parallel iterative solution of the hermite collocation equations on gpus

Parallel Iterative solution of the

Hermite Collocation Equations

on GPUs

Emmanuel N. Mathioudakis

Co-Authors : Elena Papadopoulou – Yiannis Saridakis – Nikolaos Vilanakis

TECHNICAL UNIVERSITY OF CRETE

DEPARTMENT OF SCIENCES

APPLIED MATHEMATICS AND COMPUTERS LABORATORY

73100 CHANIA - CRETE - GREECE



Talk Overview

Hermite Collocation for elliptic BVPs &

Derivation of the Collocation linear system

Development of a parallel algorithm for the

Schur Complement method on Shared

Memory Parallel Architectures

Parallel implementation on multicore computers

with GPUs



( , ) ( , ) , ( , )

( , ) ( , ) , ( , )

u x y f x y x y

u x y g x y x y

L

BBV

P

( , ) ( , ) 0 , ( , ):

( , ) ( , ) 0 , (,

, )

y y yx x xi j i j i j

y y yx x xi j i j i j

ai j

u f

u g

L

B

Hermite Collocation Method



Hermite Collocation Method…



Red – Black Collocation Linear system

with 0

( , ) ( , ) ( , ) , ( , )

( , ) ( , ) , ( , )

u x y u x y f x y x y

u x y g x y x y

2M

od

el

Pro

ble

m



Helmholtz Collocation

Matrix





Red – Black Collocation Linear system

The collocation matrix is large, sparse and enjoys no pleasant

properties (e.g. symmetric, definite)

Iterative

+

Parallel





Iterative Solution

with



Iterative Solution



Eigenvalues of Collocation matrix



Schur Complement Iterative Solution



Parallel Iterative Solution of Collocation Linear system

on Shared Memory Architectures

Uniform Load Balancing between core threads

Minimal Idle Cycles of core threads

Minimal Communication



case of ns = 2p

( )1Rx ( )

2Rx ( )

3Rx ( )

4Rx



case of ns = 2p

( )1

Bp

x

( )2

Bp

x

( )3

Bp

x

( )4

Bp

x



case of ns = 2p

V1

( )1Rx ( )

2Rx ( )

3Rx ( )

4Rx( )

1B

px

( )

2B

px

( )

3B

px

( )4

Bp

x

V2 V3 V4 V5 V6 V7 V8

V1 V2 V3 V4 V5

Mapping into a fixed size Architecture of N Cores

case of k = 2p/N even 2k virtual threads

l=(j-1)k+1,…,jk

l=2p+(j-1)k+1,…,2p+jk

j=1,…,N

( )Bl

x

( )Rl

x

V2 . . .

Pj



Parallel Schur Complement Iterative Solution



Parallel BiCGSTAB



The Dirichlet Helmholtz Problem

2100( 0.1)2with

( , ) 10 ( ) ( ) , ( , ) [0,1] [0,1]

( ) ( ) x

u x y x y x y

x x x e



HP SL390s

6 core [email protected]

24GB memory

Oracle Linux 6.3 x64

PGI 13.5 Fortran

PCI-e gen2 x16

HP SL390s – Tesla M2070 GPUs

+ 2 x



Realization on HP SL390s Tesla GPU machine Iterations / Error measurements

ns BiCGSTAB

Iterations || b – Axn||2

256 294 6.06e-11

512 589 2.85e-11

1024 1161 1.39e-11

2048 3726 9.59e-12



Realization on HP SL390s Tesla GPU machine Time measurements

Conclusions

• A new parallel algorithm implementing the Schur complement with BiCGSTAB iterative method for Hermite Collocation equations has been developed.

• The algorithm is realized on Shared Memory multi-core machines with GPUs .

• A performance acceleration of up to 30% is observed.



Future work

• Design an efficient parallel Schur complement algorithm of the Hermite Collocation equations for Multiprocessor /Grid machines with GPUs.



parallel iterative solution of the hermite collocation equations on gpus

Technology

computers laboratory

j i j i j y y yx x x

multicore computers

chania crete

b hermite collocation

hermite collocation

i j u f u g

parallel algorithm