ULTRASOUND3.
Tab. C.2 – Description des probl`emes de test non-sym´etriques.
Les m´ethodes directes de r´esolution de syst`emes lin´eaires creux sont connues pour leurs besoins m´emoire importants qui peuvent constituer une barri`ere au traitement de pro-bl`emes de grandes taille. De ce fait, les travaux effectu´es durant cette th`ese ont port´e d’une part sur l’´etude du comportement m´emoire d’un algorithme de factorisation de matrices creuses, en l’occurrence la m´ethode multifrontale, et d’autre part sur l’optimisa-tion et la minimisal’optimisa-tion de la m´emoire n´ecessaire au bon d´eroulement de la factorisal’optimisa-tion aussi bien dans un cadre s´equentiel que parall`ele. Ainsi, des algorithmes optimaux pour la minimisation de la m´emoire ont ´et´e propos´es pour le cas s´equentiel. Pour le cas pa-rall`ele, nous avons introduit dans un premier temps des strat´egies d’ordonnancement visant une am´elioration du comportement m´emoire de la m´ethode. Puis, nous les avons
´etendues pour avoir un objectif de performance tout en gardant un bon comportement m´emoire. Enfin, dans le cas o`u l’ensemble des donn´ees `a traiter a encore une taille plus importante que celle de la m´emoire, il est n´ecessaire de concevoir des approches de facto-risation out-of-core. Pour ˆetre efficaces, ces m´ethodes n´ecessitent d’une part de recouvrir les op´erations d’entr´ees/sorties par des calculs, et d’autre part de r´eutiliser des donn´ees d´ej`a pr´esentes en m´emoire pour r´eduire le volume d’entr´ees/sorties. Ainsi, une partie des travaux pr´esent´es dans cette th`ese ont port´e sur la conception de techniques out-of-core implicites adapt´ees au sch´ema des acc`es de la m´ethode multifrontale et reposant sur une modification de la politique de pagination du syst`eme d’exploitation `a l’aide d’un outil bas-niveau (MMUM&MMUSSEL).
Mots-cl´es :
Matrices creuses, m´ethodes multifrontale, techniques de renum´erota-tion, m´emoire, ordonnancement, pagination.
Abstract:
Direct methods for solving sparse linear systems are known for their large memory re-quirements that can represent the limiting factor to solve large systems. The work done during this thesis concerns the study and the optimization of the memory behaviour of a sparse direct method, the multifrontal method, for both the sequential and the par-allel cases. Thus, optimal memory minimization algorithms have been proposed for the sequential case. Concerning the parallel case, we have introduced new scheduling strate-gies aiming at improving the memory behaviour of the method. After that, we extended these approaches to have a good performance while keeping a good memory behaviour.
In addition, in the case where the data to be treated cannot fit into memory, out-of-core factorization schemes have to be designed. To be efficient, such approaches require to overlap I/O operations with computations and to reuse the data sets already in memory to reduce the amount of I/O operations. Therefore, another part of the work presented in this thesis concerns the design and the study of implicit out-of-core techniques well-adapted to the memory access pattern of the multifrontal method. These techniques are based on a modification of the standard paging policies of the operating system using a low-level tool (MMUM&MMUSSEL).
Keywords:
Sparse matrices, multifrontal method, reordering techniques, memory, scheduling, paging.