Parallel Iterative Poisson Solver for a Distributed Memory Architecture

Transcription

1 Parallel Iteratve Posso Solver for a Dstrbted Memory Archtectre Erc Dow Aerospace Comptatoal Desg Lab Departmet of Aeroatcs ad Astroatcs

2 2 Motvato Solvg Posso s Eqato s a commo sbproblem may mercal schemes otably the solto of the compressble Naver-Stokes eqatos. Ths step s typcally the most expesve of ay teratve method so a effcet Posso solver s essetal.

3 3 Problem Descrpto Posso s eqato o ay arbtrary geometry wth homogeeos Drchlet bodary codtos. 2 f x 0 x

4 Iteratve Solto Techqes I 2D Posso s eqato ca be dscretzed wth Fte Dffereces: Ths sggests the followg teratve scheme kow as the Jacob Iteratve Method: Ths s rather slow to coverge ad ca be made faster by sg the pdated vales of the solto as soo as they are avalable (Gass-Sedel Method): 4 f x 2 4 f x 2 4 f x 2 4

5 5 Iteratve Solto Techqes For very large problems (especally 3D) a drect solve s mpractcal. A desred level of accracy ca be attaed wth a teratve solver: smply stop teratg whe a desred level of accracy s acheved. Ths s ot possble wth drect solto techqes sch as LU. Potetal to save a great deal of comptatoal effort

6 6 Parallelzato Jacob method seems lke a poor choce relatve to the Gass-Sedel method: Slower to coverge Reqres twce as mch storage However Parallelzato of the Jacob method s straght forward. Iheret Data Parallelsm: The same operatos are performed o each grd pot so t makes sese to dstrbte the data amog processes. All vales ca be pdated cotemporaeosly. We eed to be more clever wth the Gass-Sedel method

7 7 Red Black Node Orderg If the sm of the row ad colm dex of a ode s eve the ode s colored red otherwse the ode s colored black. Update all of the red odes parallel sg the vales at black odes. Update all of the black odes parallel sg the vales at red odes. Restores Data Parallelsm

8 8 Dstrbtg the Data Spectral Graph Parttog: Recrsvely dvde the doma to (roghly) eqal peces.

9 9 Dstrbtg the Data Ths scheme does ot reslt a optmal partto.e. oe that creates parttos of eqal sze whle mmzg the mber of edge cts. Reslt: Large varato sze of bodary betwee sbdomas. Ths creates a commcato bottleeck ad some processes wll be watg o others to fsh commcato.

10 0 Implemetato Seral ad parallel solvers mplemeted C MPI sed for parallelzato. Each process s gve a collecto of odes to pdate At the ed of each terato each thread seds ad receves vales eeded for ext terato Call to MPI_Barrer reqred at the ed of each commcato block to prevet faster processes from racg ahead Solvers r o Beowlf clster 2 ad 4 odes

11 Reslts Seral ad parallel codes agree o steady state solto.

12 2 Reslts: Jacob Method 75 x 75 Nodes 50 x 50 Nodes

13 3 Reslts: Gass-Sedel Method 75 x 75 Nodes 50 x 50 Nodes

14 4 Coclsos Speedp hghly depedet o problem sze. Doblg the mber of grd pots each dmeso from 75 to 50 qadrples the workload of each process bt oly dobles the amot of commcato reqred. Ths explas the speedp observed. Ths s actally good ews: typcally oly se teratve solvers for very large problems. Sce the speedp seems to crease wth problem sze t makes sese to parallelze these solvers. Jacob otperforms Gass-Sedel parallel performace de to lmted commcato.

15 5 Ftre Work Mltgrd: Very effcet teratve solto techqe blt arod basc teratve solvers sch as Jacob ad Gass-Sedel. Parallel compoet s already place smply tegrate parallel solvers to create a parallel Mltgrd method. Itegrate graph parttog scheme to solver (crretly a collecto of separate MATLAB fctos). Speedp celg: Is there a maxmm attaable speedp as the problem sze creases (other tha the obvos deal oe-to-oe speedp)?

16 6 Refereces [] G. Strag Comptatoal Scece ad Egeerg. Wellesley MA:Wellesley- Cambrdge Press 2007.