SG39 Meeting May 16-17, Update on Continuous Energy Cross Section Adjustment. UC Berkeley / INL collaboration
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1 SG39 Meeting May 16-17, 2017 Update on Continuous Energy Cross Section Adjustment. UC Berkeley / INL collaboration
2 Outline Presentation of the proposed methodology (you've already seen this) Results from 239 Pu adjustment from Jezebel integral experiment: k e F28/F25 F37/F25 F49/F25 Comparison against ERANOS Conclusions
3 Continuous-energy rst order uncertainty propagation Var [R] = E max E max S R Σ [ (E) COV Σ(E), Σ(E ) ] SΣ R ( E ) de de E min E min (1) COV [Σ(E), Σ(E )] is the continuous-energy covariance matrix S R Σ (E) is the sensitivity density function for the generic response R Multi-group discretization is usually introduced here
4 Multi-group discretization of the covariance matrix Figure: Comparison between the multi-group (left) and continuous (right) 239 Pu capture cross correlation matrices adopted in the adjustment process.
5 Multi-group discretization of the covariance matrix 239 Pu capture uncert.: "continuous-energy" vs multi-group Rel. standard dev [%] "Continuous-energy" uncertainty (used by XGPT-based nuclear data assimilation) Multi-group uncertainty (used by GPT-based nuclear data assimilation) Ref. capture xs [b] Energy [ev] Figure: Comparison between 239 Pu capture cross section relative uncertainty adopted as input by the continuous and multi-group approaches.
6 Eigendecomposition of the covariance matrix COV [Σ(E), Σ(E )] = U j (E) V j U j (E ) (2) j=1 V j are the eigenvalues of the continuous energy covariance matrix corresponding to the eigenfunctions U j (E)
7 Continuous-energy uncertainty propagation (revisited) Var [R] = E max E min E max E min S R Σ (E) COV [Σ(E), Σ(E )] S R Σ (E ) de de (1) Var [R] = V j j=1 E max E min U j (E) SΣ R (E) de 2 (3)
8 Continuous-energy sensitivities Main step: calculation of integral of continuous energy sensitivity functions via Monte Carlo XGPT: SU R j = U j (E) S R Σ (E) de
9 Continuous-energy uncertainty propagation (truncated) Var [R] = ) 2 n V j (S R Uj V j j=1 j=1 ( S R U j ) 2 (4)
10 Contribution of the 239 Pu SVD bases to the uncertainties in Jezebel k eff and central reaction rate ratios (F28/F25, F37/F25, F49/F25) Basis contribution to relative variance k eff variance F28/F25 variance F37/F25 variance F49/F25 variance Index of SVD basis (sorted for each response function) Figure: Eigenfunctions contribution to the total variances in Jezebel. Response functions: k e, F28/F25, F37/F25, F49/F25. ( 239 Pu ENDF/B-VII covariances).
11 Eigenvalue decomposition lead to exponential convergence with respect to the number of the basis functions Multi-group discretization lead to slow, unpredictable convergence with respect to the number of groups Statistical eciency of Monte Carlo continuous sensitivity estimators doesn't depend on the number of eigenfunctions Statistical eciency of Monte Carlo multi-group sensitivity estimators degrades quickly when adopting ner energy grids
12 Example of basis functions from 239 Pu ENDF/B-VII SVD of 239 Pu covariance matrix - Top contributors to F28/F25 uncert. Basis #2 for F28/F25 uncertainty % of the total variance SVD relative basis function [a.u.] Elastic Capture Inelastic Fission n,2n khi mubar nubar Energy [ev]
13 Example of basis functions from 239 Pu ENDF/B-VII SVD of 239 Pu covariance matrix - Top contributors to k eff uncertainty Basis #3 for keff uncert % of the total variance pcm (rel. std) SVD relative basis function [a.u.] 0.0 Elastic Capture Inelastic Fission n,2n khi mubar nubar Energy [ev]
14 Example of basis functions from 239 Pu ENDF/B-VII SVD of 239 Pu covariance matrix - Top contributors to F28/F25 uncert. Basis #4 for F28/F25 uncertainty - 9.2% of the total variance SVD relative basis function [a.u.] 0.0 Elastic Capture Inelastic Fission n,2n khi mubar nubar Energy [ev]
15 Multi-group/GPT starting point Multi-group sensitivity coecients: S R Σ = ( S R Σ 1, S R Σ 2 S R Σ N ) (5) Prior multi-group covariance matrices: Var(Σ 1 ) COV [Σ 1, Σ 2 ] COV [Σ 1, Σ N ] COV [Σ 2, Σ 1 ] Var(Σ 2 ) COV [Σ 2, Σ N ] COV [Σ, Σ] = COV [Σ N 1, Σ 1 ] COV [Σ N, Σ 1 ] Var(Σ N ) (6)
16 Continuous-energy/XGPT starting point Eigenfunctions sensitivities: S R U = ( S R U 1, S R U 2 S R U n ) (7) Projection of the (prior) covariance matrices: V V 2 0 COV [U, U] = V n (8)
17 That's it! S R Σ and COV [Σ, Σ] are replaced by SR U and COV [U, U] The continuous-energy adjustment process follows the standard, legacy multi-group approach...
18 Adjustment parameters U = [ U1, U2 Un ] T : U = M G T [ G M G T + V e + V m ] 1 D R (9) M is the prior covariance of the continuous functions prior COV [U, U] V e and V m : matrices of the experimental and modeling errors D R contains the relative dierences between the calculated and measured experiments. [ ] T G is the matrix of the sensitivities: G = S R 1 U SR 2 U SR N U
19 U = M G T [ G M G T + V e + V m ] 1 D R (9) adjusted Σ (E) prior Σ (E) 1 + n Uj U j (E) j=1 (10)
20 Adjusted continuous energy covariance adjusted COV [U, U] via the Generalized Least Squares Method is obtained as: adjusted COV [U, U] prior COV [U, U] = = M G T [ G M G T + V e + V m ] 1 G M (11) adjusted COV [U, U] contains the correlations among the basis functions introduced by the experiments. adjusted COV [ Σ(E), Σ(E ) ] [ U 1 (E)... U n (E) ] U 1 (E ) adjusted COV [U, U]. U n (E ) (12)
21 Case study: Jezebel 239 Pu Comparison against multi-group/gpt Conclusion and future works Case study: Jezebel 239 Pu Relative experimental uncertainties k e F28/F25 F37/F25 F49/F Experimental correlation matrix k e F28/F25 F37/F25 F49/F25 k e F28/F F37/F F49/F Table: Experimental uncertainties and correlation matrix for the four considered response functions.
22 Case study: Jezebel 239 Pu Comparison against multi-group/gpt Conclusion and future works Case study: Jezebel 239 Pu Relative modeling uncertainties k e F28/F25 F37/F25 F49/F Modeling correlation matrix k e F28/F25 F37/F25 F49/F25 k e F28/F F37/F F49/F Table: Modeling uncertainties and correlation matrix for the four considered response functions.
23 Case study: Jezebel 239 Pu Comparison against multi-group/gpt Conclusion and future works Case study: Jezebel 239 Pu Exp. Calc. Calc. (this work) (WPEC-SG33) k e F28/F F37/F F49/F Table: Experimental and calculated values.
24 Case study: Jezebel 239 Pu Comparison against multi-group/gpt Conclusion and future works Continuous vs. multi-group: uncertainty reduction Prior rel. uncert. (%) Post rel. uncert. (%) multi-group XGPT multi-group XGPT k e F28/F F37/F F49/F Table: Comparison of prior (input) and post (adjusted) nuclear data uncertainties estimated by the multi-group and continuous approaches for the four response functions.
25 Case study: Jezebel 239 Pu Comparison against multi-group/gpt Conclusion and future works Continuous vs. multi-group: uncertainty reduction 20 Change in nuclear data uncertainty after XS adjustment 239 Pu elastic scattering cross section 20 Change in nuclear data uncertainty after XS adjustment 239 Pu elastic scattering cross section Prior - Multigroup (ERANOS) Adjusted - Multigroup (ERANOS) Prior - XGPT (SERPENT) Adjusted - XGPT (SERPENT) Cross section uncertainty (%) Cross section uncertainty (%) Energy [MeV] Energy [MeV] Figure: 239 Pu elastic scattering uncertainty before and after the adjustment process. Multi-group (left) and continuous energy (right) results.
26 Case study: Jezebel 239 Pu Comparison against multi-group/gpt Conclusion and future works Continuous vs. multi-group: uncertainty reduction 60 Change in nuclear data uncertainty after XS adjustment 239 Pu inelastic scattering cross section 60 Change in nuclear data uncertainty after XS adjustment 239 Pu inelastic scattering cross section Prior - Multigroup (ERANOS) Prior - XGPT (SERPENT) 50 Adjusted - Multigroup (ERANOS) 50 Adjusted - XGPT (SERPENT) Cross section uncertainty (%) Cross section uncertainty (%) Energy [MeV] Energy [MeV] Figure: 239 Pu inelastic scattering uncertainty before and after the adjustment process. Multi-group (left) and continuous energy (right) results.
27 Case study: Jezebel 239 Pu Comparison against multi-group/gpt Conclusion and future works Negative correlations Figure: 239 Pu inelastic scattering correlation matrix in the 1 kev 20 MeV energy region. Before (left) and after (center) the continuous energy adjustment process, and Prior Post dierence is shown on the right.
28 Case study: Jezebel 239 Pu Comparison against multi-group/gpt Conclusion and future works Continuous vs. multi-group: XS adjustment Nuclear data change after XS adjustment 239 Pu elastic scattering cross section 6 Relative cross section change (%) Multigroup (ERANOS) XGPT (SERPENT) Energy [MeV] Figure: 239 Pu elastic scattering cross section before and after the adjustment process. Multi-group (red) and continuous energy (black) results.
29 Case study: Jezebel 239 Pu Comparison against multi-group/gpt Conclusion and future works Continuous vs. multi-group: XS adjustment Nuclear data change after XS adjustment 239 Pu inelastic scattering cross section 0-5 Relative cross section change (%) Multigroup (ERANOS) XGPT (SERPENT) Energy [MeV] Figure: 239 Pu inelastic scattering cross section before and after the adjustment process. Multi-group (red) and continuous energy (black) results.
30 Case study: Jezebel 239 Pu Comparison against multi-group/gpt Conclusion and future works Continuous vs. multi-group: Post C/E Prior C/E Post C/E multi-group 1 XGPT multi-group XGPT k e F28/F F37/F F49/F Table: Comparison of prior and post C/E estimated by the multi-group and continuous approaches for the four response functions.
31 Case study: Jezebel 239 Pu Comparison against multi-group/gpt Conclusion and future works Conclusions Main goal: new methodology for continous-energy XS adjustment Shorten the distance between evaluators and Monte Carlo users (?) Enable the adoption of integral experiments in a simple, eective and timely way ( 35 Cl (n, p), 233 U (n, γ)... )
32 Case study: Jezebel 239 Pu Comparison against multi-group/gpt Conclusion and future works Conclusions Main goal: new methodology for continous-energy XS adjustment Shorten the distance between evaluators and Monte Carlo users (?) Enable the adoption of integral experiments in a simple, eective and timely way ( 35 Cl (n, p), 233 U (n, γ)... ) First tests are promising... we need to move to broader case studies. Anyone wants to help/contribute??? In the resonance region, resonance parameters XS sensitivities (after MF-32 decompositions) and scattering radii are the basis functions for the continuous adjustment
33 Case study: Jezebel 239 Pu Comparison against multi-group/gpt Conclusion and future works Lessons learned (random thoughts) and ongoing works Please, leave MF-32 in the ENDF les In the future, storing MF-33 in the form of eigenvectors/eigenvalues might save, memory, CPU, and headaches Now working on secondaries distribution adjustment... Legendre or double dierential? Next step: URR adjustment (this might take some time!)
34 Questions? Suggestions? Ideas?
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