A Hybrid Variance Reduction Method Based On Gaussian .

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Department of NUCLEAR ENGINEERINGA Hybrid Variance Reduction MethodBased on Gaussian Process forNuclear Reactor AnalysisZeyun Wu, Qiong Zhang, and Hany Abdel-KhalikDepartment of Nuclear EngineeringNorth Carolina State UniversityNovember 2nd, 2011Page 1 of 1417

Department of NUCLEAR ENGINEERINGIntroduction Deterministic method is fast but lack of flexibility and inaccurate Monte-Carlo (MC) method is universal and more physical reliablebut time consuming Hybrid deterministic-MC methods have being recently gettingmore and more interest to researchers. Deterministic models solution (both forward and adjoint) isemployed to bias source particles and assign appropriateimportance map to MC models to accelerate MC simulation andreduce the variance. Some current developed hybrid approaches:–––––Variational variance reduction (Densmore & Larsen 2003)Correction method (Becker et al. 2007)FW-CADIS (Wagner et al. 2007)Talley linear combination (Solomon et al. 2009)Coarse mesh finite difference (Lee et al. 2009)Page 2 of 17

Department of NUCLEAR ENGINEERINGChallenges Multiple responses application– Importance for different responses areexpected to be different– Adjoint calculation needs to performindividually for each response– Computational overheads becomeunacceptable with the increase of responses. Global and uniform variance reduction in thewhole phase spacePage 3 of 17

Department of NUCLEAR ENGINEERINGFW-CADIS ApproachL* (φi* ) Ri φ Adjoint deterministic model: Adjoint solutions are employed to bias particlesource distribution and weight window map Pseudo response - combine multiple responseswith linear combination and the weight for eachresponse is assigned as the reverse of the forwardsolutionsm ξ**ξ Ri / φiL (φ ) φi 1Page 4 of 17

Department of NUCLEAR ENGINEERINGMotivations of GP Approach Importance for different responses are expected to becorrelated albeit they are different. Resulting responses uncertainties are expected to becorrelated– Given m responses, let r denote number of independentcorrelations.– Bias MC particles towards r (rather than m) independentcorrelations Gaussian Process (GP) approach is developed on theseideasPage 5 of 17

Department of NUCLEAR ENGINEERINGCorrelations? Given m random variables: X 1 ,., X m Correlations are described by:cov ( X i , X j ) E ( X i X iμ )( X j X μj ) Ci , jC WΣ 2 WT Wr Σ 2r WrT W [ w1 . wm ] R m m Σ diag {σ 1 ,., σ m } R Wr [ w1 . wr ] Rm mm r Σ r diag {σ 1 ,., σ r } R r rPage 6 of 17X2w2σ2w1σ1X1

Department of NUCLEAR ENGINEERINGGaussian Process (GP) Approach Radiation transport may be treated as a GaussianProcess1 If responses correlations (covariance matrix) can beconstructed and effective rank r can be estimated,one can reduce it to identify r uncorrelated pseudoresponses r pseudo responses are formed in GP approachmξ jGP wi , j Ri , j 1,., ri 11M.KENNEDY and A. O’HAGAN, “Bayesian calibration of computer model,” Journal of the RoyalStatistical Society, 63, 3, 425–464 (2001).Page 7 of 17

Department of NUCLEAR ENGINEERINGGP Approach - EstimatingResponses Covariance MatrixKTR [ R1 R2 . Rm ] Rm represent a vector of the mLetresponses of interest representing m random Gaussiank mprocesses. Denote{ R i }i 1 , k 1, 2 , ., N as N realizations ofthese random processes. The covariance between the tworesponses R i and R j is given by:1 Ncov( Ri , R j ) limRik Rˆ i R kj Rˆ j N N 1k 1()()The covariance information between all pairs of m responsesmay also be represented by a symmetric covariance matrixC R m m such that:Cij cov( Ri , R j ). The SVD form of this matrix is:rK KTK KC WΣ W σ wi wi σ i2 wi wiT Crm2Ti 12ii 1Page 8 of 17

Department of NUCLEAR ENGINEERINGNumerical ApplicationsPage 9 of 17

Department of NUCLEAR ENGINEERINGCase Study 1: BWR Assembly Model Code: MAVRICsequences in SCALEpackage BWR Assembly, 7x7array of fuel pins withvarious enrichments Fixed sourcesubcritical system Total 27 neutron and19 photon energygroup library areappliedPage 10 of 17

Department of NUCLEAR ENGINEERINGCase Study 2: PWR Core Model X-Y view of the coreloading pattern withdetails assemblydescribed on the side Total 193 fuelassemblies (blueregion) laid out a17x17 grid schemeand surrounded bylight water (redregion) Two types of fuelassemblies aredesigned: UO2 fuelassembly and a UO2Gd2O3 fuel assembly.Page 11 of 17

Department of NUCLEAR ENGINEERINGEstimate of the Effective Rank forCovariance MatrixRelative response error expectaion (100%)2101SVD of Covariance Matrix:0K KC WΣ W σ i2 wi wiT10m210Ti 1Truncated SVD Approximation:-110K KCr σ i2 wi wiTri 1-210-31005101520Rank numberSearch Criterion:2530C Cr δPage 12 of 17

Department of NUCLEAR ENGINEERINGRelative Uncertainty Comparison for Thermal Flux(GP vs. FW-CADIS, Assembly Model)Thermal Flux Distribution (n/cm2-s)112233y-indexy-indexReduced Uncertainty ndex2.55363.5744x 10a. Standard Deviation Reductionb. Mean Thermal Flux Distributionσ iFW-CADIS σ iGP 100%Metric: ε i FW-CADISσiPage 13 of 17

Department of NUCLEAR ENGINEERINGRelative Variance (%)Global Uniformity of Variance(GP vs. FW-CADIS, Assembly y-indexa. FW-CADIS ApproachPage 14 of 17567123456x-indexb. GP Approach7

Department of NUCLEAR ENGINEERINGRelative Uncertainty Comparison for Thermal Flux(GP vs. FW-CADIS, Core Model)Thermal Flux Distribution (n/cm2-s)22446688y-indexy-indexReduced Uncertainty (%)101012121414161656010x-index6515570500a. Standard Deviation Reduction10x-index600 70015800 900 1000 1100b. Mean Thermal Flux Distributionσ iFW-CADIS σ iGP 100%Metric: ε i σ iFW-CADISPage 15 of 17

Department of NUCLEAR ENGINEERINGRelative Variance (%)Global Uniformity of Variance(GP vs. FW-CADIS, Core . FW-CADIS Approach15101010205x-indexb. GP ApproachPage 16 of 17

Department of NUCLEAR ENGINEERINGConclusions Number of independent correlations are much smaller than numberof responses, when responses are required everywhere in phasespaceGP assumption provides one way to take advantage of responsesuncertainties correlations and the deterministic models can beemployed to identify correlationsSimple numerical experiments show that GP approachsuccessfully gain better convergence in MC simulation comparingto FW-CADIS approachThis idea could be extended to other hybrid deterministic-MCtechniquesA hybrid between GP and FW-CADIS methodology is suggested toreach their combined benefits in the futurePage 17 of 17

Journal of the Royal Statistical Society, 63, 3, 425–464 (2001). . array of fuel pins with various enrichments Fixed source subcritical system . group library are applied. Page 11 of 17 Department of NUCLEAR ENGINEERING Case Study 2: PWR Core Model X-Y view of the core loadin

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