F.Yuasa at ACAT2002
Multidimensional Integration
Package: DICE
and
its parallelization
F.Yuasa / KEK
K.Tobimatsu / Kogakuin Univ.
S.Kawabata / KEK
ACAT2002
24-28, Jun. 2002 at MSU, Moscow
F.Yuasa at ACAT2002
BASES
•
•
•
•
•
Multidimensional Integration Package
Stratified and Importance sampling method
Singular function can be integrated
Up to 100 dimensions
Heavily used in GRACE framework
F.Yuasa at ACAT2002
When singularities go along a diagonal line,
we need appropriate variable transformation.
y
Y
x
(x , y)
X
(X , Y)
DICE
F.Yuasa at ACAT2002
• Developed by K.Tobimatsu and S.Kawabata
– First version of DICE in 1992
– Research Reports of Kogakuin Univ. No.72
(1992)
• Divide the integral region into 2Ndim hypercubes
• Two kinds of sampling method
• DICE Input
– Ndim, Expected Error, # of Sampling points,
Maximum division level, Maximum # of
iteration
F.Yuasa at ACAT2002
Ndim=2
How to divide Hypercube
regular
Level = 2
Level = 3
Regular sampling and random sampling
F.Yuasa at ACAT2002
Example 1
I1 
1

0
2x 2
 dx1dx 2 (x  x 1)2   2
1
2
0
1
F.Yuasa at ACAT2002
F.Yuasa at ACAT2002
Example 2
x  (1 x  x )
I2    dx1dx 2 2
2
2
(x1  x 2  a )  
1 1
1
1
2
2
2
1
2 2
2
2
F.Yuasa at ACAT2002
F.Yuasa at ACAT2002
Example 3
 x  x  x  (1 x  x  x )
I3     dx1dx 2 dx 3
2
2
2
2
(x1  x 2  x 3  a )  
1 1 1
1 1 1
2
1
2
2
2
3
2
1
2 2
2
2
2
3
F.Yuasa at ACAT2002
F.Yuasa at ACAT2002
Example 4
 (1 x  x  x  x )
I3      dx1dx 2dx 3dx 4 f  2 2 2 2 2 2 2 ,
(x1  x 2  x 3  x 4  a )  
1 1 1 1
1 1 1 1
4
2
1
2
2
Rx x x x ,
and
2
1
2
2
2
3
2
4
f (R)  (R 1/4)(R 1/2) (R  3/4)
2
2
3
2
4
F.Yuasa at ACAT2002
f (R)  (R 1/4)(R 1/2) (R  3/4)
2

 10
a  0.8
1
 10
2
a  0.8
F.Yuasa at ACAT2002
Results of I4
Package
DICE-mpi
ParInt1.1
BASES
Analytical
results
Eps =10**(-1)
(3.1408
+-0.0029)E-02
0.031216955
+-0.0016138
(3.174411
+-0.047391)E-02
Eps =10**(-2)
(-1.0662
+-0.0011)E-02
-0.010710149
+-0.00055466
(-1.056945
+-0.035239)E-02
0.0314313
-0.0106773
F.Yuasa at ACAT2002
Results of I4 (part2)
Package
Eps =10**(-3)
Eps =10**(-4)
DICE-mpi
(-1.6070
+-0.0023)E-02
-0.01593609
0.01293747
(-1.614249
+-0.009806)E-02
-0.0160761
(-1.6676
+-0.0024)E-02
-0.0114787661
0.0206822789
(-1.639091
+-0.033053)E-02
-0.0166246
ParInt1.1
BASES
Analytical
results
F.Yuasa at ACAT2002
Example 5
• More complicated Integrand
• # of dimensions = 4
• # of lines in FORTRAN = about 300 lines
F.Yuasa at ACAT2002
Results of Example5
Package
Result
# of Sample
points
DICE-mpi
1 processor
(1.0638+-0.0011)E-13
24798768
ParInt1.1
1 processor
(1.0622+-0.0529)E-13
2100000110
BASES
(1.064086+-0.000337)E- 9996350
13
F.Yuasa at ACAT2002
Results of Example5 (part2)
Package
Result
# of Sample
points
DICE-mpi
1 processor
( -1.1529+-0.0019)
E-13
24798768
ParInt1.1
1 processor
(-1.1452+-0.0652)E-13
2100000110
BASES
(-1.154076+-0.000681)E- 9996350
13
F.Yuasa at ACAT2002
Results of Example5 (part3)
Package
Result
# of Sample
points
DICE-mpi
1 processor
( -8.8675+-0.0872)E-15
0.98%
33956096
ParInt1.1
1 processor
We did not try
BASES
We did not try
F.Yuasa at ACAT2002
Parallelization
• We use MPI for the parallelization.
• Parallelization is useful for higher
dimensional integrand
• Parallelization is useful for complicated
integrand
• Example 5 is calculated by the parallelized
DICE
F.Yuasa at ACAT2002
Speed Up
Example5
# of CPUs 1
CPU time 3232.80
[sec]
2
1648.40
4
839.00
8
431.68
Speed Up 1.00
1.96
3.85
7.49
F.Yuasa at ACAT2002
Summary
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•
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We have developed DICE.
DICE is available to Vector Processor.
DICE is available to Parallel Processor.
We have used MPI for parallelization.
For the complicated integrand, parallelization
shows good scalability.