Joint ICTP-IAEA Workshop on Nuclear Reaction Data for Advanced Reactor Technologies May 2010

Transcription

1 Joint ICTP-IAEA Workshop on Nuclear Reaction Data for Advanced Reactor Technologies 3-14 May 2010 Monte Carlo Approaches for Model Evaluation of Cross Sections and Uncertainties CAPOTE R. IAEA Vienna AUSTRIA

2 Monte Carlo approaches for Model evaluation of cross sections and uncertainties Roberto Capote and Donald Smith (*) International Atomic Energy Agency, NAPC - Nuclear Data Section (*) Argonne National Laboratory

3 OUTLINE Introduction: Monte Carlo method UMC formulation (normal distributions) UMC sampling: Brute Force, Metropolis Toy models: Linear UMC convergence Toy models: Ratio data Log transformation Non-normal PDFs (& examples): 1) lognormal distribution; 2) uniform interval distribution; 3) exponential distribution. Results and discussions for each PDF 2/26

4 Nuclear Data Evaluation process RIPL Model Input Local input EXFOR Select & update Experim. data C4 EMPIRE Modeling code Resonance Evaluation Trial ENDF File Model vs. Experim. Final evaluated File in ENDF-6 GLSQ fit (GANDR) Model C4 + PRIOR 3/26

5 i 1 MONTE CARLO METHOD D.L. Smith, Covariance Matrices for Nuclear Cross-Sections Derived from Nuclear Model Calculations. Report ANL/NDM-159, Argonne National Laboratory, 2005 K K k 1 ik Vij i j i j i,j - energy indexes Monte Carlo calculation of covariance first tested by A. Koning Monte Carlo prior + GANDR (GLS) D.W. Muir, GANDR project (IAEA), Online at www-nds.iaea.org/gandr/. A. Trkov and R. Capote, Cross-Section Covariance Data, Th-232 evaluation for ENDF/B-VII.0 (MAT=9040 MF=1 MT=451); Pa-231 and Pa-233 evaluations for ENDF/B-VII.0 (MAT=9133 and 9137 MF=1 MT=451), National Nuclear Data Center, BNL ( 15 December /26

6 Merging of Model Calculated and Experimental Results More (a.k.a. Auto Repair Shop Solution) Donald L. Smith Argonne National Laboratory WPEC 2007 SG24

7 UNIFIED MONTE CARLO (UMC) D.L. Smith, A Unified Monte Carlo Approach to Fast Neutron Cross Section Data Evaluation, Proceedings of the 8th International Topical Meeting on Nuclear Applications and Utilization of Accelerators, Pocatello, July 29 August 2, 2007, p BAYES THEOREM & PRINCIPLE OF MAXIMUM ENTROPY p(σ) = C x L(y E,V E σ) x p 0 (σ σ C,V C ) p 0 (σ σ C,V C ) ~ exp{-(½)[(σ σ C ) T (V C ) -1 (σ σ C )]} L(y E,V E σ) ~ exp{-(½)[(y y E ) T ( V E ) -1 (y y E )]}, y=f (σ) y E, V E : measured quantities with n elements y C, V C : calculated using nuclear models with m elements UMC based on p(σ), GLS on the peak of the distribution 6/26

8 UMC sampling schemes BF approach: A set of independent {σ} METROPOLIS approach: An stochastic Markov chain {σ} distributed following p(σ) If p(σ )> γ p(σ(t)) then σ(t+1)=σ ; else σ(t+1)=σ(t) p 0 (σ σ C,V C ) ~ exp{-(½)[(σ σ C ) T (V C ) -1 (σ σ C )]} 7/26

9 Central Limit Theorem 8/26

10 Metropolis algorithm 9/26

11 800 Usual normal rng (Sum of 12 random number) Metropolis algorithm Gaussian fit, R= Arbitrary Units X 10/26

12 SPREAD OF SAMPLED VALUES 11/26

13 LINEAR MODEL: y = σ Node Model Expt σ E (%) Expt / Model Comments % within error % within error % marginal % discrepant % marginal % within BIG error 7 6 No exp.data covexp(3,1) = covexp(1,3) = 0.2 σ E (1) σ E (3) - weak correlation covexp(5,2) = covexp(2,5) = 0.8 σ E (2) σ E (5) - strong correlation 12/26

14 MODEL DATA & CORRELATION ~ 95% correlation 13/26

15 UMC convergence Chisq Convergence GLS UMC: BF UMC: Metropolis Red.Chisq E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08 1.E+09 Histories 50 Chisq Convergence GLS UMC: BF UMC: Metropolis Red.Chisq E+05 1.E+06 1.E+07 1.E+08 1.E+09 Histories 14/26

16 RATIO CASE MODEL EXPERIM / MODEL y1=210 +/- 63 (30%) y2=32 +/- 9.6 (30%) Cov(1,2) = 0.95 EXPERIM f1=y1= / (30%) f2=y2/y1= / (5%) ~ 43 Cov(1,2) = 0.

17 EXPERIM MODEL COMBINED 16/26 5% exp. ratio unc., 95% model correl COMBINED GLS UMC 195 +/- 44 (22%) 151 +/- 37 (25%) /- 7 (18%) 30 +/- 7 (22%)

18 GLS FAILURE: ANALYSIS 5% exp. ratio unc. 95% model correlation 5% exp. ratio unc. no model correlation Quantity Node 1 Node 2 Ratio BF/GLS METR/GLS METR/BF Quantity Node 1 Node 2 Ratio BF/GLS METR/GLS METR/BF % exp. ratio unc. 95% model correlation Quantity Node 1 Node 2 Ratio BF/GLS METR/GLS METR/BF /26

19 LOG TRANSFORMATION 18/26

20 SUMMARY UMC is a viable tool for cross-section data evaluation Metropolis sampling scheme is recommended for UMC calculations When data values are cross sections or cross sections and integral (spectrum-averaged) cross sections, GLS and UMC are equivalent so GLS is recommended If ratio or other explicitly non-linear data are introduced UMC may be preferable to GLS 19/26

21 PDF 1. Lognormal distribution There is a problem associated with utilizing normal distributions if the uncertainties in the variables are large (e.g., typically > 30%). q(z) = (C/z) exp[-(ln z ν) 2 /(2σ 2 )] where 0 < z < +, mean value = <z> = exp[ν + (σ 2 /2)], variance = V z = exp(2ν + 2σ 2 ) exp(2ν + σ 2 ). Best approach: x = ln z Lognormal PDF in z a normal PDF in x. If the variable z has an uncertainty E z, then the uncertainty in E x =E z /z = relative error in z. 20/26 See paper by Capote R and Smith D in NDST2010 (Korea)

22 Normal Lognormal CASE DATA NUMBER Experim Model UMC (x+ 0.2) GLS (x - 0.2) DATA NUMBER 21/26

23 RESULTS 1 st data point ( MOD =80%, exp =90%) 10 Experim Model UMC (normal) GLS (normal) UMC (logn) GLS (logn) DATA NUMBER GLS (normal) GLS(logn) Diff(%) /26

24 PDF 2. Uniform distribution a,b 0 (cross sections) Uniform distribution [1 / (b-a)] if a < z < b, q(z) = 0 otherwise Mean value <z> = (a+b)/2; Variance E z2 = [(b-a) 2 /12] Rel. Error = (b-a)/(a+b)/ 3 If a=0 => Rel. Error = 1/ 3 = 57.7%!!! 23/26

25 Uniform normal (no correlations) <z> = (a+b)/2 1 st point uniform 2 nd point normal Case 2 (strong overlap): a C1 = 2, b C1 = 40 a E1 = 5, b E1 = 60 SOLUTION: a S1 = 5 and b S1 = 40. UMC analysis <σ 1 > = (44.9%) GLS analysis <σ 1 > = (36.5%) Case 1 (identical intervals): a C1 = 10, b C1 = 40, σ C2 = 100 (20%) a E1 = 10, b E1 = 40, σ E2 = 85 (15%) SOLUTION: a S1 = 10 and b S1 = 40. UMC analysis <σ 1 > = (34.6%) <σ 2 > = (12.0%) GLS analysis <σ 1 > = (24.5%) <σ 2 > = (12.0%) Case 3 (weak overlap): a C1 = 10, b C1 = 20 a E1 = 16, b E1 = 40 SOLUTION: a S1 = 16 and b S1 = 20. UMC analysis <σ 1 > = (6.4%) GLS analysis <σ 1 > = (15.7%) 24/26

26 PDF 3. exponential normal (no correl) A cross section estimate is provided but no uncertainty is given Exponential distribution q(z) = C exp(-z/<z>) where 0 < z < and <z> > 0 ; Mean value = <z>, Variance V z = <z 2 > - <z> 2 = E z2 = <z> 2. Case 1: UMC GLS σ C1 = (30.00%), σ C2 = 22 (no uncert) (100%) σ E1 = (61.68%), σ E2 = 34 (no uncert) (100%) SOLUTION: UMC analysis <σ 1 > = (21.1%); <σ 2 > = ( 9.9%) GLS analysis <σ 1 > = (21.1%) ; <σ 2 > = (72.3%) The posterior function p includes the product of two simple exponentials. This product equals the exponential factor exp(-σ 2 /σ m2 ), where (1/σ m2 )= (1/σ C2 )+(1/σ E2 ). Thus, for the input values given, one obtains σ m2 = /26

27 CONCLUDING REMARKS Covariance Workshop, Port Jefferson 2008 UMC is a viable tool for cross-section data evaluation. UMC Metropolis sampling scheme is recommended. When data values are cross sections or cross sections and integral (spectrum-averaged) cross sections, GLS and UMC are equivalent so GLS is recommended. If ratio or other explicitly non-linear data are introduced UMC may be preferable to GLS. GLS is equivalent to UMC for PDFs closely resembling symmetric normal distributions with small to modest uncertainties. UMC results are more reliable for broad or strongly asymmetric distributions. The lognormal distribution can be used to good advantage in certain situations involving large uncertainties. Use of the uniform and exponential distributions discouraged.