WPEC 2016 meetings
OECD/NEA Paris,
1011 May 2016
Continuous energy adjustments:
a potential breakthrough(?)
Manuele Auero & Massimiliano Fratoni
dreaming about continuous-energy cross sections adjustment
thanks to:
G. Palmiotti and M. Salvatores
eXtended Generalized Perturbation Theory (XGPT)
Sensitivity analysis and uncertainty quantication adopting
continuous-energy functions.
First results of XS adjustment via XGPT
Is it possible to produce adjusted continuous energy XS?
Manuele Auero UC Berkeley
Continuous energy adjustments
aaa
Generalized Perturbation Theory capabilities
Eect of a perturbation of the parameter x on the response R :
R ≡ dR /R
Sx
dx /x
Considered response functions:
R
hΣ1 , φi
hΣ2 , φi
†
φ , Σ1 φ
=
†
φ , Σ2 φ
E [e1 ]
R =
E [e2 ]
R
R
= ke
=
Eective multiplication factor
Reaction rate ratios
Bilinear ratios (Adjoint-weighted quantities)
Something else
Manuele Auero UC Berkeley
Continuous energy adjustments
aaa
Generalized sensitivities: a couple of examples
Flattop-Pu
Manuele Auero UC Berkeley
Continuous energy adjustments
aaa
Generalized sensitivities: a couple of examples
Popsy (Flattop) - F28/F25 - Pu-239 - chi total
F28/F25 sensitivity - 10 generations - ENDF/B-VII
0,3
Sensitivity per lethargy unit
0,2
Extended SERPENT-2
TSUNAMI-1D
0,1
0
-0,1
-0,2
-0,3
4
10
5
6
10
10
7
10
Energy (eV)
Manuele Auero UC Berkeley
Continuous energy adjustments
aaa
Generalized sensitivities: a couple of examples
Popsy (Flattop) - Leff - Pu-239 - fission
Effective prompt lifetime sensitivity - 8-16 generations - ENDF/B-VII
0,2
Sensitivity per lethargy unit
0,1
0
-0,1
-0,2
Extended SERPENT-2
TSUNAMI-1D (EGPT)
-0,3
-0,4
3
10
4
10
5
10
Energy (eV)
Manuele Auero UC Berkeley
6
10
7
10
Continuous energy adjustments
aaa
Generalized sensitivities: a couple of examples
No bi-linear ratio sensitivity available in Scale (?)...
Adopting EGPT for comparison against Serpent
∂
`e
Sx
∂
Sx e
`
=
=
∂ ke /ke
∂ a 1/v
∂ x /x
∂ ke /ke
∂ x /x
∂ a 1/v
Sx e
`
'
,
,
∂ ke /ke
∂ a 1/v
∂ Sxke
∂ a 1/v
∂ ke /ke
=
∂ ke /ke
∂ a 1/v
∂ a 1/v
∗ S ke
− Sxke
∗k
e − ke
x
ke
Manuele Auero UC Berkeley
Continuous energy adjustments
aaa
Generalized sensitivities: a couple of examples
Flattop - keff - U-238 - elastic scattering
Effective multiplication factor sensitivity - 10 generations - ENDF/B-VII
0.06
Sensitivity per lethargy unit
0.04
0.02
0.00
-0.02
ela
Ext. Serpent -- Ela Scatt. XS ( f0)
st
ela
Ext. Serpent -- 1 Leg. moment ( f1)
nd
-0.04
ela
Ext. Serpent -- 2 Leg. moment ( f2)
nd
ela
Ext. Serpent -- 3 Leg. moment ( f3)
st
-0.06
ela
ERANOS -- 1 Leg. moment ( f1)
4
10
5
10
Energy (eV)
Manuele Auero UC Berkeley
6
10
7
10
Continuous energy adjustments
aaa
Generalized sensitivities: a couple of examples
PWR pin cell
Manuele Auero UC Berkeley
Continuous energy adjustments
aaa
Generalized sensitivities: a couple of examples
UAM TMI-1 PWR cell - F28/F25 - U-238 - disappearance
F28/F25 sensitivity - 10 generations - ENDF/B-VII
0.15
Sensitivity per lethargy unit
Extended SERPENT-2
TSUNAMI-1D
Difference (S-T)
0.1
0.05
0
-0.05
-2
10
0
10
2
10
Energy (eV)
Manuele Auero UC Berkeley
4
10
6
10
Continuous energy adjustments
aaa
Generalized sensitivities: a couple of examples
UAM TMI-1 PWR cell - αcoolant - U-238 - disappearance
coolant void reactivity coeff. sensitivity - 4 generations - ENDF/B-VII
0.6
Extended SERPENT-2
TSUNAMI-1D
Difference (S-T)
Sensitivity per lethargy unit
0.5
0.4
0.3
0.2
0.1
0
-0.1
-2
10
0
10
2
10
Energy (eV)
Manuele Auero UC Berkeley
4
10
6
10
Continuous energy adjustments
aaa
Generalized sensitivities: a couple of examples
UAM TMI-1 PWR cell - αcoolant - U-235 - nubar total
coolant void reactivity coeff. sensitivity - 4 generations - ENDF/B-VII
1
Extended SERPENT-2
TSUNAMI-1D
Difference (S-T)
0.8
Sensitivity per lethargy unit
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-2
10
0
2
10
10
4
10
Energy (eV)
Manuele Auero UC Berkeley
Continuous energy adjustments
aaa
Continuous energy vs. multi-group
Cross sections
Covariance matrices
20
Energy [MeV]
2
0.2
0.02
0.02
0.2
2
20
Energy [MeV]
Manuele Auero UC Berkeley
Continuous energy adjustments
aaa
Continuous-energy function sensitivity approach
Continuous energy uncertainty propagation formula:
EZmax EZmax
Var [R ]
R
SΣ (E ) · COV
=
Σ(E ) , Σ(E 0 ) · SΣR
E
0
dE dE
0
Emin Emin
It is hard to solve double integrals with Monte Carlo transport.
Legacy approach: multi-group discretization
+ sum over bin-averaged sensitivities
New approach: eigenvalue expansion
+ sum over continuous-energy integrated sensitivities
Manuele Auero UC Berkeley
Continuous energy adjustments
aaa
Eigenvalue decomposition of the covariance matrix
COV [Σ(E ) , Σ(E )] =
0
∞
X
j
j=
Manuele Auero UC Berkeley
U (E ) · V · U (E 0)
1
j
Continuous energy adjustments
j
aaa
Eigenvalue decomposition of the covariance matrix
COV [Σ(E ) , Σ(E )] ∼
0
n
X
j
j=
Manuele Auero UC Berkeley
U (E ) · V · U (E 0)
1
j
Continuous energy adjustments
j
aaa
Continuous-energy function sensitivity approach
Continuous energy uncertainty propagation formula:
EZmax EZmax
Var [R ]
R
SΣ (E ) · COV
=
Σ(E ) , Σ(E 0 ) · SΣR
E
0
dE dE
0
Emin Emin
COV
0
Σ(E ) , Σ(E ) ∼
n
X
j =1
Var [R ]
∼
n
X

j ·
Uj (E ) · V
j · U (E 0 )
j
2
EZmax
R
j =1

Uj (E ) · SΣ (E ) dE 
Var [R ]
∼
V
V
Emin
n
X
j =1
Manuele Auero UC Berkeley

j · SR 2
Uj
Continuous energy adjustments
aaa
Singular Value Decomposition
Original
image
SVD/POD
5 basis functions
Multi-group
5 energy groups
A=imread("massimo.png"); [A, map]=gray2ind(A,255);
[U, S, V]=svd(A);
A_SVD_5 = U(:,1:5) * S(1:5,1:i) * V(:,1:5)0 ;
imwrite(A_SVD_5, gray(255), "massimo_5.png");
Manuele Auero UC Berkeley
Continuous energy adjustments
aaa
Singular Value Decomposition
Original
image
SVD/POD
Multi-group
10 basis functions 10 energy groups
A=imread("massimo.png"); [A, map]=gray2ind(A,255);
[U, S, V]=svd(A);
A_SVD_10 = U(:,1:10) * S(1:10,1:i) * V(:,1:10)0 ;
imwrite(A_SVD_10, gray(255), "massimo_10.png");
Manuele Auero UC Berkeley
Continuous energy adjustments
aaa
Singular Value Decomposition
Original
image
SVD/POD
Multi-group
20 basis functions 20 energy groups
A=imread("massimo.png"); [A, map]=gray2ind(A,255);
[U, S, V]=svd(A);
A_SVD_20 = U(:,1:20) * S(1:20,1:i) * V(:,1:20)0 ;
imwrite(A_SVD_20, gray(255), "massimo_20.png");
Manuele Auero UC Berkeley
Continuous energy adjustments
aaa
Singular Value Decomposition
Original
image
SVD/POD
Multi-group
40 basis functions 40 energy groups
A=imread("massimo.png"); [A, map]=gray2ind(A,255);
[U, S, V]=svd(A);
A_SVD_40 = U(:,1:40) * S(1:40,1:i) * V(:,1:40)0 ;
imwrite(A_SVD_40, gray(255), "massimo_40.png");
Manuele Auero UC Berkeley
Continuous energy adjustments
aaa
Singular Value Decomposition
Original
image
SVD/POD
Multi-group
80 basis functions 80 energy groups
A=imread("massimo.png"); [A, map]=gray2ind(A,255);
[U, S, V]=svd(A);
A_SVD_80 = U(:,1:80) * S(1:80,1:i) * V(:,1:80)0 ;
imwrite(A_SVD_80, gray(255), "massimo_80.png");
Manuele Auero UC Berkeley
Continuous energy adjustments
aaa
Two simple case studies
HMF-64
PMF-35
Manuele Auero UC Berkeley
Continuous energy adjustments
aaa
Random cross sections
3000 dierent 208 Pb ENDF les from TENDL-2013
Random MF2-MT151 (resonances), MF3-MT1, MF3-MT2 (elastic),
MF3-MT51-58,91 (inelastic), and MF3-MT102 (n, γ ) processed
with NJOY
3000 ACE les with random cross sections
The random continuous energy XS reect the UNCERTAINTIES
and their CORRELATIONS (according to TENDL-2013)
Continuous-energy covariances reconstructed from random XS
Manuele Auero UC Berkeley
Continuous energy adjustments
aaa
Random cross sections
208
Pb elastic xs [b]
2
10
Reference
1
10
-2
-2
-2
7×10
-2
8×10
Energy [MeV]
9×10
-1
1×10
1
10
208
Pb elastic xs [b]
6×10
0
10
-1
3×10
-1
-1
4×10
5×10
-1
6×10
Energy [MeV]
Manuele Auero UC Berkeley
Continuous energy adjustments
aaa
Random cross sections
208
Pb elastic xs [b]
2
10
10 random evaluations
Reference
1
10
-2
-2
-2
7×10
-2
8×10
Energy [MeV]
9×10
-1
1×10
1
10
208
Pb elastic xs [b]
6×10
0
10
-1
3×10
-1
-1
4×10
5×10
-1
6×10
Energy [MeV]
Manuele Auero UC Berkeley
Continuous energy adjustments
aaa
Random cross sections
208
Pb elastic xs [b]
2
10
50 random evaluations
Reference
1
10
-2
-2
-2
7×10
-2
8×10
Energy [MeV]
9×10
-1
1×10
1
10
208
Pb elastic xs [b]
6×10
0
10
-1
3×10
-1
-1
4×10
5×10
-1
6×10
Energy [MeV]
Manuele Auero UC Berkeley
Continuous energy adjustments
aaa
Random cross sections
208
Pb elastic xs [b]
2
10
500 random evaluations
Reference
1
10
-2
-2
-2
7×10
-2
8×10
Energy [MeV]
9×10
-1
1×10
1
10
208
Pb elastic xs [b]
6×10
0
10
-1
3×10
-1
-1
4×10
5×10
-1
6×10
Energy [MeV]
Manuele Auero UC Berkeley
Continuous energy adjustments
aaa
Proper Orthogonal Decomposition of Nuclear Data
We want a set of orthogonal basis functions

Σei (E ) = Σ0 (E ) · 1 +
Manuele Auero UC Berkeley
n
X
j =1
bΣ,j so that:

αij · bΣ,j (E )
Continuous energy adjustments
aaa
Proper Orthogonal Decomposition of Nuclear Data
Rel. basis function [a.u.]
Basis # 3
Rel. basis function [a.u.]
Rel. basis function [a.u.]
3 basis functions from the POD of 208 Pb (n, ela)
Basis # 4
Basis # 5
2
0
10
Ref. elastic xs [b]
Ref. elastic xs [b]
Ref. elastic xs [b]
10
2
1
10
1
10
10
1
10
0
10
Basis # 3
0
10
Energy [MeV]
1
10
Basis # 4
-2
6×10
-2
7×10
Manuele Auero UC Berkeley
-2
8×10
Energy [MeV]
-2
9×10
-1
1×10
Basis # 5
-1
10
Continuous energy adjustments
0
10
Energy [MeV]
aaa
Proper Orthogonal Decomposition of Nuclear Data
2
2
Energy [MeV]
20
Energy [MeV]
20
0.2
0.02
0.02
0.2
Continuous
Energy
0.2
Energy [MeV]
2
20
0.02
0.02
SVD/POD
Multi-group
20 basis functions >100 ene g.
0.2
2
20
Energy [MeV]
Manuele Auero UC Berkeley
Continuous energy adjustments
aaa
XGPT+POD: calculating sensitivities to CE basis functions
We need to calculate the eect on the response
due to a perturbation on Σ equal to bΣ,j
SbRΣ, =
j
dR /R
=
dbΣ,j
EZmax
R
bΣ,j (E ) · SΣR (E ) dE
Emin
The calculation of SbRΣ,j is the main innovation of
XGPT (implemented in Serpent).
Manuele Auero UC Berkeley
Continuous energy adjustments
aaa
XGPT+POD: uncertainty propagation
From the Proper Orthogonal Decomposition of Nuclear data...
Σi (E ) ' Σei (E ) = Σ0 (E ) ·
!
n
X
1+
αij · bΣ,j (E )
j =1
...and the basis functions sensitivity coecients
approximate the response function
RΣi ' RfΣi = RΣ0 ·
Manuele Auero UC Berkeley
RΣ
i
SbR
Σ,j
, we can
for each random XS
n
X
1+
αij · SbRΣ,j
Σi
!
j =1
Continuous energy adjustments
aaa
XGPT+POD: uncertainty propagation
Estimating the ke distribution in the simple case study
ke Σi ' kf
e Σi = ke Σ0 ·
n
X
1+
αij · SbkΣ,ej
!
j =1
The ke for all the N (3000) random XS Σi were calculated
in a single Serpent run (ACE le for Σ0 ) with n =
bases
50
The XGPT+POD results are compared to TMC results
(3000 separate Serpent runs)
Manuele Auero UC Berkeley
Continuous energy adjustments
aaa
XGPT+POD: uncertainty propagation in
PMF-35
PMF-35
Manuele Auero UC Berkeley
Continuous energy adjustments
aaa
XGPT+POD: uncertainty propagation in
PMF-35 ke
PMF-35 - keff uncertainty - XGPT + POD vs. TMC
500
Total Monte Carlo
XGPT + POD
Number of counts per bin [-]
400
300
200
100
0
-1000
-500
0
500
keff - keff [pcm]
Manuele Auero UC Berkeley
1000
1500
2000
Continuous energy adjustments
aaa
XGPT+POD: uncertainty propagation in
PMF-35 ke
PMF-35 - keff estimates - XGPT + POD vs. TMC
2000
keff - keff (XGPT + POD) [pcm]
1500
1000
500
0
-500
-1000
-1000
-500
0
500
1000
keff - keff (Independent MC runs) [pcm]
Manuele Auero UC Berkeley
1500
2000
Continuous energy adjustments
aaa
XGPT+POD: uncertainty propagation in
HMF-64
PMF-35
Manuele Auero UC Berkeley
Continuous energy adjustments
aaa
XGPT+POD: uncertainty propagation in
HMF-64 ke
HMF-64 - keff uncertainty - XGPT + POD vs. TMC
500
Total Monte Carlo
XGPT + POD
Number of counts per bin [-]
400
300
200
100
0
-4000
-2000
0
2000
keff - keff [pcm]
Manuele Auero UC Berkeley
4000
6000
Continuous energy adjustments
aaa
XGPT+POD: uncertainty in
PMF-35 & HMF-64
Table: Standard deviation, skewness and kurtosis of the PMF-35 ke
distribution from TENDL-2013 208 Pb cross section data.
Methos
Standard deviation
skewness
kurtosis
TMC
426 pcm
0.81
3.62
XGPT
423 pcm
0.80
3.58
Table: Standard deviation, skewness and kurtosis of the HMF-64 ke
distribution from TENDL-2013 208 Pb cross section data.
Standard deviation
skewness
kurtosis
TMC
1326 pcm
0.74
3.49
XGPT
1371 pcm
0.81
3.65
Manuele Auero UC Berkeley
Continuous energy adjustments
aaa
Advanced Lead Fast Reactor European Demonstrator
Developed within the European FP7 LEADER project
ALFRED, is a small-size (300MWth) pool-type LFR.
Manuele Auero UC Berkeley
Continuous energy adjustments
aaa
Advanced Lead Fast Reactor European Demonstrator
Developed within the European FP7 LEADER project
171 FAs are subdivided into two radial zones with dierent
plutonium fractions
Manuele Auero UC Berkeley
Continuous energy adjustments
aaa
ke
uncertainty in
ALFRED
ALFRED - keff uncertainty - XGPT vs. TMC
keff distribution from
208
Pb cross sections uncertainty (from TENDL-2013)
500
Total Monte Carlo
XGPT
Number of counts per bin [-]
400
300
200
100
0
-900
-600
-300
0
300
keff - keff [pcm]
Manuele Auero UC Berkeley
600
900
1200
Continuous energy adjustments
aaa
Representativity study
ALFRED vs. HMF-64
HMF-64 / ALFRED correlation --
208
Pb XS uncertainties
2000
XGPT + POD (2 serpent runs)
TMC (6000 independet Serpent runs)
keff - keff ALFRED [pcm]
1000
0
-1000
-2000
-4000
-2000
0
2000
keff - keff HEU-MET-FAST-064 [pcm]
Manuele Auero UC Berkeley
4000
Continuous energy adjustments
6000
aaa
Representativity study
ALFRED vs. HMF-64
HMF-64 / ALFRED correlation --
208
Pb XS uncertainties
2000
XGPT + POD (2 serpent runs)
TMC (6000 independet Serpent runs)
keff - keff ALFRED [pcm]
1000
0
-1000
-2000
17
18
19
20
21
22
Effective prompt lifetime HEU-MET-FAST-064 [ns]
Manuele Auero UC Berkeley
23
Continuous energy adjustments
24
aaa
Cross section sensitivity to resonance parameters
Cross sections derivatives are calculated numerically via NJOY
Perturbation of
238
U resonance parameters -- Γγ @6.67 eV
Sensitivity of capture XS (MT102)
XS sensitivity [-]
1.0
0.8
0.6
0.4
0.2
0.0
4
Reference XS [b]
10
2
10
0
10
-6
10
-5
10
Energy [MeV]
Manuele Auero UC Berkeley
Continuous energy adjustments
aaa
Cross section sensitivity to resonance parameters
R ≡ dR /R
Denition of sensitivity coecient: Sx
ke
SΓ
γ
Z
=
k
S e
dx /x
Z
σcapture
(E )·dE +
σcapture (E )·SΓγ
k
Sσ e
elastic
(E )·SΓσγelastic (E )·dE +...
Multi-group discretization is usually introduced here...
The new method avoids any discretization
Manuele Auero UC Berkeley
Continuous energy adjustments
aaa
Case study PWR MOX 2D pin cell
Material compositions and geometry specications from:
Benchmarks for uncertainty analysis in modelling (UAM) for the
design, operation and safety analysis of LWRs
Case: GEN-III PWR MOX 2D pin cell
Pu content in fuel 3.7%
Manuele Auero UC Berkeley
Continuous energy adjustments
aaa
Verication against direct perturbation: 238 U
Perturbation of
238
U resonance parameters -- Γn @6.67 eV
UAM GEN-III MOX-3.7% 2D Pin HZP -- Γn effect on keff
9
Effective multiplication factor -- keff [-]
1.108
Direct perturbation (independent runs, 10 particles)
1.107
1.106
1.105
1.104
1.103
0
-2
2×10
-2
-2
-2
4×10
6×10
8×10
Relative perturbation on Γn @6.67 eV
Manuele Auero UC Berkeley
-1
1×10
Continuous energy adjustments
aaa
Verication against direct perturbation: 238 U
Perturbation of
238
U resonance parameters -- Γn @6.67 eV
UAM GEN-III MOX-3.7% 2D Pin HZP -- Γn effect on keff
0
9
Direct perturbation (independent runs, 10 particles)
7
XGPT (single run, 10 particles)
-3
-1×10
-3
∆keff [-]
-2×10
-3
-3×10
-3
-4×10
-3
-5×10
0
-2
2×10
-2
-2
-2
4×10
6×10
8×10
Relative perturbation on Γn @6.67 eV
Manuele Auero UC Berkeley
-1
1×10
Continuous energy adjustments
aaa
Verication against direct perturbation: 238 U
238 U @6.67eV
ke
SΓ
γ
k
SΓ e
n
Direct perturbation
−2
10
−4.603 ×
±9.9 × 10−4
−4.392 × 10−2
±9.9 × 10−4
GPT
−4.469 × 10−2
±1.4 × 10−4
−4.512 × 10−2
±1.6 × 10−4
Sensitivities are very large
GPT is more ecient: 10
7 vs 109 particles, smaller err.
All GPT sensitivities calculated in a single run
Manuele Auero UC Berkeley
Continuous energy adjustments
aaa
Verication against direct perturbation: 239 Pu
Perturbation of
239
Pu resonance parameters -- Γf @0.2956 eV
Cross sections sensitivities (3% Γf relative perturbation)
XS sensitivity [-]
0.5
0.0
-0.5
MT102
MT18
-1.0
4
Reference XS [b]
10
MT102
MT18
3
10
2
10
1
10
0
10
-7
-6
10
10
-5
10
Energy [MeV]
Manuele Auero UC Berkeley
Continuous energy adjustments
aaa
Verication against direct perturbation: 239 Pu
Perturbation of
239
Pu resonance parameters -- Γγ @0.2956 eV
Cross sections sensitivities (3% Γγ relative perturbation)
XS sensitivity [-]
1.0
0.5
0.0
MT102
MT18
-0.5
Reference XS [b]
-1.0
4
10
MT102
MT18
3
10
2
10
1
10
0
10
-7
-6
10
10
-5
10
Energy [MeV]
Manuele Auero UC Berkeley
Continuous energy adjustments
aaa
Verication against direct perturbation: 239 Pu
Perturbation of
239
Pu resonance parameters -- Γn @0.2956 eV
Cross sections sensitivities (3% Γn relative perturbation)
XS sensitivity [-]
0.4
0.2
0.0
-0.2
MT102
MT18
MT2
-0.4
-0.6
4
Reference XS [b]
10
MT102
MT18
MT2
3
10
2
10
1
10
0
10
-7
-6
10
10
-5
10
Energy [MeV]
Manuele Auero UC Berkeley
Continuous energy adjustments
aaa
Verication against direct perturbation: 239 Pu
Perturbation of
239
Pu resonance parameters @0.2956 eV
UAM GEN-III MOX-3.7% 2D Pin HZP -- Γn , Γf , Γγ effect on keff
-3
6×10
-3
4×10
Γf -- direct perturbation
-3
∆keff [-]
2×10
Γn -- direct perturbation
Γγ -- direct perturbation
0
Γf -- XGPT
Γn -- XGPT
Γγ -- XGPT
-3
-2×10
-3
-4×10
-3
-6×10
0
-2
-2
1×10
2×10
Relative perturbation on Γn , Γf , Γγ @0.2956 eV
Manuele Auero UC Berkeley
-2
3×10
Continuous energy adjustments
aaa
Verication against direct perturbation: 239 Pu
239 Pu @0.295eV
ke
SΓ
γ
k
SΓ e
n
k
SΓ e
f
Direct perturbation
−2
10
−1.832 ×
±9.9 × 10−4
−2
1.495 × 10
±9.9 × 10−4
−2
1.859 × 10
−
±9.9 × 10 4
GPT
−1.835 × 10−2
±3.8 × 10−4
−2
1.495 × 10
±1.6 × 10−4
−2
1.857 × 10
−
±4.1 × 10 4
Sensitivities are very large
GPT is more ecient: 10
7 vs 109 particles, smaller err.
All GPT sensitivities calculated in a single run
Manuele Auero UC Berkeley
Continuous energy adjustments
aaa
Fancy sensitivities (scattering radius)
Perturbation of
238
U scattering radius
Elastic scattering cross section
XS sensitivity [-]
6.0
4.0
2.0
0.0
-2.0
-4.0
4
Reference XS [b]
10
2
10
0
10
-2
10
-6
10
-5
10
Energy [MeV]
Manuele Auero UC Berkeley
-4
10
Continuous energy adjustments
aaa
Fancy sensitivities (negative energy resonances)
Perturbation of
239
Pu resonance parameters -- Γfb @-0.2194 eV
Cross sections sensitivities (5% Γfb relative perturbation)
XS sensitivity [-]
0.2
0.1
0.0
-0.1
MT102
MT18
-0.2
5
Reference XS [b]
10
4
10
3
10
2
10
MT102
MT18
1
10
0
10
-10
10
-9
10
-8
10
-7
10
Energy [MeV]
Manuele Auero UC Berkeley
-6
10
-5
10
Continuous energy adjustments
aaa
Continuous energy adjustment
1
Generate continuous-energy basis functions
2
Project the uncertainties on these bases (MF-32,
SVD of the continuous-energy covariance matrices
XS derivatives for resonance parameters, scatt. radius & co.
Sensitivity to nuclear model parameters?
MF-33, etc.)
3
Run Serpent-XGPT once for each system
Get the uncertainty for each response function
Var [R ]
∼
n
X
j =1
V
j·
S
R
U
2
j
Manuele Auero UC Berkeley
Continuous energy adjustments
aaa
Continuous energy adjustment
4
Get
5
Solve the GLLS problem to get:
adjusted
COV [V , V], ∆αUj
6
Reconstruct the adjusted Σ
adj
C /E , Var [R ], V
j
, and
Σ (E ) 'prior Σ (E ) ·
Manuele Auero UC Berkeley
S
1+
R
U
j
n
P
j=
1
!
∆αUj · Uj (E )
Continuous energy adjustments
aaa
Continuous energy adjustment
7
Reconstruct the adjusted
prior
adj
=
COV [Σ(E ) , Σ(E 0)] ∼
n
P
j=
COV [Σ(E ) , Σ(E 0)] ∼ U
U1(E ) U2(E ) . . .
COV [Σ , Σ]
adj
1
U (E ) · V · U (E 0)
j
j
j
COV [V , V] U


U1(E )
U (E )
2

COV [V , V] 
U (E )
 3 
T
adj
..
.
Manuele Auero UC Berkeley
Continuous energy adjustments
aaa
Adjustment via XGPT
Very simple case study
Only one system: Jezebel
Only one response function: ke
Only one isotope: 239 Pu
Covariances from ENDF/B-VII.0 (multigroup!)
Manuele Auero UC Berkeley
Continuous energy adjustments
aaa
Adjustment via XGPT
SVD of
239
Pu XS cov. matrix & XGPT - Jezebel keff uncertainty
Basis #1 for keff uncert. - 49.5% of the total variance - 594 pcm (rel. std)
Rel. basis function [a.u.]
MT2
MT4
MT18
MT102
4
10
5
6
10
10
7
10
Energy [eV]
Manuele Auero UC Berkeley
Continuous energy adjustments
aaa
Adjustment via XGPT
SVD of
239
Pu XS cov. matrix & XGPT - Jezebel keff uncertainty
Basis #2 for keff uncert. - 20.2% of the total variance - 379 pcm (rel. std)
Rel. basis function [a.u.]
MT2
MT4
MT18
MT102
4
10
5
6
10
10
7
10
Energy [eV]
Manuele Auero UC Berkeley
Continuous energy adjustments
aaa
Adjustment via XGPT
SVD of
239
Pu XS cov. matrix & XGPT - Jezebel keff uncertainty
Basis #3 for keff uncert. - 14.1% of the total variance - 316 pcm (rel. std)
Rel. basis function [a.u.]
MT2
MT4
MT18
MT102
4
10
5
6
10
10
7
10
Energy [eV]
Manuele Auero UC Berkeley
Continuous energy adjustments
aaa
Adjustment via XGPT
Contribution of the
239
Pu XS bases to the Jezebel keff uncert.
-4
10
-5
Contribution to keff rel. variance [-]
10
-6
10
-7
10
-8
10
-9
10
-10
10
0
10
20
30
40
50
60
Sorted basis index
Manuele Auero UC Berkeley
70
80
90
100
Continuous energy adjustments
aaa
Adjustment via XGPT (SQUADRA output)
(E -C)/ C (%) BEFORE AND AFTER ADJUSTM .
#
EXPERIMENT
1
JEZEBEL KEFF
BEFORE
AFTER
CHANGE
0.014
0.001
-0.013
#
EXPERIMENT
BEFORE
AFTER
CHANGE
1
JEZEBEL KEFF
0.950
0.196
-0.754
EXPERIMENT UNCERT . (%) BEFORE AND AFTER ADJUSTM .
NUCL . DATA UNCERT . BEFORE AND AFTER ADJUSTM .
NUCLEAR DATA
EIGENV_INEL
EIGENV_INEL
EIGENV_KHI
EIGENV_INEL
EIGENV_INEL
EIGENV_KHI
EIGENV_CAPT
EIGENV_KHI
1
2
1
4
5
2
1
3
NUC . DATA CHANGE
BEFORE
AFTER
CHANGE
-14.1
-2.5
-2.1
-0.9
-0.5
-0.4
0.5
0.2
1274.8
463.0
600.3
342.4
218.7
247.4
884.6
115.4
820.4
429.4
583.0
336.3
215.7
246.2
883.9
114.9
-454.4
-33.6
-17.3
-6.1
-3.0
-1.2
-0.7
-0.5
Manuele Auero UC Berkeley
Continuous energy adjustments
aaa
Strong negative correlation in the inelastic covariance
Manuele Auero UC Berkeley
Continuous energy adjustments
aaa
Adjustment via XGPT
PROBLEMS/OPEN ISSUES:
Continuous-energy cov. matrices are not available
Covariances with unphysical negative eigenvalues
Few cross-terms in the covariances
Inelastic: MT4 versus MT51-91
Resonances: MF32 or MF33
Angular distributions:
P
n
or Σ(µ, E )
Ongoing: sensitivity for S (α, β) and URR
Manuele Auero UC Berkeley
Continuous energy adjustments
aaa
Adjustment via XGPT
MAIN GOAL:
Simplify (and reduce) the steps between:
the data (XS and covariances)...
...and their use (SA/UQ and adjustment).
Manuele Auero UC Berkeley
Continuous energy adjustments
aaa
Acknowledgments
Thanks to:
A. Bidaud (LPSC Grenoble)
D. Rochman (PSI)
A. Sartori (SISSA Trieste)
Jaakko & Serpent developers team
Manuele Auero UC Berkeley
Continuous energy adjustments
aaa
THANK YOU FOR THE ATTENTION
QUESTIONS? SUGGESTIONS? IDEAS?
The bay area from the Berkeley hills (multi-group version).
O-line steps
Generate the optimal bases via POD (from random XS):
Load N random ACE les for the selected isotope
Score the rel. di. of the XS on the unionized e-grid
Build the (weighted) correlation matrix K ∈ RN ×N
Solve [S, V] = EIG(K) for the rst n eigenvalues
Reconstruct the bases and store them in a cache-friendly way
Generate the optimal bases via SVD (from cov. matrices):
Should you already have the relative covariance matrices, the
bases can be obtained directly via SVD:
Solve [U, S, V] = SVD(COV) for the rst n eigenvalues
The o-line steps need to be done just once
Manuele Auero UC Berkeley
Continuous energy adjustments
aaa
CPU-time & memory
TMC
CPU-time
Memory
∝
N
GPT + COV
small
∝
∝
# of coll.
# of coll.
Manuele Auero UC Berkeley
XGPT + POD
×λ×
pop
∝
small ∝ n
n × λ × pop
Continuous energy adjustments
aaa
XGPT+POD: uncertainty propagation in
PMF-35 ke
PMF-35 - keff estimates - XGPT errors
keff errors (XGPT + POD) [pcm]
60
40
20
0
-20
-40
-60
1.000
1.005
1.010
1.015
1.020
keff (Independent MC runs) [pcm]
Manuele Auero UC Berkeley
1.025
1.030
Continuous energy adjustments
aaa
XGPT+POD: uncertainty propagation in
Scaled eigenvalues of the POD of
208
PMF-35
Pb cross sections
3
Scaled eigenvalue [a.u.]
1×10
2
1×10
1
1×10
0
1×10
0
10
20
30
40
50
Basis number
Manuele Auero UC Berkeley
Continuous energy adjustments
aaa