Serpent UGM 2015 Knoxville, 13 16 October...

Serpent UGM 2015

Knoxville,

13�16 October 2015

Implementation of new adjoint-basedmethods for sensitivity analysis anduncertainty quanti�cation in Serpent

Manuele Au�ero & Massimiliano Fratoni

UC Berkeley

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Generalized perturbation theoryXGPT & Proper Orthogonal Decomposition

Unfortunately, the nuclear data we use everyday are a�ectedby uncertainties.

As nuclear data users, we should assess the impact ofuncertainties on our results and conclusions.

Uncertainty propagation, data assimilation and cross sectionadjustment were the territory of deterministic codes...but today is the Monte Carlo era (right?).

We implemented in Serpent and tested new methods foruncertainty quanti�cation (UQ).

Manuele Au�ero � UC Berkeley Serpent UGM � October 14, 2015

mailto:[email protected]

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Generalized Perturbation Theory (GPT)Available in deterministic codes from the 60s (Gandini, 1967)GPT in Serpent not discussed in details here. You can have a look at SerpentUGM14

presentation or at Annals of Nuclear Energy, 85, 2015. A collision history approach...

eXtended Generalized Perturbation Theory (XGPT)Newly developed. Continuous energy approach.Ongoing testing & optimization.



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Collision history approachUncertainty propagation

Generalized Perturbation Theory capabilities

E�ect of a perturbation of the parameter x on the response R :

SRx ≡

dR/R

dx/x

Considered response functions:

R = ke� E�ective multiplication factor

R =〈Σ1,φ〉〈Σ2,φ〉

Reaction rate ratios

R =

⟨φ†,Σ1φ

⟩⟨φ†,Σ2φ

⟩ Bilinear ratios (Adjoint-weighted quantities)

R =? Something else



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A collision history-based approach to GPT calculations



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Particle's weight perturbation

Weight conservation

w0 punbiased =∑

w∗ pbiased

w∗ ' w

0 ·(1 +

dΣn,2n

Σn,2n

)·(1 +

dΣs

Σs

)·(1−

dΣf

Σf

)·(1 +

dΣs

Σs

)·(1−

dΣc

Σc

)·

·(1 +

dΣf

Σf

)·(1 +

dΣs

Σs

)·(1 +

dΣf

Σf

)...



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Particle's weight perturbation

∂wn

∂x/x' wn·

α∑g=(α−λ)

((n,g)

ACCx − (n,g)REJx

)α = present generationλ = number of propagation generations

ACCx = accepted events x in the history of the particle nREJx = rejected events x



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Example: Monte Carlo sensitivity for bilinear ratios

R =

⟨φ†,Σ1φ

⟩⟨φ†,Σ2φ

⟩Examples:

βe� =

⟨φ†,

1

ke�χd ν̄dΣfφ

⟩⟨φ†,

1

ke�χt ν̄tΣfφ

⟩ `e� =

⟨φ†,

1

vφ

⟩⟨φ†,

1

ke�χt ν̄tΣfφ

⟩

αcoolant = −⟨φ†,Σt,coolantφ

⟩⟨φ†,

1

ke�χt ν̄tΣfφ

⟩



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R ′ =

⟨φ† + ∆φ†, (Σ1 + ∆Σ1) (φ+ ∆φ)

⟩⟨φ† + ∆φ†, (Σ2 + ∆Σ2) (φ+ ∆φ)

⟩∆R

R=

⟨φ†,∆Σ1φ

⟩⟨φ†,Σ1φ

⟩ −⟨φ†,∆Σ2φ

⟩⟨φ†,Σ2φ

⟩ +

⟨φ†,Σ1∆φ

⟩⟨φ†,Σ1φ

⟩ −⟨φ†,Σ2∆φ

⟩⟨φ†,Σ2φ

⟩ +

⟨∆φ†,Σ1φ

⟩⟨φ†,Σ1φ

⟩ −⟨

∆φ†,Σ2φ⟩

⟨φ†,Σ2φ

⟩

SRx =

⟨φ†,

∂Σ1

∂x/xφ

⟩⟨φ†,Σ1φ

⟩ −

⟨φ†,

∂Σ2

∂x/xφ

⟩⟨φ†,Σ2φ

⟩ +

+

⟨φ†,Σ1

∂φ

∂x/x

⟩⟨φ†,Σ1φ

⟩ −

⟨φ†,Σ2

∂φ

∂x/x

⟩⟨φ†,Σ2φ

⟩ +

⟨∂φ†

∂x/x,Σ1φ

⟩⟨φ†,Σ1φ

⟩ −

⟨∂φ†

∂x/x,Σ2φ

⟩⟨φ†,Σ2φ

⟩



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Indirect terms...

E�ect of perturbation on the forward �ux:⟨φ†,Σ1

∂φ

∂x/x

⟩=∑

n∈α

∑t ∈ n

∂wn

∂x/x· `tΣ1 ·

1

wn

∑k ∈ d

(γ)n

wk

E�ect of perturbation on the adjoint �ux:

⟨∂φ†

∂x/x,Σ1φ

⟩=∑

n∈α

∑t ∈ n

wn · `tΣ1

1

wn

∑k ∈ d

(γ)n

∂wk

∂x/x−

1

wn2

∂wn

∂x/x

∑k ∈ d

(γ)n

wk

Sum of indirect terms (rewritten as function of neutrons in generation α + γ ):⟨φ†,Σ1

∂φ

∂x/x

⟩+

⟨∂φ†

∂x/x,Σ1φ

⟩=

∑k ∈ (α+γ)

[wk

( ∑t ∈ (−γ)k

`tΣ1

)∂wk/wk

∂x/x

]



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Generalized response functions

If the quantity R can be estimated as the ratio of two generic MonteCarlo responses

R =E [e1]

E [e2]

the sensitivity coe�cient of R with respect to x can be obtained as:

SRx =

COV

[e1 ,

history∑(ACCx − REJx)

]E [e1]

−COV

[e2 ,

history∑(ACCx − REJx)

]E [e2]



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What was that?

For example, x can be the value of a cross section Σ (e.g., elasticscattering) of a given nuclide at the energy E :

x = Σ(E )

In that case, SRΣ (E ) can be calculated from the accepted and

rejected scattering events in the collision history:

SRΣ (E) =

COV

e1 , history∑ (ACCΣ,E − REJΣ,E

)E [e1]

−

COV

e2 , history∑ (ACCΣ,E − REJΣ,E

)E [e2]

This is a continuous energy estimator but...



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Unfortunately, energy discretization is required

In continuous energy Monte Carlo transport, the probability ofoccurrence of a collision at the exact energy E is ZERO

For this reason, energy-resolved sensitivity pro�les are obtained (asin deterministic codes) by calculating group-wise integrals of SR

Σ :

SRΣ,g =

Eg+1∫Eg

SRΣ (E ) dE

This is easily obtained by scoring the collisions occurred at energiesbetween Eg and Eg+1.



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Bilinear ratios sensitivities: a couple of examples

Flattop-Pu



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103

104

105

106

107

Energy (eV)

-0,4

-0,3

-0,2

-0,1

0

0,1

0,2

Sens

itivi

ty p

er le

thar

gy u

nit

Extended SERPENT-2TSUNAMI-1D (EGPT)

Popsy (Flattop) - Leff - Pu-239 - fissionEffective prompt lifetime sensitivity - 8-16 generations - ENDF/B-VII



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PWR pin cell



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10-2

100

102

104

Energy (eV)

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Sens

itivi

ty p

er le

thar

gy u

nit

Extended SERPENT-2TSUNAMI-1DDifference (S-T)

UAM TMI-1 PWR cell - αcoolant

- U-235 - nubar totalcoolant void reactivity coeff. sensitivity - 4 generations - ENDF/B-VII



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10-2

100

102

104

106

Energy (eV)

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Sens

itivi

ty p

er le

thar

gy u

nit

Extended SERPENT-2TSUNAMI-1DDifference (S-T)

UAM TMI-1 PWR cell - αcoolant

- U-238 - disappearancecoolant void reactivity coeff. sensitivity - 4 generations - ENDF/B-VII



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Uncertainty propagationFirst-order uncertainty propagation formula (sandwich rule)

Var [R] = SRx Cov [x ]

(SRx

)T



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Continuous energy transport, multi-group sens. & UQ

Choosing the multi-group energy grid is not an easy task(for both covariance matrices and sensitivity pro�les)

The e�ciency of any Monte Carlo sensitivity estimatordegrades quickly with �ner discretizations

You could never know if your energy grid is OK for uncertaintypropagation, unless you repeat everything with a new grid

What about higher moments of the uncertain responsefunction distribution?

Multi-group & sandwich rule were the obvious choices fordeterministic codes. Still a good choice for Monte Carlo?



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Starting from the Total Monte Carlo approachProjecting ND uncertainties onto a set of CE basis functions

Can we do it better?Yes we can...

We developed a new method, called Monte Carlo eXtended Generalized

Perturbation Theory (XGPT)... I know, it's a boring name (suggestions?)

XGPT for nuclear data uncertainty obtained a �side e�ect� of theongoing development of adjoint capabilities for multiphysicsproblems at UC Berkeley (not discussed here)

Main goals:

Continuous energy approach to uncertainty propagationNo �expert user� tasks (e.g., choice of energy grid)Estimate higher moments of the responses distributionsFaster than classic GPT+Covariance



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The Total Monte Carlo approach

The Total Monte Carlo approach (TMC, Rochman et al., 2011)

Random ENDF �les generated from uncertainties in nuclear modelsparameters (e.g., TENDL libraries)

Continuous-energy cross sections (ACE �les) produced via NJOY

No major approximations in the uncertainty propagation process

Some disadvantages (not only CPU time)



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A simple case study

A small sphere of 239Pu with 5 mm of 208Pb re�ector

Only 208Pb uncertainties are considered (data from TENDL-2013)



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Random cross sections

3000 di�erent 208Pb ENDF �les from TENDL-2013

Random MF2-MT151 (resonances), MF3-MT1, MF3-MT2 (elastic)and MF3-MT102 (n, γ) processed with NJOY

3000 ACE �les with random elastic scattering and (n, γ) XS

The random continuous energy XS re�ect the UNCERTAINTIESand their CORRELATIONS (according to TENDL-2013)



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6×10-2

7×10-2

8×10-2

9×10-2

1×10-1

Energy [MeV]

101

102

208 P

b el

asti

c xs

[b] Reference

3×10-1

4×10-1

5×10-1

6×10-1

Energy [MeV]

100

101

208 P

b el

asti

c xs

[b]



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6×10-2

7×10-2

8×10-2

9×10-2

1×10-1

Energy [MeV]

101

102

208 P

b el

asti

c xs

[b] 10 random evaluations

Reference

3×10-1

4×10-1

5×10-1

6×10-1

Energy [MeV]

100

101

208 P

b el

asti

c xs

[b]



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6×10-2

7×10-2

8×10-2

9×10-2

1×10-1

Energy [MeV]

101

102

208 P

b el

asti

c xs


Reference

3×10-1

4×10-1

5×10-1

6×10-1

Energy [MeV]

100

101

208 P

b el

asti

c xs

[b]



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TMC results for ke� � 3000 Serpent runs

-400 -200 0 200 400 600keff - keff [pcm]

0

100

200

300

400

500

Num

ber

of c

ount

s pe

r bi

n [-

]

Total Monte Carlo

Pu239 & Pb208 sphere - keff uncertainty - TMCkeff distributions from Pb208 elastic scattering and capture xs uncertainty (from TENDL-2013)

CPU time: 3000 runs × 11 min. × 20 cores



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Reconstructing continuous energy covariance matrices

Processing the independent ACE �les, XS standard deviation and correlationmatrices are obtained as continuous function of the incident neutron energy.

0

20

40

60

Rel

. sta

ndar

d de

v [%

]

Pb208 nuclear data uncertainty - Random evaluations approach208

Pb elastic scattering and capture xs - random MF2 MT151 & MF3 MT(1),2,102 from TENDL-2013

10-2

10-1

100

101

Energy [MeV]

100

101

102

Ref

. ela

stic

xs

[b]



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0

20

40

60

Rel

. sta

ndar

d de

v [%

]

Pb208 nuclear data uncertainty - Random evaluations approach208

Pb elastic scattering and capture xs - random MF2 MT151 & MF3 MT(1),2,102 from TENDL-2013

6×10-2

7×10-2

8×10-2

9×10-2

1×10-1

Energy [MeV]

101

102

Ref

. ela

stic

xs

[b]



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0.02

0.2

2

20

0.02 0.2 2 20

En

erg

y [

MeV

]

Energy [MeV]

Continuous energy Multi-group (JANIS-4.0)



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eXtendend Generalized Perturbation TheoryProper Orthogonal Decomposition of Nuclear Data uncertainties

The main steps:

Process N random continuous energy XS Σi

Build an optimal set of n CE orthogonal basis functions bj

Project the ND uncertainties onto this set of basis functions

Run a single Serpent simulation with the reference XS Σ0

Calculate the sensitivities of each response R to the bases bj

Reconstruct the expected responses RΣifor each XS Σi

Obtain the distribution of the uncertain response R



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Proper Orthogonal Decomposition (POD)Singular Value Decomposition (SVD)

Let's talk about something else for a while...

*Standard test image for image processing from http://sipi.usc.edu/database/

Digital image compression



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Proper Orthogonal Decomposition (POD)Singular Value Decomposition (SVD)

Let's talk about something else for a while...

*Standard test image for image processing from http://sipi.usc.edu/database/

Digital image compressionManuele Au�ero � UC Berkeley Serpent UGM � October 14, 2015


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How to build a reduced-order approximation of the imageHow to build a reduced-order approximation of the covariance matrix

Matrix eigendecomposition

A = UΣV∗



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A = UΣV∗

A ∼ UdΣdV∗d



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Singular Value Decomposition

Originalimage

SVD/POD5 basis functions

Multi-group5 energy groups

A=imread("lena.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)′;imwrite(A_SVD_5, gray(255), "lena_5.png");



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Originalimage




[U, S, V]=svd(A);




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Proper Orthogonal Decomposition of Nuclear DataLet's go back to 208Pb uncertainties...

...there is much more room for data compression in thecovariance matrices (physics-based correlations)

Proper Orthogonal Decomposition of Nuclear Data

We want a set of orthogonal basis functions bΣ,j so that:

Σ̃i (E ) = Σ0 (E ) ·

1 +n∑

j=1

αji · bΣ,j (E )



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3 basis functions from the POD of 208Pb (n, ela)

Rel

. bas

is f

unct

ion

[a.u

.] Basis # 3

100

101

Energy [MeV]

100

101

Ref

. ela

stic

xs

[b]

Basis # 3

Rel

. bas

is f

unct

ion

[a.u

.] Basis # 4

6×10-2

7×10-2

8×10-2

9×10-2

1×10-1

Energy [MeV]

101

102

Ref

. ela

stic

xs

[b]

Basis # 4

Rel

. bas

is f

unct

ion

[a.u

.] Basis # 5

10-1

100

Energy [MeV]

100

101

102

Ref

. ela

stic

xs

[b]

Basis # 5



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Correlation matrices after POD and reconstruction

0.02

0.2

2

20

0.02 0.2 2 20

En

erg

y [

MeV

]

Energy [MeV]

n = 2

0.02

0.2

2

20

0.02 0.2 2 20

En

erg

y [

MeV

]

Energy [MeV]

n = 5

0.02

0.2

2

20

0.02 0.2 2 20

En

erg

y [

MeV

]

Energy [MeV]

n = 10

0.02

0.2

2

20

0.02 0.2 2 20

En

erg

y [

MeV

]

Energy [MeV]

n = 20



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0.02

0.2

2

20

0.02 0.2 2 20

En

erg

y [

Me

V]

Energy [MeV]

ContinuousEnergy

0.02

0.2

2

20

0.02 0.2 2 20

En

erg

y [

Me

V]

Energy [MeV]


Multi-group>100 ene g.



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XGPT+POD: calculating sensitivities to CE basis functions

We need to calculate the e�ect on the response Rdue to a perturbation on Σ equal to bΣ,j

SRbΣ,j

=dR/R

dbΣ,j=

Emax∫Emin

bΣ,j (E ) · SRΣ (E ) dE



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XGPT+POD: calculating sensitivities to CE basis functions

Using the collision history approach, SRbΣ,j

can be estimated

from the covariance between the terms of R and the collisionsweighted by bΣ,j (E)

SRbΣ,j

=

COV

[e1 ,

history∑GbΣ,j

]E [e1]

−COV

[e2 ,

history∑GbΣ,j

]E [e2]



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XGPT+POD: uncertainty propagation

From the Proper Orthogonal Decomposition of Nuclear data...

Σi (E ) ' Σ̃i (E ) = Σ0 (E ) ·

(1 +

n∑j=1

αji · bΣ,j (E )

)

...and the basis functions sensitivity coe�cients SRbΣ,j

, we can

approximate the response function RΣifor each random XS Σi

RΣi' R̃Σi

= RΣ0·

(1 +

n∑j=1

αji · S

RbΣ,j

)



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Estimating the ke� distribution in the simple case study

ke�Σi' k̃e�Σi

= ke�Σ0·

(1 +

n∑j=1

αji · S

ke�bΣ,j

)

The ke� for all the N (3000) random XS Σi were calculatedin a single Serpent run (ACE �le for Σ0) with n = 50 bases

The XGPT+POD results are compared to TMC results(3000 separate Serpent runs)



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-400 -200 0 200 400 600 800keff - keff (Independent MC runs) [pcm]

-400

-200

0

200

400

600

800

keff -

kef

f (

XG

PT

+ P

OD

) [p

cm]

Pu239 & Pb208 sphere - keff estimates - XGPT + PODkeff distributions from Pb208 elastic scattering and capture xs uncertainty (from TENDL-2013)



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-400 -200 0 200 400 600keff - keff [pcm]

0

100

200

300

400

500

Num

ber

of c

ount

s pe

r bi

n [-

]

Total Monte CarloXGPT + POD

Pu239 & Pb208 sphere - keff uncertainty - XGPT + POD vs. TMCkeff distributions from Pb208 elastic scattering and capture xs uncertainty (from TENDL-2013)



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Moment TMC XGPT + POD

Standard deviation [pcm] 165.8 164.0Skewness [-] 0.81 0.79Kurtosis [-] 3.60 3.55



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XGPT+POD CPU-time

Real-time demonstration



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Conclusions

The XGPT+POD method for uncertainty quanti�cationhas been implemented in Serpent

Uncertainty propagation w/o multi-group energydiscretization of sensitivities and covariances

Estimation of higher moments of response distributions

Straight-forward & simple implementation

Useful also for multi-group GPT+COV

Not implemented (yet) for URR and S(α,β)



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Next steps

Testing and veri�cation required!!! (ongoing)

Continuous energy cross sections adjustment

UQ in multiphysics applications (coupled neutronics/CFD)

Higher order sensitivities

Beyond uncertainties: ROMs for coupled problems



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Acknowledgments

Thanks toA. Bidaud (LPSC Grenoble)

D. Rochman (PSI)A. Sartori (SISSA Trieste)

for precious discussions



THANK YOU FOR THE ATTENTION

The bay area from the Berkeley hills (multi-group version).

QUESTIONS? SUGGESTIONS? IDEAS?

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Backup-slides

O�-line steps

Generate the optimal bases via POD (from random XS):

Load N random ACE �les for the selected isotopeScore the rel. di�. of the XS on the unionized e-gridBuild the (weighted) correlation matrix K ∈ RN×N

Solve [S, V] = EIG(K) for the �rst n eigenvaluesReconstruct 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, thebases 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



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Backup-slides

POD: eigenvalues

0 10 20 30 40 50Basis number

1×100

1×101

1×102

1×103

Scal

ed e

igen

valu

e [a

.u.]

Scaled eigenvalues of the POD of 208

Pb cross sections



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Backup-slides

STD convergence: XGPT+POD vs. TMC

0 1000 2000 3000Sample number

0

50

100

150

200

250

keff s

ampl

e st

d. d

ev. [

pcm

]

GPT + PODTMC

Pu239 & Pb208 sphere - convergence of TMC std. dev.keff distribution from Pb208 elastic scattering and capture xs uncertainty (from TENDL-2013)

TMC

0 10 20 30 40 50Basis number

0

50

100

150

200

250

keff s

td. d

ev. [

pcm

]

XGPT + PODTMC (sample # 3000)

Pu239 & Pb208 sphere - convergence of XGPT std. dev.keff distribution from Pb208 elastic scattering and capture xs uncertainty (from TENDL-2013)

XGPT+POD



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Backup-slides

Skewness convergence: XGPT+POD vs. TMC


0

0.2

0.4

0.6

0.8

1

1.2

1.4

keff s

ampl

e sk

ewne

ss [

-]

GPT + PODTMCTMC (sample # 3000)

Pu239 & Pb208 sphere - convergence of TMC skewnesskeff distribution from Pb208 elastic scattering and capture xs uncertainty (from TENDL-2013)

TMC

0 10 20 30 40 50Basis number

0

0.2

0.4

0.6

0.8

1

1.2

1.4

keff s

kew

ness

[-]


Pu239 & Pb208 sphere - convergence of XGPT skewnesskeff distribution from Pb208 elastic scattering and capture xs uncertainty (from TENDL-2013)

XGPT+POD



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Backup-slides

Kurtosis convergence: XGPT+POD vs. TMC


0

1

2

3

4

5

6

keff s

ampl

e ku

rtos

is [

-]

GPT + PODTMCTMC (sample # 3000)

Pu239 & Pb208 sphere - convergence of TMC kurtosiskeff distribution from Pb208 elastic scattering and capture xs uncertainty (from TENDL-2013)

TMC

0 10 20 30 40 50Basis number

0

1

2

3

4

5

6

keff k

urto

sis

[-]


Pu239 & Pb208 sphere - convergence of XGPT kurtosiskeff distribution from Pb208 elastic scattering and capture xs uncertainty (from TENDL-2013)

XGPT+POD



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Backup-slides

CPU-time & memory

TMC GPT + COV XGPT + POD

CPU-time ∝ N small ∝ # of coll. small ∝ n

Memory � ∝ # of coll. × λ × pop ∝ n × λ × pop



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