1. variance reduction techniques to improve efficiency of calculation B
PHITS
Multi-Purpose Particle and Heavy Ion Transport code System
variance reduction techniques to improve efficiency of calculation B
January 2017 revised
title

2. Contents of Lecture Neutron deep penetration calculation
[weight window]
　Weight windows
　[weight window generator]
Function of automatically production of [weight window]

3. Concept of weight in Monte Carlo calculation
Example：track length tally
Weight： Importance of the particle in Monte Carlo simulation
always to be 1 for normal calculation*
Artificially increase the probability of rare event occurrences.
Frequency distribution per a history cannot be calculated,
e.g. [t-deposit] with output = deposit, NO MORE event generator!
*Not the case for low-energy neutron transport simulation

4. Difference between cell importance and weight window methods
“Cell importance method” assign a single value of weight for each cell.
“Weight window method” assign allowed weight range (window) for each cell and each energy group.
Efficient simulation with focusing on the important energy region, such as high-energy neutron.
WU=2.5
WU=2.5
I1=1
I2=3
WU=1.25
W1=1
W=1
W1=1
Weight
Weight
Weight
WU=0.83
WL=0.5
WL=0.5
W2=1/3
WL=0.25
W2=0.5
region1
region2
WL=0.17
region1
region2
region1
region2
Number of particle 1 　→
energy group1
energy group2
I2/I1=3
Number of particle 1 → 1 1→
W1/W2=2
Cell importance method
Weight window method

5. Parameters Related to Weight Window
wupn: Ratio of the upper and lower limits of allowed weight (D=5.0)
mxspln: Maximum number of split per event (D=5.0)
mwhere: Timing for considering weight window (D=0)
　　　　　　　　-1: reaction, 0: both, 1: crossing surface
These parameters should be defined in [parameters] section.
See manual “4.2.4　Cut off time, cut off weight, and weight window” in more detail.

6. Check the Trajectory of Single Particle
Let’s execute “weight-hist.inp”, and check the neutron trajectory
Air
Water
（r= 10cm，
Depth = 5cm×2）
5MeV
neutron
weight-trajectory.eps

7. Set [Weight Window] Section
Activate the 1st [weight window] section in weight-hist.inp, and execute PHITS!
weight-hist.inp
[weight window]
part = neutron
reg ww1
Lower limit of allowed weights, WL, for each cell is defined in [weight window]
Upper limit of allowed weights, WU, is automatically set to WL x wupn (D=5)
Set value in each region
Same weight window for all energies
WU=5 x WL =1.25
W1=1
Weight
WL=0.25
WL=0.25
Cell 1
Cell 2
weight-trajectory.eps
Weight of neutron is within the weight window range -> No split and No Russian Roulette

8. Let’s Decrease Weight Window in Cell 2
weight-hist.inp
[weight window]
part = neutron
reg ww1
Decrease the lower limit of allowed weights in cell 2, and execute PHITS!
WU=1.25
W1=1
WU=0.625
Weight
WL=0.25
WL=0.125
Cell 1
Cell 2
Particle Split
weight-trajectory.eps
Neutron is divided into 2 because its weight (=1) is higher than the upper limit (=0.625)

9. Let’s Decrease Weight Window in Cell 2 EXTREMELY!
weight-hist.inp
[weight window]
part = neutron
reg ww1
Decrease the lower limit of allowed weights in cell 2 EXTREMELY, and execute PHITS!
WU=1.25
W1=1
Weight
WL=0.25
WU=0.025
WL=0.005
Cell 1
Cell 2
weight-trajectory.eps
Divided into neutrons of mxspln(D=5) by each event (crossing surface or reaction)
It is better to set 2~3 for max weight window ratio between neighboring cells

10. Set different Weight Window for Each Energy Group
weight-hist.inp
[weight window]
part = neutron
eng = 2 e5
reg ww ww2
Let’s “off” the 1st [weight window], and
activate the 2nd one & execute PHITS
Maximum energy for each group
2 energy groups are defined
But the weight window are the same for each group
WU=1.25
WU=1.25
W1=1
W1=1
WU=0.625
WU=0.625
Weight
Weight
WL=0.25
WL=0.25
WL=0.125
WL=0.125
Cell 1
Cell 2
Cell 1
Cell 2
Particle Split
Particle Split
E < 0.01 MeV
0.01 MeV < E < 105MeV
weight-trajectory.eps
Neutron is divided into 2 because its weight (=1) is higher than the upper limit (=0.625)
(Same as 2 pages before)

11. Increase Weight Window Only for Low-Energy Neutrons
weight-hist.inp
[weight window]
part = neutron
eng = 2 e5
reg ww ww2 * *
Increase weight window for low-energy neutrons by 10 times, and execute PHITS
WU=12.5
WU=6.25
WU=1.25
WL=2.5
Low-energy neutron is killed
WL=1.25
W1=1
W1=1
WU=0.625
Weight
Weight
WL=0.25
WL=0.125
Cell 1
Cell 2
Russian
Roulette
Cell 1
Cell 2
Particle Split
E < 0.01 MeV
0.01 MeV < E < 105MeV
weight-trajectory.eps
Reduce the production of low-energy neutrons,
and concentrate on high-energy neutron transport simulation
Get poor and better statistics at shallower and deeper locations, respectively

12. Example of calculation using [weight window]
Execute PHITS using weight.inp, and check your CPU time
weight.inp
Geometry, source, history are the same as imp.inp
[weight window]
set: c10[1.0]
part = neutron
eng = 2
1.0e e5
reg ww ww2
1 c10*2.5**(-1) 2.5**(-1)
2 c10*2.5**(-2) 2.5**(-2)
3 c10*2.5**(-3) 2.5**(-3)
・・・
Cylindrical Concrete
(r = 50cm, depth = 1.8 m)
14MeV
neutron
Depth of each cell = 15 cm
total cpu time = sec
Decrease weight window of neighboring cells by 1/2.5 for particle split
Set same weight window for all energy groups (because c10=1.0)
weight-dose-xz.eps
weight-dose-xz_err.eps

13. Example of calculation using [weight window]
Increase the weight window for low-energy neutrons by 10 times, and execute PHITS
weight.inp
[weight window]
set: c10[10.0]
part = neutron
eng = 2
1.0e e5
reg ww ww2
1 c10*2.5**(-1) 2.5**(-1)
2 c10*2.5**(-2) 2.5**(-2)
3 c10*2.5**(-3) 2.5**(-3)
・・・
Most low-energy neutrons are immediately killed when they produced, because of Russian Roulette
Low-energy neutrons have short range, so statistical uncertainties of deeper locations do not change that much
total cpu time = >
14.53 sec
weight-dose-xz.eps
weight-dose-xz_err.eps

14. Comparison of CPU time total cpu time = 19.53 sec
Normal calculation (No [importance], No [weight window])
total cpu time = sec
Use [importance]
total cpu time = sec
Use [weight window]; same value for all energies
total cpu time = sec
Use [weight window]; different values for low- and high-energy neutrons
total cpu time = sec

15. Dose Reduction Rate in Concrete
90 cm
180 cm
Good agreement except for deeper location doses obtained from default setting
You can improve the efficiency of your calculation by appropriately setting weight window for each neutron energy group

16. Contents of Lecture Neutron deep penetration calculation
[weight window]
　Weight windows
　[weight window generator]
Function of automatically production of [weight window]

17. Weight Window Generator [t-WWG]
Tally to generate appropriate weight window automatically in each cell
mesh=reg only because [weight window] allows mesh=reg.
【Procedure of automatically generation】
To generate the weight window, track length of particle normalized by a volume in each cell is calculated (flux: 1/cm2). Same method with [t-track].
It is necessary to define a volume in each cell in [volume] section.
[ T - WWG ]
mesh = reg
reg = {1-12}
part = neutron
e-type = 1
ne = 2 e5
axis = wwg
file = WWG.out
The weight window is given by a flux in each cell normalized by the maximum flux.
In a cell where flux is zero, the weight window is given by the minimum flux with finite value.
This procedure applies to each particle and energy bin.

18. Neutron Deep Penetration Calculation with WWG
Calculate neutron transport in thick shield and dose rate distribution to depth
Execute “imp.inp” and check geometry.
WWG.inp
[ P a r a m e t e r s ]
icntl =
maxcas =
maxbch = …
14MeV
neutron
weight-dose-xz.eps
100 cm radius x 180 cm thick cylinder

19. Check Output file of [t-WWG]
Let’s execute “WWG.inp” with icntl=0, and check “WWG.out”.
WWG.out
Initial value of weight window
WWG.inp
[ T - WWG ]
mesh = reg
reg = {1-12}
part = neutron
e-type = 1
ne = 2 $ 2 energy bin e5
axis = wwg
file = WWG.out
[ Weight Window ]
set: c71[0.0] c72[c E-08] c73[ E-05]
set: c74[0.0] c75[c E-08] c76[ E-05] part = neutron
eng = 2
E E+05
reg ww ww2
1 ( E-05+c71)/c73 ( E-05+c74)/c76
2 ( E-05+c71)/c73 ( E-05+c74)/c76
3 ( E-05+c71)/c73 ( E-05+c74)/c76
4 ( E-05+c71)/c73 ( E-06+c74)/c76
5 ( E-06+c71)/c73 ( E-06+c74)/c76
6 ( E-06+c71)/c73 ( E-07+c74)/c76
7 ( E-07+c71)/c73 ( E-07+c74)/c76
8 ( E-07+c71)/c73 ( E-08+c74)/c76
9 ( E-08+c71)/c73 ( E+00+c75)/c76
10 ( E-08+c71)/c73 ( E+00+c75)/c76
11 ( E+00+c72)/c73 ( E+00+c75)/c76
12 ( E+00+c72)/c73 ( E+00+c75)/c76
Minimum flux
Maximum flux
weight-dose-xz_err.eps
Without weight window
WWG gives minimum value in the cell where flux is zero.

20. Particle Deep Penetration Calculation with WWG
Method for efficient deep penetration of particles using results of [t-WWG]
In the 1st transport calculation, the initial weight window are generated by [t-WWG].
In the 2nd, the weight window are updated by initial values of weight window.
After the 3rd, calculations are repeated until particles are transported to overall, and then you can obtain the appropriate weight window.
　After checking the weight window, you can adjust the total history.

21. Recalculation using Output Results of [t-WWG]
Activate the infl: {WWG.out} in “WWG.inp”, and execute PHITS
WWG.out
The weight window obtained by the 2nd calculation
Total history = 1,500
[ Weight Window ]
set: c71[0.0] c72[c E-09] c73[ E-05]
set: c74[0.0] c75[c E-10] c76[ E-05]
part = neutron
eng = 2
E E+05
reg ww ww2
1 ( E-05+c71)/c73 ( E-05+c74)/c76
2 ( E-05+c71)/c73 ( E-05+c74)/c76
3 ( E-05+c71)/c73 ( E-05+c74)/c76
4 ( E-05+c71)/c73 ( E-06+c74)/c76
5 ( E-06+c71)/c73 ( E-06+c74)/c76
6 ( E-06+c71)/c73 ( E-07+c74)/c76
7 ( E-06+c71)/c73 ( E-07+c74)/c76
8 ( E-07+c71)/c73 ( E-08+c74)/c76
9 ( E-07+c71)/c73 ( E-08+c74)/c76
10 ( E-08+c71)/c73 ( E-09+c74)/c76
11 ( E-09+c71)/c73 ( E-09+c74)/c76
12 ( E-09+c71)/c73 ( E-10+c74)/c76
Use the 1st weight window
“WWG.out” is updated every execution
Do not change input file name “WWG.inp”
weight-dose-xz_err.eps
There is no region where flux is zero.

22. Execute 3rd Calculation
Let’s execute 3rd calculation and check distribution to depth and radial directions.
Total history = 1,500
Use the 2nd weight window.　
“WWG.out” is updated every execution.
weight-dose-reg.eps
Dose reduction rate in concrete
weight-dose-xz_err.eps
＊Statistic of photon dose in deeper location is not enough because WWG is not applied to photon.
As the same weight window is used in a cell,
the large difference among fluxes in a cell causes a region with poor statistic in a cell.

23. Divide Cell to Radial Direction
Let’s execute “WWG2.inp” with cells divided to radial direction and check geometry.
Calculate the effective dose rate in cell 60 (r= cm, z= cm).
WWG2.inp
[ P a r a m e t e r s ]
icntl =
maxcas =
maxbch =
emin(2) = E-10
dmax(2) =
file(6) = phits.out
file(7) = c:/phits/data/xsdir.jnd
file(14) = c:/phits/data/trxcrd.dat
$ infl: {WWG2.out} …
weight-dose-xz.eps
Divide cell into 5 parts to radial direction
R=0～20cm, 20～40cm, 40～60cm,60～80cm, 80～100 cm

24. Particle Transport Calculation in Geometry with Cell Divided to Radial Direction
Let’s execute 1st calculation using “WWG2.inp” with “icntl=0” and generate the initial weight window.
Total history = 1,500
WWG2.out
[ Weight Window ]
set: c71[0.0] c72[c E-09] c73[ E-04]
set: c74[0.0] c75[c E-08] c76[ E-03]
part = neutron
eng = 2
E E+05
reg ww ww2
1 ( E-04+c71)/c73 ( E-03+c74)/c76
2 ( E-04+c71)/c73 ( E-04+c74)/c76
3 ( E-04+c71)/c73 ( E-04+c74)/c76
4 ( E-04+c71)/c73 ( E-05+c74)/c76 ……
weight-dose-xz_err.eps
Without weight window

25. Divide cell to transport particles overall efficiently
Particle Transport Calculation in Geometry with Cell Divided to Radial Direction
Let’s activate the infl: {WWG2.out} in “WWG2.inp” and execute 2nd and 3rd calculations.
weight-dose-xz_err.eps
weight-dose-xz_err.eps
weight-dose-xz_err.eps
Results of 2nd calculation
Results of 3rd calculation
【reference】
Results of 3rd calculation with “WWG.inp”
Divide cell to transport particles overall efficiently

26. Increase the number of history and check effective dose rate
Change maxbch=5 in WWG2.inp, increase the number of history, and execute calculation
Calculate dose rate in cell 60(r= cm, z= cm)
Check dose rate in “weight-dose-reg.out”
weight-dose-reg.out
x: Serial Num. of Region
y: dose [(\muSv/h)/(source/sec)]
p: xlin ylog afac(0.8) form(0.9)
h: x n n y(neutron ),l3 n
# num reg volume neutron r.err
E E
E E
E E
E E
E E
E E
E E
E E
E E
E E
E E
E E
weight-dose-xz_err.eps
＊omit photon dose rate

27. Task： Deep Penetration Calculation for 14 MeV Photon
Let’s change the [source] section in “WWG2.inp” to use 14 MeV photon with isotropic direction and calculate photon dose rate in cell 60.
Firstly, generate the initial weight window with [t-WWG].
Initial history number (maxcas=1500, maxbch=1)
Disable “infl command”
In [T-WWG], particle is photon, energy bin is 0-1MeV and 1-105MeV.
WWG2.inp for this task
[ P a r a m e t e r s ]
icntl =
maxcas =
maxbch = 1 …
$ infl: {WWG2.out}
[ T - WWG ]
mesh = reg
reg = {1-60}
part = photon
e-type = 1
ne = 2 e5 …
[ S o u r c e ]
s-type = 1
proj = photon
e0 = …
dir = all
weight-dose-xz_err.eps
Without weight window

28. Task： Deep Penetration Calculation for 14 MeV Photon
Activate infl command
and use the initial weight window
weight-dose-xz_err.eps
Warning due to many particle productions
Without weight window
Small values of weight window are set in a cell where photon does not come,
and the number of split particle increases.
It is better to increase the number of history and transportation of particles to generate the initial weight window.

29. Task： Deep Penetration Calculation for 14 MeV Photon
Change History number (maxcas=30000, maxbch=1), disable infl command and generate the initial weight window
Activate infl command and calculate dose rate in cell 60 with the initial weight window
weight-dose-xz_err.eps
With the initial weight window
weight-dose-xz_err.eps
Cell60　8.27E-11(μSv/h)/(source/sec)
Statistical error
Without weight window

30. Summary
“Weight window method” assign allowed weight range (window) for each cell and each energy group.
[t-WWG] generate appropriate weight window automatically in each cell and energy.
The weight window is given by a flux in each cell normalized by the maximum flux.
It is necessary to define a volume in each cell in [volume] section.
It is convenient for particle transport calculation to read output file of [t-WWG]（= the [weight window] section） by the infl command except for the 1st calculation.
Note that warning message due to many particle productions appears when statistics of the 1st calculation is very bad.