Sept.2011 Transactions of Nanjing University of Aeronautics
&.
Astronautics
Vo
l.
28
No. 3
SOLVERS
FOR
SYSTEMS
OF
LARGE
SPARSE
LINEAR
AND
NONLINEAR
EQUATIONS BASED ON
MULTI-GPUS
Liu
Sha
1
,
Zhong
Chengwen
1
•
2
, Chen
Xiaopeng
3
O.
National Key Laboratory of
Sc
ience and Technology on Aerodynamic
De
sign and Research ,
Northwestern Polytechnical University, Xi' an , 710072 , P. R. China;
2.
Center for High Performance Computing, Northwestern Polytechnical University , Xi' an , 710072 , P. R. China;
3.
School of Mechanics, Civ
i\
Engineering and Architecture , Northwestern
Polytechnical University
, Xi' an , 710072 , P. R. China)
Abstract: Numerical treatment of engineering application problems often eventually results in a solution of sys-
tems of linear or nonlinear equations. The solution process using digital computational devices usually takes
tremendous time due to the extremely large size encountered
in
most real-world engineering applications.
So
,
practical solvers for systems of linear and nonlinear equations based on multi graphic process units
(GPUs)
are
proposed in order to accelerate the solving process. In the linear and nonlinear solvers
, the preconditioned bi-con-
jugate gradient stable (PBi-CGstab) method and the Inexact Newton method are used to achieve the fast and sta-
ble convergence behavior. Multi-GPUs are utilized to obtain more data storage that large size problems need.
Key
words: general purpose graphic process unit
(GPGPU);
compute unified device architecture
(CUDA);
sys-
tem of linear equations; system of nonlinear equations; Inexact Newton method; bi-conjugate gradi-
ent stable (Bi-CGstab) method
CLC number: TP391 Document code:A Article
ID:
1005-1120(2011)03-0300-09
INTRODUCTION
Mathematica1
modeling
of
engineering
prob-
1ems
often
1eads
to
systems
of
1inear
or
non1inear
equations.
The
solution
of
such
resulting
equa-
tions
utilizing
numerica1
too1s
via
digita1
computa-
tiona1
devices
is
usually
of
very
time-consuming
,
because
most
rea1-world
engineering
app1ications
are
often
of
extremely
1arge
size
for
computation.
In
this
paper
,
with
the
Inexact
Newton
method
and
the
preconditioned
bi-conjugate
gradient
sta-
b1e
(PBi-CGstab)
method
,
linear
and
non1inear
solvers
based
on
muti
graphic
process
units
(GPUs)
are
proposed
for
1arge
sca1e
prob1ems.
Genera1
purpose
GPU
(GPGPU)
technique
denotes
the
imp1ementation
of
genera1
purpose
computing
by
using
programmab1e
GPUs[
I].
It
has
been
wide1y app1ied
in
many
computationa1
areas
owing
to
its
more
powerfu1
floating
calcu1a-
tion
abilities
and
wider
bandwidth
compared
with
the
traditiona1
CPU[2-3].
Furthermore
,
the
inher-
ent
sing1e-instruction-multip1e-data
(SIMD)
mechanism
for
GPGPU
operation
renders
this
technique
suitab1e
for
massive1y
10aded
ca1cu1a-
t
lO
ns.
A seria1
of
GPU-based
linear
a1gebra
opera-
tions
were
proposed
by
Krüger
et
a1
in
2003[4].
The
first
GPU-based
conjugate
gradie
l}
t
solver
for
unstructured
matrices
was
proposed
by
Bo
1z
et
aF5].
Buatois
et
a1
proposed
a
genera1
sparse
lin-
ear
sol
ver
using
CG
method
in
2 0 0
9
∞.
Cevahir
et
a1
deve10ped
a
fast
CG
based
on
GPUs
with
some
nove1
optimization
techniques[7].
These
arti-
cles
show
great
speedup
ratio
of
GPU
to
CPU.
In
early
work
,
Zhong
and
Liu
proposed
a
fast
solver
which
has
a
great
speedup
ratio
about
30
on
sing1e
GPU[8].
In
this
paper
,
multi-GPUs
are
used
to
obtain
more
data
storage
space
that
1arge
size
prob1ems
need.
In
the
case
of
1inear
solver
,
the
Bi-CGstab
method
can
afford
to
solve
the
sys-
Received date: 2010-10-13; revision received date: 2011-03-10
E-mail:virgilius@mai
l.
nwpu.edu.cn
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