Presentation PG-4037,Fast modal analysis with NX Nastran and GPUs, by Leonard Hoffnung at the AMD Developer Summit (APU13) November 11-13, 2013.

1. FAST
MODAL
ANALYSIS
WITH
NX
NASTRAN
AND
GPUS
LEONARD
HOFFNUNG
SIEMENS
PLM
SOFTWARE

2. INTRODUCTION
INTRO
/
FREQ
RESP
/
MODES
/
CONCLUSIONS

3. ABOUT
NX
NASTRAN
! Industry
standard
finite
element
package
from
Siemens
PLM
! Analysis
opSons
include:
‒ Stress,
vibraSon,
structural
failure
‒ Heat
transfer,
acousScs,
rotor
dynamics,
and
more
! Advanced
numerical
capabiliSes
and
proven
scalability:
‒ Problem
sizes
approaching
1
billion
dofs
‒ SMP
to
24
cores
‒ DMP
to
2048
nodes
3
|
FAST
MODAL
ANALYSIS
WITH
NX
NASTRAN
AND
GPUS
|
NOVEMBER
12,
2013
|
CONFIDENTIAL

4. MODAL
FREQUENCY
RESPONSE
OVERVIEW
NASTRAN
SOL
111
! Bread
and
buer
industrial
computaSon:
modal
frequency
response
! Widely
used
in
automoSve
&
aerospace
to
determine
response
under
varying
excitaSons
‒ OpSmize
weight,
rigidity
‒ Minimize
noise,
resonance
! Two
phase
calculaSon
more
efficient
than
direct:
‒ Modal
analysis
‒ Frequency
response
calculaSon
4
|
FAST
MODAL
ANALYSIS
WITH
NX
NASTRAN
AND
GPUS
|
NOVEMBER
12,
2013
|
CONFIDENTIAL

5. MODAL
FREQUENCY
RESPONSE
COMPUTATIONAL
STEPS
! EigensoluSon
-­‐-­‐
ℎ
normal
modes
of
𝑓× 𝑓
structural
matrices:
​ 𝐾↓𝑓𝑓 ​Φ↓𝑓ℎ =​ 𝑀↓𝑓𝑓 ​Φ↓𝑓ℎ ​Λ↓ℎℎ
! Frequency
response
-­‐-­‐
ℎ×ℎ
complex
linear
soluSon
at
each
of
𝑛 𝑟𝑒𝑠𝑝
frequencies:
(​ 𝐾↓ℎℎ +​ 𝜔↓𝑘 𝑖 ​ 𝐵↓ℎℎ −​ 𝜔↓𝑘↑2 ​ 𝑀↓ℎℎ )​ 𝑥↓𝑘 =​ 𝑏↓𝑘 , 𝑘=1,…, 𝑛𝑟𝑒𝑠𝑝
! All
parameters
large
in
typical
customer
usage:
‒ 𝑓-­‐size
10-­‐30M
for
model
fidelity
‒ ℎ-­‐size
10-­‐60K
for
modal
accuracy
‒ 𝑛𝑟𝑒𝑠𝑝
20K
for
detailed
response
graph
5
|
FAST
MODAL
ANALYSIS
WITH
NX
NASTRAN
AND
GPUS
|
NOVEMBER
12,
2013
|
CONFIDENTIAL

6. PERFORMANCE
CASE
STUDY
PR
MODEL
–
FREQUENCY
RESPONSE
COST
! Shell
dominated
SOL
111
model
‒ 245K
degrees
of
freedom
( 𝑓-­‐size)
‒ 1200
eigenpairs
(ℎ-­‐size)
‒ 20K
frequency
responses
( 𝑛𝑟𝑒𝑠𝑝)
! EigensoluSon
Sme:
30
minutes
! Frequency
response:
127
minutes
! Frequency
response
cost
𝑂( 𝑛𝑟𝑒𝑠𝑝 ∗​ℎ↑3 )
‒ EsSmated
run
Sme
in
decades
as
ℎ→60 𝐾
6
|
FAST
MODAL
ANALYSIS
WITH
NX
NASTRAN
AND
GPUS
|
NOVEMBER
12,
2013
|
CONFIDENTIAL

7. PERFORMANCE
CASE
STUDY
CUSTOMER
BENCHMARK
! More
typical
industrial
model:
‒ 11
million
degrees
of
freedom
( 𝑓-­‐size)
‒ Shell
dominated
model
‒ Approximately
3000
eigenpairs
(ℎ-­‐size)
‒ 300
frequency
responses
( 𝑛𝑟𝑒𝑠𝑝)
! Frequency
response
expensive,
but
modal
calculaSon
sSll
expensive
even
with
RDMODES:
‒ Modal
calculaSon:
375
minutes
‒ Frequency
response
Sme:
22
minutes
! Need
to
improve
performance
in
both
phases
7
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FAST
MODAL
ANALYSIS
WITH
NX
NASTRAN
AND
GPUS
|
NOVEMBER
12,
2013
|
CONFIDENTIAL

8. FREQUENCY
RESPONSE
INTRO
/
FREQ
RESP
/
MODES
/
CONCLUSIONS

9. FREQUENCY
RESPONSE
IMPLEMENTATION
DETAILS
OF
ORIGINAL
METHOD
! NX
Nastran
implementaSon
uses
symmetric
𝐿 𝐷​ 𝐿↑𝑇
factorizaSon
and
forward-­‐backward
subsStuSon:
For
𝑘=1,…, 𝑛𝑟𝑒𝑠𝑝
Assemble
𝐴=​ 𝐾↓ℎℎ +​ 𝜔↓𝑘 𝑖​ 𝐵↓ℎℎ −​ 𝜔↓𝑘↑2 ​ 𝑀↓ℎℎ
Factor
𝐴= 𝐿𝐷​ 𝐿↑𝑇
Solve
​ 𝑥↓𝑘 =​ 𝐴↑−1 ​ 𝑏↓𝑘 =​ 𝐿↑− 𝑇 ​ 𝐷↑−1 ​ 𝐿↑−1 ​ 𝑏↓𝑘
End
for
! NX
Nastran
sparse
factorizaSon
difficult
to
adapt
to
GPU:
‒
Disk
oriented
‒ Tuned
for
sparse
matrices
‒ Symmetric
pivoSng
required
for
stability
(indefiniteness)
9
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FAST
MODAL
ANALYSIS
WITH
NX
NASTRAN
AND
GPUS
|
NOVEMBER
12,
2013
|
CONFIDENTIAL

10. FREQUENCY
RESPONSE
IMPLEMENTATION
DETAILS
OF
REVISED
METHOD
! For
GPU
code,
use
LU
factorizaSon
instead:
For
𝑘=1,…, 𝑛𝑟𝑒𝑠𝑝
Assemble
𝐴=​ 𝐾↓ℎℎ +​ 𝜔↓𝑘 𝑖​ 𝐵↓ℎℎ −​ 𝜔↓𝑘↑2 ​ 𝑀↓ℎℎ
Factor
𝐴= 𝐿𝑈
Solve
​ 𝑥↓𝑘 =​ 𝐴↑−1 ​ 𝑏↓𝑘 =​ 𝑈↑−1 ​ 𝐿↑−1 ​ 𝑏↓𝑘
End
for
! OpenCL
port
of
LAPACK
zgesv
available
with
clMAGMA
and
clBLAS
‒ In
core
storage
‒ Dense
oriented
(okay
for
this
applicaSon)
‒ Benefit
mainly
in
factorizaSon
step
(cubic
operaSon
count)
10
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FAST
MODAL
ANALYSIS
WITH
NX
NASTRAN
AND
GPUS
|
NOVEMBER
12,
2013
|
CONFIDENTIAL

11. FREQUENCY
RESPONSE
IMPLEMENTATION
LINEAR
SOLVER
SELECTION
STRATEGY
! Original
NX
Nastran
sparse
symmetric
solver
‒ Spills
to
disk,
requires
minimal
memory
‒ Minimizes
flops
by
uSlizing
symmetry
‒ Takes
advantage
of
sparsity
! Improved
SMP
method
(system462=1
in
NXN9.0)
‒ In
core,
based
on
LAPACK
zsytrf/zsytrs
‒ Efficient
parallelizaSon
of
𝑛 𝑟𝑒𝑠𝑝
loop
‒ Large
memory
requirements
! OpenCL
method
(to
appear
in
NXN9
MP)
‒ In
core,
based
on
clMAGMA
zgesv (LU
factorizaSon)
‒ USlizing
GPU
for
best
performance
11
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FAST
MODAL
ANALYSIS
WITH
NX
NASTRAN
AND
GPUS
|
NOVEMBER
12,
2013
|
CONFIDENTIAL

12. FREQUENCY
RESPONSE
INITIAL
PERFORMANCE
COMPARISON
! Test
machine:
‒ Magny-­‐Cours
2.1
GHz,
24
cores
‒ 32GB
memory
‒ 4GB
TahiS
GPU
! GPU
roughly
40%
faster
than
24-­‐way
SMP
Model
Modes
e10k
1785
e20k
3631
e30k
5576
e40k
2:24:00
2:09:36
serial
1:55:12
smp=8
1:40:48
smp=24
1:26:24
GPU
1:12:00
0:57:36
0:43:12
7646
0:28:48
0:14:24
0:00:00
12
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FAST
MODAL
ANALYSIS
WITH
NX
NASTRAN
AND
GPUS
|
NOVEMBER
12,
2013
|
CONFIDENTIAL
e10k
e20k
e30k
e40k

13. FREQUENCY
RESPONSE
–
FURTHER
IMPROVEMENTS
SINGLE
PRECISION
ARITHMETIC
! Use
single
precision
on
GPU
for
improved
performance
‒ Higher
flop
rate
(typically
4-­‐5
Smes)
‒ Lower
memory
uSlizaSon
‒ (larger
dimension
problems
possible)
‒ Beer
scaling
with
larger
systems
‒ Single
precision
disadvantage:
lower
precision
‒ Accuracy
acceptable
for
most
engineering
purposes
‒ (largest
relaSve
error
of
​10↑−5 )
1
Double
precision
0.1
0.01
0.001
0.0001
0.00001
0.000001
0.0000001
1E-­‐08
13
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FAST
MODAL
ANALYSIS
WITH
NX
NASTRAN
AND
GPUS
|
NOVEMBER
12,
2013
|
CONFIDENTIAL
Single
precision
RelaSve
error

14. FREQUENCY
RESPONSE
–
FURTHER
IMPROVEMENTS
SINGLE
PRECISION
ACCURACY
AND
PERFORMANCE
! 40-­‐50%
reducSon
in
run
Sme
0:17:17
! Largest
example
only
possible
in
single
precision
0:14:24
Double
Single
0:11:31
0:08:38
Model
Modes
e10k
1785
0:05:46
e20k
3631
0:02:53
e30k
5576
e40k
7646
e60k
12088
14
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FAST
MODAL
ANALYSIS
WITH
NX
NASTRAN
AND
GPUS
|
NOVEMBER
12,
2013
|
CONFIDENTIAL
0:00:00
e10k
e20k
e30k
e40k
e60k

15. FREQUENCY
RESPONSE
–
FURTHER
IMPROVEMENTS
MATRIX
SUMMATION
ON
GPU
! Perform
addiSon
of
matrices
at
each
frequency
on
GPU
(assembly
step)
𝐴=​ 𝐾↓ℎℎ +​ 𝜔↓𝑘 𝑖​ 𝐵↓ℎℎ −​ 𝜔↓𝑘↑2 ​ 𝑀↓ℎℎ
! I.e.
store
​ 𝐾↓ℎℎ , ​ 𝐵↓ℎℎ , ​ 𝑀↓ℎℎ
in
GPU
buffers
and
sum
using
zaxpy/saxpy kernels:
𝐴≔​ 𝐾↓ℎℎ
𝐴≔ 𝐴+​ 𝜔↓𝑘 𝑖 ​ 𝐵↓ℎℎ
𝐴≔ 𝐴−​ 𝜔↓𝑘↑2 ​ 𝑀↓ℎℎ
! Minimizes
data
transfer
to/from
main
memory
! AddiSonal
GPU
memory
consumpSon
15
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FAST
MODAL
ANALYSIS
WITH
NX
NASTRAN
AND
GPUS
|
NOVEMBER
12,
2013
|
CONFIDENTIAL

16. FREQUENCY
RESPONSE
–
FURTHER
IMPROVEMENTS
MATRIX
SUMMATION
ON
GPU
PERFORMANCE
! Double
precision
best
result
(e30k):
‒ Time
reduced
30%
from
6:52
to
4:50
‒ 2x
faster
than
best
CPU
Sme
0:12:58
0:11:31
0:10:05
0:08:38
! Single
precision
best
result
(e40k):
‒ Time
reduced
22%
from
6:23
to
4:58
‒ 4x
faster
than
best
CPU
Sme
0:07:12
Double
Double
+
zaxpy
Single
Single
+
caxpy
0:05:46
0:04:19
0:02:53
! Best
scaling
with
largest
problems
‒ Limited
by
GPU
memory
0:01:26
0:00:00
e10k
16
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FAST
MODAL
ANALYSIS
WITH
NX
NASTRAN
AND
GPUS
|
NOVEMBER
12,
2013
|
CONFIDENTIAL
e20k
e30k
e40k

17. MODAL
ANALYSIS
INTRO
/
FREQ
RESP
/
MODES
/
CONCLUSIONS

18. MODAL
ANALYSIS
WITH
RDMODES
OVERVIEW
! RDMODES
–
proprietary
high-­‐performance
approximate
eigensolver
! Tuned
for
typical
customer
use
cases:
‒ Larger
models
(10
million+
dofs)
‒ Many
modes
(300+)
‒ Accelerated
computaSon
when
few
output
dofs
required
‒ Sufficient
accuracy
for
frequency
response
calculaSons
! Performance
up
to
20x
faster
than
Lanczos
! Demonstrated
DMP
scalability
to
2048
nodes
18
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FAST
MODAL
ANALYSIS
WITH
NX
NASTRAN
AND
GPUS
|
NOVEMBER
12,
2013
|
CONFIDENTIAL

19. MODAL
ANALYSIS
WITH
RDMODES
COST
BREAKDOWN
! RDMODES
method
comprised
of
mulSple
smaller
operaSons
–
five
areas
listed
below
! Costs
for
customer
benchmark:
! Dense
operaSons
good
candidates
for
GPU
‒ FactorizaSon,
eigensoluSon
19
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FAST
MODAL
ANALYSIS
WITH
NX
NASTRAN
AND
GPUS
|
NOVEMBER
12,
2013
|
CONFIDENTIAL
Wall
?me
Sparse
factorizaSon
18:40
Dense
factorizaSon
24:00
Sparse
eigensoluSon
9:33
Dense
eigensoluSon
‒ 11
million
dofs
‒ Shell
dominated
‒ 3000
modes
below
400
Hz
‒ 300
frequency
responses
Opera?on
65:00
Reduced
(dense)
eigensoluSon
21:16
Total
250:06

20. RDMODES
FACTORIZATION
CLASSIFICATION
! Fairly
large
quanSty
of
each
type
! Sparse
factorizaSons:
‒ Typically
too
large
to
treat
efficiently
as
dense
‒ NXN
mulSfrontal
solver
very
efficient
‒ Efficient
sparse
soluSon
on
GPU
difficult
(acSve
research)
! Dense
factorizaSons:
‒ Model
dependent,
typically
small
‒ Symmetric
posiSve
definite,
may
use
clMAGMA
dposv
‒ Candidate
for
GPU
20
|
FAST
MODAL
ANALYSIS
WITH
NX
NASTRAN
AND
GPUS
|
NOVEMBER
12,
2013
|
CONFIDENTIAL

21. RDMODES
FACTORIZATION
DENSE
FACTORIZATION
COST
COMPARISON
! Dense
factorizaSon
wall
Smes
‒ Costs
include
factorizaSon
and
miscellaneous
assembly
Dense
factoriza?on
?mes
0:25:55
0:23:02
! As
with
frequency
response,
GPU
suitable
above
0:20:10
threshold
NXN
0:17:17
‒ Threshold
of
5000
for
this
example
! Dense
in
core
methods
helpful
LAPACK
GPU
0:14:24
0:11:31
0:08:38
0:05:46
! GPU
ineffecSve
for
this
model
‒ (all
linear
soluSons
relaSvely
small)
0:02:53
0:00:00
Serial
21
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FAST
MODAL
ANALYSIS
WITH
NX
NASTRAN
AND
GPUS
|
NOVEMBER
12,
2013
|
CONFIDENTIAL
SMP=24

22. RDMODES
EIGENSOLUTION
CLASSIFICATION
! Sparse
eigensoluSons:
‒ Large
number
‒ Sparse,
relaSvely
large
dimension
‒ Inexpensive
with
NXN
sparse
eigensolvers
! Dense
eigensoluSons:
‒ Large
number
‒ Dense,
small-­‐medium
dimension
‒ Candidate
for
GPU
! Reduced
eigensoluSon:
‒ Only
one
instance
‒ Dense,
fairly
large,
many
modes
‒ Strong
candidate
for
GPU
22
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FAST
MODAL
ANALYSIS
WITH
NX
NASTRAN
AND
GPUS
|
NOVEMBER
12,
2013
|
CONFIDENTIAL

23. RDMODES
EIGENSOLUTION
DENSE
SOLUTION
METHODS
! Householder
type
soluSon
for
real
symmetric
problem
(dsyev):
‒ Reduce
to
tridiagonal:
‒ Eigenvalues
of
tridiagonal:
‒ Compute
eigenvectors:
‒ Then
​ 𝑄↑𝑇 𝐴𝑄= 𝑇
​ 𝑍↑𝑇 𝑇𝑍=Λ
Φ= 𝑄𝑍
𝐴Φ=ΦΛ
! Efficient
choice
for
dense
problems,
and/or
many
eigenvectors
needed
‒ High
memory
consumpSon
! Transform
generalized
eigenvalue
problem
as
follows:
‒ Factor:
‒ Solve:
‒ Generalized
eigensoluSon:
𝑀= 𝐿​ 𝐿↑𝑇
​ 𝐿↑−1 𝐾​ 𝐿↑− 𝑇 𝑋= 𝑋Λ
𝐾(​ 𝐿↑− 𝑇 𝑋)= 𝑀(​ 𝐿↑− 𝑇 𝑋)Λ
23
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ANALYSIS
WITH
NX
NASTRAN
AND
GPUS
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NOVEMBER
12,
2013
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24. RDMODES
EIGENSOLUTION
DENSE
EIGENSOLUTION
SCALABILITY
! Dimensions
range
from
2800
to
8800
‒ Dense
problems,
modes
variable
! GPU
beneficial
for
larger
sizes
! Total
Smes
(serial)
-­‐-­‐
50%
reducSon:
‒ 56:29
‒ 15:30
‒ 7:29
(all
Lanczos)
(all
LAPACK)
(using
GPU)
2:24:00
Serial
0:14:24
0:01:26
Lanczos
LAPACK
GPU
0:00:09
0:00:01
2000
2:24:00
4000
8000
SMP=24
0:14:24
0:01:26
! Total
Smes
(SMP)
–
36%
reducSon:
‒ 52:22
‒ 4:41
‒ 3:00
(all
Lanczos)
(all
LAPACK)
(using
GPU)
Lanczos
LAPACK
GPU
0:00:09
0:00:01
2000
24
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ANALYSIS
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NX
NASTRAN
AND
GPUS
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NOVEMBER
12,
2013
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4000
8000

25. RDMODES
EIGENSOLUTION
GPU
SUPPORT
! Householder
methods
well
suited
(as
expected)
! Larger
dimension
dense
problems
benefit
from
the
GPU
‒ And
are
the
most
Sme
consuming
! Send
most
expensive
problems
to
GPU
! Threshold
set
to
3800
for
this
test
‒ Note:
opSmal
threshold
depends
on
hardware
and
SMP
25
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FAST
MODAL
ANALYSIS
WITH
NX
NASTRAN
AND
GPUS
|
NOVEMBER
12,
2013
|
CONFIDENTIAL

26. RDMODES
EIGENSOLUTION
MOST
SIGNIFICANT
COST
COMPONENTS
! Reduced
eigensoluSon
‒ Not
ideally
suited
to
NXN
Lanczos
eigensolver
‒ Unique,
but
large
(14K
dofs)
‒ Many
eigenvectors
needed
‒ GPU
30%
speedup
(both
SMP
and
serial)
! GPU
in
RDMODES
conclusions
‒ Dense
and
reduced
eigensoluSons
benefit
‒ Threshold
for
dense
eigensoluSon
‒ Dense
factorizaSon
benefits
from
LAPACK:
lile
addiSonal
benefit
on
GPU
! Sparse
methods
not
supported
yet
Reduced
Eigensolu?on
0:57:36
NXN
LAPACK
GPU
0:50:24
0:43:12
0:36:00
0:28:48
0:21:36
0:14:24
0:07:12
0:00:00
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NX
NASTRAN
AND
GPUS
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NOVEMBER
12,
2013
|
CONFIDENTIAL
Serial
SMP=24

27. RDMODES
AND
FREQUENCY
RESPONSE
BENCHMARK
PERFORMANCE
RESULTS
! SMP=24,
customer
benchmark
8:24:00
Frequency
response
7:12:00
! Compared
to
NXN
system:
‒ Frequency
response
3x
faster
‒ Reduced
eigensoluSon
2.8x
faster
‒ FactorizaSon
28%
faster
‒ Dense
eigensoluSon
9x
faster
‒ 30%
reducSon
in
total
run
Sme
Reduced
eigensoluSon
6:00:00
4:48:00
Dense
eigensoluSon
3:36:00
FactorizaSon
2:24:00
Other
1:12:00
! Compared
to
LAPACK:
‒ Frequency
response
3x
faster
‒ Reduced
eigensoluSon
2x
faster
‒ 10%
reducSon
in
total
run
Sme
0:00:00
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NXN
LAPACK
GPU

28. RDMODES
EIGENSOLUTION
SINGLE
PRECISION
! Performance
advantages
with
single
precision
eigensoluSon
‒ As
with
linear
soluSon
in
frequency
response,
single
precision
faster
on
GPU
‒ Lower
GPU
memory
consumpSon
‒ (larger
problems)
! Dense
eigensoluSons
(customer
benchmark)
–
35-­‐40%
speedup:
Double
precision
Single
precision
7:01
4:16
SMP=24
3:41
2:23
Serial
! Reduced
eigensoluSon
also
benefits
–
20%
speedup:
‒ 3:05
to
2:29
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AND
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12,
2013
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CONFIDENTIAL

29. CONCLUSIONS
INTRO
/
FREQ
RESP
/
MODES
/
CONCLUSIONS

30. CONCLUSIONS
! Significant
benefit
with
GPU
for
certain
computaSon
types
‒ Frequency
response
calculaSon
2x-­‐3x
faster,
dense
eigensoluSon
2x
faster
‒ AddiSonal
35-­‐50%
improvement
possible
with
single
precision
‒ 30%
lower
turnaround
Sme
for
typical
customer
benchmark
! Efficient
dense
matrix
algebra
on
GPU
with
clMath,
clMAGMA
! Many
thanks
to:
Ben-­‐Shan
Liao,
Wei
Zhang
(Siemens
PLM),
Antoine
Reymond
(AMD)
Thank
you!
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31. DISCLAIMER
&
ATTRIBUTION
The
informaSon
presented
in
this
document
is
for
informaSonal
purposes
only
and
may
contain
technical
inaccuracies,
omissions
and
typographical
errors.
The
informaSon
contained
herein
is
subject
to
change
and
may
be
rendered
inaccurate
for
many
reasons,
including
but
not
limited
to
product
and
roadmap
changes,
component
and
motherboard
version
changes,
new
model
and/or
product
releases,
product
differences
between
differing
manufacturers,
sotware
changes,
BIOS
flashes,
firmware
upgrades,
or
the
like.
AMD
assumes
no
obligaSon
to
update
or
otherwise
correct
or
revise
this
informaSon.
However,
AMD
reserves
the
right
to
revise
this
informaSon
and
to
make
changes
from
Sme
to
Sme
to
the
content
hereof
without
obligaSon
of
AMD
to
noSfy
any
person
of
such
revisions
or
changes.
AMD
MAKES
NO
REPRESENTATIONS
OR
WARRANTIES
WITH
RESPECT
TO
THE
CONTENTS
HEREOF
AND
ASSUMES
NO
RESPONSIBILITY
FOR
ANY
INACCURACIES,
ERRORS
OR
OMISSIONS
THAT
MAY
APPEAR
IN
THIS
INFORMATION.
AMD
SPECIFICALLY
DISCLAIMS
ANY
IMPLIED
WARRANTIES
OF
MERCHANTABILITY
OR
FITNESS
FOR
ANY
PARTICULAR
PURPOSE.
IN
NO
EVENT
WILL
AMD
BE
LIABLE
TO
ANY
PERSON
FOR
ANY
DIRECT,
INDIRECT,
SPECIAL
OR
OTHER
CONSEQUENTIAL
DAMAGES
ARISING
FROM
THE
USE
OF
ANY
INFORMATION
CONTAINED
HEREIN,
EVEN
IF
AMD
IS
EXPRESSLY
ADVISED
OF
THE
POSSIBILITY
OF
SUCH
DAMAGES.
ATTRIBUTION
©
2013
Advanced
Micro
Devices,
Inc.
All
rights
reserved.
AMD,
the
AMD
Arrow
logo
and
combinaSons
thereof
are
trademarks
of
Advanced
Micro
Devices,
Inc.
in
the
United
States
and/or
other
jurisdicSons.
SPEC
is
a
registered
trademark
of
the
Standard
Performance
EvaluaSon
CorporaSon
(SPEC).
Other
names
are
for
informaSonal
purposes
only
and
may
be
trademarks
of
their
respecSve
owners.
31
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