Optimizing LCLS2 taper profile with genetic
algorithms: preliminary results
X. Huang, J. Wu, T. Raubenhaimer, Y. Jiao,
S. Spampinati, A. Mandlekar, G. Yu
2/29/2012
2/29/2012
1
An Overview of Multi-Objective Genetic Algorithms
• Multi-objective optimization
– Goal: to find the Pareto optimal set
– Traditional approach: Weighted sum of objectives and its variants.
– Evolutionary approach: converge to the Pareto front in one run.
• Genetic algorithms
– Manipulate a set of solutions (a population) toward the optimal front
with operations that simulate biological evolution.
– Three operators
• Selection – apply the evolution pressure toward the optimal front
• Crossover – create new solution (child) by combining two solutions
(parents)
• Mutation – alters an existing solution to create a new one.
2/29/2012
2
• Pros and Cons
– Obtain global optimum (more likely) despite complexity of the
problem.
– Optimize multiple objectives simultaneously.
– Easy to apply constraints.
– But it can be much slower than gradient-based methods.
2/29/2012
3
Domination and the Pareto set
2/29/2012
4
The NSGA-II algorithm
• NSGA (non-dominated sorting genetic algorithm) -II
Selection (of parents)
Crossover
Mutation
K. Deb, IEEE Transtions On Evolutionary Computation Vol 6, No 2,April 2002
2/29/2012
5
NSGA-II with parallel computation
• Use Matlab script for control and processing
– The algorithm is implemented in matlab
– Post-processing is in matlab
• Parallel computation via submitting multiple jobs to a
cluster
– Use file input/output as communication between external program
(Genesis) and matlab.
– I/O time limits the average number of nodes in use when
computation time is short.
35 min per generation with up to 60 processors, or 4.5 s per evaluation, up from 20 s
for individual evaluation.
However the speed gain from parallel computing will be much higher for timedependent runs.
2/29/2012
6
LCLS2 Taper Optimization
• Undulator tapering is required for LCLS2 to reach TW
power because of SASE saturation.
• Taper profile optimization is critical to capture as many
electrons as possible in coherent emission.
– Exploration of profile models is necessary.
• Should phase between undulator segments be included in
optimization?
2/29/2012
7
Taper Models Considered
For
z  z0
Basic 8 variables
Aw ( z)  Aw0[1  a( z  z0 )b ]
Cubic 9 variables
Aw ( z)  Aw0[1  a( z  z0 )  b( z  z0 )2  c( z  z0 )3 ]
Quartic 8 variables
Aw ( z)  Aw0[1  a( z  z0 )2  b( z  z0 )4 ]
Focusing scheme
K 0 , z  z1
K ( z )  K 0 (1  r1 z ), z1  z  z2 ,
K 0 (1  r2 z ), z  z2
Adding phase shift variables to the above models. So far we only varied the first
few phase shifts after exponential growth.
2/29/2012
8
GA setup
• Objectives: 2
– Power
– “Emittance”: beam size x divergence at the exit, a convenient way
to introduce diversity
• Population: 600
• Termination condition: about 100 generations or
converged.
• Evolving mutation and crossover probability
2/29/2012
9
The basic 8 variable model (0118)
gen 1
gen 11
gen 21
gen 41
gen 61
gen 81
gen 103
1.8
power (TW)
1.6
1.4
1.2
1
0.8
0.6
0.06
0.08
(a, z0)
2/29/2012
0.1
0.12
0.14
0.16
emittance
(b, K0)
0.18
0.2
0.22
0.24
Aw  Aw0 [1  a(
param
eter
low
a
0.01
z0
10
b
1.1
K0
20
r1
-0.005
z1
20
r2
-0.01
z2-z1
0
(r1, z1)
z  z0 b
) ]
Lu  z0
high
delta best
0.3
0.001 0.1043
40
0.2
13.1
3.3
0.01
2.0359
40
0.1
34.4
0.005 0.00005 0.0018
80
0.2
80.0
0.01 0.00005 0.0061
70
0.2
28.9
(r2, z2-z1)
10
The basic 8 variable model with 7 phase shifts (0115b)
gen 1
gen 16
gen 31
gen 46
gen 61
gen 76
gen 91
gen 100
1.8
1.6
power (TW)
1.4
1.2
1
0.8
0.6
0.4
0.06
0.07
(a, z0)
2/29/2012
0.08
0.09
0.1
0.11
0.12
emittance
0.13
(b, K0)
0.14
0.15
0.16
Introduce phase shifts in gaps
following undulators 5 to 11.
param
eter
low
a
0.01
z0
10
b
1.1
K0
20
r1
-0.005
z1
20
r2
-0.01
z2-z1
0
(r1, z1)
high
delta best
0.3
0.001
0.114
40
0.2
16.8
3.3
0.01
2.072
40
0.1
34.9
0.005 0.00005 0.0008
80
0.2
74.3
0.01 0.00005 0.0022
70
0.2
9.3
(r2, z2-z1)
11
power (TW)
The cubic model (9 variables) (0119)
01192012
1.8
Aw ( z )  Aw0 [1  a(
L0  100 m
1.6
1.4
1.2
gen 1
gen 11
gen 21
gen 41
gen 61
gen 81
gen 101
1
0.8
0.6
0.05
0.1
(z0, a1)
2/29/2012
0.15
0.2
0.25
emittance
0.3
(a2, a3)
0.35
z  z0
z  z0 2
z  z0 3
)  b(
)  c(
) ]
L0
L0
L0
parame
ter
z0
a1
a2
a3
K0
r1
z1
r2
z2-z1
Low
10
-0.1
0.001
-0.1
20
-0.01
20
-0.01
0
high
40
0.1
0.3
0.1
40
0.01
80
0.01
70
delta best
0.2
18.8
0.001
0.0118
0.001
0.0551
0.001
0.0538
0.1
27.9
0.0005
-0.005
0.2
38.1
0.0005
-0.009
0.2
66.8
0.4
(K0, r1)
(z1,r2)
12
The quadratic and quartic model (0112)
1.8
gen 1
gen 21
gen 41
gen 61
gen 81
gen 104
power (TW)
1.6
1.4
Aw ( z )  Aw0 [1  a(
z  z0 2
z  z0 4
)  b(
) ]
Lu  z0
Lu  z0
1.2
1
0.8
0.06
0.07
(a, z0)
2/29/2012
0.08
0.09
0.1
0.11 0.12
emittance
0.13
(b, K0)
0.14
0.15
0.16
(r1, z1)
(r2, z2-z1)
13
Summary of time-independent results
case
Nvar
inc in 10 emittance
capture
generation population max Power gen
(um)
taper ratio ratio
#
"01182012" basic 8
"01152012b
"
8+7
"01152012b" no
phase
#
TW
%
um
%
103
600
1.760
0.20%
0.0753
0.075
43.0% 1+a x^b
100
600
1.830
0.27%
0.0790
0.0816
41.1% from random
0.0816
35.1%
1.563
"01212012" phase 7
109
600
1.805
0.00%
0.0751
0.0762
based on 01182012
43.4% @ gen 47, 1.753 TW
"01192012" cubic 9
100
600
1.743
0.00%
0.0702
0.0722
44.3% 1+a x+b x^2+c x^3
"01202012" 9+7
115
600
1.842
0.31%
0.0794
0.0804
42.0%
0.0804
34.7%
0.0783
42.1% 1+a x^2 + b x^4
"01202012" no phase
"01122012" quartic 8
2/29/2012
1.521
104
600
1.799
0.00%
0.0757
14
Effects of phase shift variables
comparison of phase variables 2/2/2012
0.3
Based on case 0118.
0.2
phase
8+7
7 phase
9+7
0.1
0
-0.1
-0.2
1
2
3
4
wiggler index-4
5
6
7
Inside undulators, phase rotation and energy loss both change. In the gaps,
the two can be decoupled. Can this improve the performance?
2/29/2012
15
0.7
1.2
0.6
1
0.5
0.8
0.4
0118: basic
0119: cubic
0112: quartic
0.4
0.2
20
40
60
z (m)
80
100
0
0
120
n
0.15
 (nm)
0.155
80
100
120
0
0.5
1
1.5
2
s (um)
2.5
50
2.4
2.35
2.3
0118: basic
0119: cubic
0112: quartic
2.25
0.145
60
z (m)
0118: basic
0119: cubic
0112: quartic
2.2
0
20
40
60
z (m)
80
100
3
3.5
4
0118: basic
0119: cubic
0112: quartic
2.45
b (bunching factor)
P( ) (arb. units)
-10
40
1.2
0.8
2.5
0118: basic
0119: cubic
0112: quartic
-5
20
1.4
1
0118: basic
0119: cubic
0112: quartic
0.1
0
0
10
1.6
0.3
0.2
10
1.8
bn (bunching factor)
0.6
2
Power (TW)
1.4
bn
Power (TW)
Time dependent results with the taper profiles
0
-50
0
20
40
60
z (m)
80
100
120
The three model attain similar power. More study is needed to understand the results.
Taper profile slightly shifted (detuned to maximize for average power for the slices) to maintain
high power (but not optimized)
2/29/2012
16
Time dependent simulation with phase shifts
2
0118: basic
0121: phase only
0115b: basic+phase
Power (TW)
0.6
n
1
0.8
0118: basic
0121: phase only
0115b: basic+phase
b
Power (TW)
1.5
0.4
0.5
1.5
1
0118: basic
0121: phase only
0115b: basic+phase
0.2
0
0
20
40
60
z (m)
80
100
0
0
120
20
40
60
z (m)
80
100
0.5
0
120
1
2
s (um)
3
4
40
-10
10
0.146
0.148
0.15
0.152
 (nm)
0.154
0.156
2.45
2.4
2.35
2.3
n
-5
10
b (bunching factor)
0118: basic
0121: phase only
0115b: basic+phase
bn (bunching factor)
P( ) (arb. units)
2.5
0118: basic
0121: phase only
0115b: basic+phase
2.25
20
40
60
z (m)
0
80
100
0118: basic
0121: phase only
0115b: basic+phase
-20
-40
0
2.2
0
20
20
40
60
z (m)
80
100
120
The effects of phase shifts are not conclusive from results we got so far.
2/29/2012
17
Summary
• All cases without phase shifts converge to solutions with
similar beam power and taper ratio, with a capture ratio of
about 43%.
• Phase shifts only slightly increase beam power. But they
can considerably change capture ratio (e.g., from 35% to
41%).
• We will continue the exploration
– Other taper profile models
– Introduce other objective functions
– More time dependent studies
2/29/2012
18