International Conference on Reactor Physics, Nuclear Power: A Sustainable Resource Casino-Kursaal Conference Center, Interlaken, Switzerland, September 14-19, 2008

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A quasi-regression method for automated cross-section parameterization: PBMR benchmark example

Pavel M. Bokova, *, Rian H. Prinslooa, Djordje I. Tomaševića, Frederik Reitsmab

a Necsa, Pretoria 0001, South Africa b Pebble Bed Modular Reactor Pty (Ltd), Centurion, South Africa

Abstract

The problem of parameterization of few-group neutron cross-sections is addressed. The objective of the work is to create an automated tool for parameterization, consisting of the following essential parts: approximation of a smooth function of an arbitrary number of dimensions; built-in approximation error characterization; capability of analysis for the obtained dependencies; and library optimization. A methodology fulfilling the requirements above and based on randomized quasi-regression is considered. In the present work the parameterization technique is illustrated with an example of generating a library for the OECD PBMR-400 MW transient benchmark. With some small changes the methodology can potentially be applied for any reactor type as a powerful diagnostic tool of analysis for cross-section dependencies. Results show that a practical library representation can be obtained with accurate error estimations

* Corresponding author, pbokov@necsa.co.za Tel: +27 (012) 305 5624; Fax: +27 (012) 305 5166.

1. Introduction

Few-group homogenized cross-sections are typically pre-calculated, parameterized and stored in representations such as tables or polynomial libraries. This approach is followed since the cost of online cross-section calculations during steady state, but especially during transient calculations, is computationally very expensive. The construction of such multi-dimensional parameterized libraries requires an in depth knowledge and understanding of the physics of the problem and involves extensive analysis of cross-section dependencies. The practicality of such a library is measured by its associated accuracy, cross-section reconstruction time and storage requirements.

In this work a method is developed for, and applied to, the automatic generation of a parameterized few-group cross-section library. The method employs randomized quasi-regression, which aims to globally approximate a smooth, black-box function, dependent upon a near arbitrary number of state parameters. Coupled with the Analysis of Variance (ANOVA) global sensitivity analysis method, this approach provides:

• the powerful ability to automatically determine and quantify the importance of cross-section dependencies upon different state parameters within the model,

• the ability to capture appropriate cross-term dependencies,

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• the capability to identify and exclude irrelevant state parameters from the model,

• a way to carry out all the above mentioned manipulations and calculations, at the same time providing a consistent estimate for the approximation error.

This methodology can potentially be applied to any type of reactor (e.g. Bokov and Prinsloo, 2007, discuss a similar technique adapted and applied for the Material Test Reactor (MTR)). In this paper we focus on the generation of a library for the OECD PBMR-400 MW transient benchmark (Reitsma et al., 2006), and compare our methodology to the existing table lookup (using linear interpolation) library supplied with the benchmark specification. The performance of both libraries was evaluated by comparing the reconstructed cross sections to the reference solution from the Tispec code (Reitsma et al., 2006). The unique advantages and features of the quasi-regression method are explained and illustrated.

2. Methodology

2.1. Methodology summary

The cross-section library is generated using the quasi-regression (QR) parameterization technique in conjunction with a global method of functional and Sensitivity Analysis (SA) called Analysis of Variance (ANOVA).

Quasi-regression is a Monte Carlo (MC) method for constructing approximations to functions, based on observations sampled from the domain of the function to be approximated (An and Owen, 2001). Due to its generality and flexibility, QR is particularly suitable for approximations of smooth continuous “black box” dependencies, such as few-group neutron cross-sections. The first attempt to use QR for cross-section parameterization was made by Bokov and Prinsloo (2007) for a MTR-type application.

In this work, a modified version of quasi-regression (Lemieux and Owen, 2002), called randomized quasi-regression (RQR), is used. The randomization increases the calculational cost of parameterization, but aims to overcome the principal drawback of QR, namely the lack of an easy way to provide an estimation of approximation error.

Global Sensitivity Analysis (SA) and Analysis of Variance (ANOVA) are tools devised tp provide unambiguous information on the general structure of complicated multi-parameter functions by quantifying the variation in the output variables to the variation of the input variables. One of the most efficient SA techniques is based on the global Sensitivity Indices (SI), developed by Sobol’ (1993) which in turn derive from an ANOVA-type of a high dimensional model representation. The method of global SA can be applied to both linear and highly nonlinear functions.

The parameterization methodology presented in this paper includes the following essential components:

• Approximation, including the generation of a set of basis functions and an estimation of the corresponding regression coefficients.

• Sampling strategy, allowing both optimal coverage of the parameter space and on-line error estimation for any particular regression coefficient (as well as for the overall approximation).

• Optimization (shrinkage) of the set of basis functions, with the goal to reduce the library size and to improve the efficiency of cross-section reconstruction.

• Analysis of the importance of different state parameters in the constructed dependencies and rejection of non-relevant state parameters from the model.

Each of these components is discussed in more detail below.

2.2. Approximation

Consider dependencies of a physical value F (e.g. few-group homogenized neutron cross-section) on d state parameters ix (where 1,2,...,i d= )

within a rectangular multidimensional problem domain : , 1 d d

i i iD a x b i d= ∈ ≤ ≤ ≤ ≤x ℝ . We

require ( )F x to be a smooth continuous function

within dD . The rectangular region dD is usually mapped, for convenience, onto a unit hypercube

dH = : 0 1, 1 diu i d∈ ≤ ≤ ≤ ≤u ℝ . The function

( )F u is then approximated by a linear combination

* ( ) ( )F β ψΩ ∈Ω=∑ k kk

u u (1)

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of orthonormal multivariate basis functions ( )ψ k u :

( ) ( )dH

dψ ψ δ=∫ k m kmu u u (2)

built as a tensor product of orthonormal univariate functions:

1( ) ( ).

i

d

k iiuψ φ

== ∏k u (3)

In our work the univariate orthonormal basis functions ( )

ik iuφ are constructed from conventional

Legendre polynomials ( )kL x by rescaling its argu-

ment from the interval [-1,1] to the interval [0,1]

and then by renormalizing the obtained function,

thus giving: ( ) 2 1 (2 1)i ik i i k iu k L uφ = + − . It is easy

to prove that the resulting multidimensional basis function ( )ψ k u satisfy the condition (2), where the

Kronecker symbol with the vector indices signify

1 1...

d dk m k mδ δ δ= × ×km . The index vectors , d+∈k m ℤ

contain d nonnegative integers representing the orders of Legendre polynomials. Different combinations of indices ik will lead to a countably

infinite set ∞Ω of index vectors.

In practice we use finite sets ∞Ω ⊂ Ω of

approximation term index vectors k constructed as different combinations of indices ik +∈ℤ resulting

in 0 0B≤k , 1 1B≤k and B∞ ∞≤k , where

0B , 1B , B∞ are pre-defined integers. The norms

0 011

i

d

ki >==∑k , 1 1

d

iik

==∑k ,

1max i

i dk∞ ≤ ≤

=k

of the integer vector k are called rank, degree and order, correspondingly (An and Owen, 2001).

Note that, any particular term in approximation formula (1) can be characterized explicitly by its index vector ∞∈ Ωk , hence, for the sake of brevity,

later on in this paper we do not make a distinction between sets of terms used in approximations and sets of corresponding index vectors.

The regression coefficients minimizing the least-square error can then be calculated as follows (An and Owen, 2001):

( ) ( ) .dH

F dβ ψ= ∫k ku u u (4)

The problem of approximation has thus been reduced to a problem of evaluating integrals over a multidimensional domain. Monte Carlo (MC) methods are usually the most efficient way of evaluating such integrals, and hence the estimators for the regression coefficients are calculated as an average

( ) ( )( )

1

1ˆn

nm m

m

Fn

β ψ=

= ∑k ku u (5)

on a set of sample points 1,...,n nP = u u , uniformly

distributed over dH (henceforth denoted ( )d

m U Hu ∼ ). Note that in this paper we use the

convention that the hat symbol over some variable means its numerical (e.g. MC) estimation.

2.3. Sampling strategy

The efficiency of MC methods is determined by the properties of the set nP . In the case of the

conventional version of Monte Carlo technique mu

are pseudo-random points. Many practical studies have proven that the efficiency of integration is better if the sample points mu belong to so-called

Low-Discrepancy Sequences (LDS). In this case the integration method is termed quasi-Monte Carlo (QMC). Specifically, so-called Sobol’ LDS which are in many respects superior to other LDS (Kucherenko and Shah, 2007), are used in our work.

The main disadvantage of QMC is that there is no simple way to estimate the integration error. In order to overcome this drawback, a special technique called Randomized Quasi-Regression (RQR) has been developed (Lemieux and Owen, 2002). The main idea behind this technique is to

make 1R − random replications , 1, ,,...,n r r n rP = u u

of the initial LDS applying fixed random shifts

( )dr U Hs ∼ :

( ), mod 1m r m r= +u u s (6)

( 1,..., 1,r R= − 1,...,m n= ). Here mod 1x means

x x− , where x is the greatest integer less or equal to x . This operation is applied coordinate-

wise: ( ), , , , mod 1i m r i m i ru u s= + ( 1,...,i d= ). Note,

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that replications ,n rP inherit the crucially important

properties of the initial LDS, namely that each such replication constitute a LDS of uniformly distributed points over the unit hypercube dH .

Then, using Eq. (5) we calculate estimates ( ),

ˆ nrβk

( 0,1,..., 1r R= − ), where the case 0r = corresponds to the initial LDS. We can therefore introduce a pooled βk estimate for regression coefficient βk as

an average over all estimations, i.e.:

1( ) ( )

,0

1 ˆ ,R

n nr

rRβ β

−

=

= ∑k k (7)

characterized by variance Var β k , which can be

estimated with sample variance as follows:

( ) ( )1 2

( ) ( ) ( ),

0

1 ˆVar .1

Rn n n

rrR R

β β β−

=

= − − ∑k k k (8)

The sample variance ( )Var[ ]nβk characterizes

the error of estimation of the corresponding regression coefficient.

2.4. Shrinkage

When no information about the function ( )F u

is available, the number of terms used in Eq. (1) may be very high. It is therefore important to select a subset ′Ω ⊆ Ω of terms which substantially contribute to the approximation, i.e. “shrink” the library. This not only reduces the library size, but also significantly improves the calculational time of cross-section reconstruction.

2.4.1 Analysis of variance and error control The shrinkage can be done using ANOVA

principles (see discussion by An and Owen (2001) and by Lemieux and Owen (2002)). According to these principles, the contribution (and hence importance) of each particular term ≠k 0 (i.e. excluding the constant term) to the total variance of the function ( )F u is equal to the square of the

corresponding regression coefficient:

[ ] 2

\Var ( ) ,F β

∞∈Ω=∑ kk 0

u (9)

where the variance is introduced as follows:

[ ] ( ) ( )22Var ( ) ( ) ( ) .

d dH HF F d F d≡ −∫ ∫u u u u u (10)

The contributions of individual terms in Eq. (9) are independent, which is due to the orthogonality of the basis functions ( )ψ k u .

From Eqs. (9) and (10) it follows that the Mean Squared Error (MSE)

* 2MSE [ ( ) ( )]dH

F F d′Ω Ω≡ −∫ u u u (11)

of the approximation introduced due t the truncation of the infinite set ∞Ω to a finite set ∞Ω ⊂ Ω

(hereafter named truncation error) is

[ ] 2 2\MSE Var ( ) ,F β βΩ ∈Ω ∉Ω= − =∑ ∑k 0 k k ku (12)

i.e. it is equal to the sum of the squared regression coefficients of terms which are not included in the approximation formula. In practice it is convenient to use the Lack Of Fit (LOF):

LOF MSE/ Var[ ( )],F= u (13)

which describes a part of the function variance not explained by the approximation function.

Eq. (9) and Eq. (12) lead us towards a library shrinkage approach including a consistent, error controlled criterion for rejection of unimportant terms. Assume that values Var[ )]F(u and 2βk can

be assessed independently. We can then assess LOF .Ω Suppose that we want to shrink a library in

such a way that the error of approximation due to this shrinkage does not exceed some predefined target value *LOF∆ . Therefore, all the small terms in the approximation formula can be rejected until the sum of their contributions (each contribution is characterized by 2LOF / Var[ )]Fβ∆ =k k (u ) is less

than *LOF∆ . The remaining terms form a subset

∞′Ω ⊆ Ω ⊂ Ω . The approximation based on ′Ω is

characterized by the error *LOF LOF LOF .′Ω Ω≤ + ∆

It is important to highlight that, due to the orthogonality of terms in Eq. (1), no further regression calculations are required in order to perform shrinkage. The same argument holds for the addition of extra terms, which can be independently

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estimated. The sum in Eq. (1) can therefore be extended to include further terms if ANOVA shows that the initial set Ω is not sufficient to approximate the function with the given accuracy (i.e., if LOFΩ

is too big).

2.4.2 Practical error control Inevitable numerical errors should be taken into

account in practical calculations, thus leading to some corrections in the error control approach discussed above. In this section we briefly address these issues.

Built-in error control based ANOVA requires

estimates for Var[ ( )]F u and 2βk . Var[ ( )]F u may

be assessed either using (with slight modifications to account for randomization) the recurrence formulas implemented by An and Owen (2001) or numerical integration in Eq. (10). As discussed by Lemieux and Owen (2002), one should use a bias-corrected estimator for 2βk , constructed in the following way:

( ) ( ) ( )1 222 ( ) ( )

,0

1 ˆ .1 1

Rn n

rr

R

R R Rβ β β

−

== −

− −∑k k k (14)

The easiest and fastest way for an inline approximation error characterization can be based on the discrepancy between estimates of the function variance and the estimated contribution of terms included in the approximation (see Eq. (11))

2\VD | Var[ ( )] |F β′Ω ′∈Ω= −∑ k 0 ku , (15)

which we denote as the Variance Defect (VD). Our analysis (which is not given for the sake of brevity) showed that the VD can be used for practical ANOVA, but gives a poor (and often very pessimistic) estimation for the MSE. Indeed, if the regression coefficients are estimated with some error ε β β= −k k k then the error of approximation

becomes the sum of the integration and of the truncation errors (Bokov and Prinsloo, 2007):

2 2MSE .ε β′ ′ ′Ω ∈Ω ∉Ω= +∑ ∑k k k k (16)

It can be easily demonstrated that as 0ε →k ,

| MSE VD |′Ω′Ω − can be much larger than the

theoretical error 2ε′∈Ω∑ kk following from Eq. (16).

An alternative (and more exact) approach towards the approximation error characterization is based on the MSE definition (Eq. (11)) and consists in an on-line comparison of the function value prediction by the approximation function, i.e.

* ( ) ( )ˆ ( ) ( )n nF β ψ′Ω ′∈Ω=∑ k kk

u u and of the sample

function value ,( )m rF u , averaged over recent

0N m R= × samples (from 0 0 1n n m≡ − + to n ),

where 0 2m n= as recommended by An and

Owen (2001):

0

1( )*( 1) 2

, ,0

1 ˆMSE [ ( ) ( )] .n Rn

mm r m r

m n r

F FN

−−

′Ω ′Ω= =

= −∑∑ u u (17)

( )

MSEn

′Ω can be utilized for the calculation of

the estimator ( ) ( ) ( )

LOF MSE / Var [ ( )]n n n

F′ ′Ω Ω= u and

then used in approximation term optimization as described above. The ANOVA-based error control and library optimization is consistent, but not necessarily the only way of characterization of the approximation accuracy. Other practical estimators can be calculated on-line, such as the Average Relative Error (ARE)

0

1( )

,0

1ARE RE

n Rn

m rm n rN

−

= == ∑∑ (18)

and the Maximum Relative Error (MaxRE)

0

( )

,, 0 1

MaxRE max REn

m rn m n r R≤ ≤ ≤ ≤ −

= , (19)

where *( 1), , , ,RE | [ ( ) ( )] / ( ) |m

m r m r m r m rF F F−′Ω= −u u u .

Note that, Eq. (16) can also be interpreted as a balance between the error occurring due to the rejection of a particular term ( 2β∝ k ) and its

retention ( 2ε∝ k ). Randomization permits us to

estimate the squared value of the integration error 2εk according to Eq. (8). Hence, in order to improve

the quality of the approximation, the rejection criterion (discussed above) can be modified in a way that not only insignificant ( 2 Var[ ( )]Fβk u≪ ), but

poorly estimated terms (2 2β ε<k k ), are also rejected.

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2.5. Relevance of different state parameters

Optimization of the cross-section library can include more than the rejection of irrelevant terms in an approximation, but also rejection of irrelevant state parameters. This becomes important when a large number of state parameters is used in the model, which is the case if complicated and/or unknown physical dependencies are involved. Therefore, reduction of the set of model input variables to a subset of significantly contributing parameters is of high importance from a practical point of view.

Then the same ANOVA principles may be used to determine the relevance of state parameters, or combinations thereof, in the model. It can be done using Sensitivity Indices (SI) introduced by Sobol’ (1993). An analysis, based on the global SI, gives us qualitative and unambiguous information on the importance of the different subsets of input parameters to the output variance of ( ).F u

In practice we are usually not very interested in the importance of the different combinations of state parameters, but rather in whether a particular state parameter iu can be excluded from the model (i.e.

fixed at its nominal value) without compromising the accuracy of the approximation. The important SI in this case are iS and tot

iS , which can be calculated

as follows (Lemieux and Owen, 2002):

[ ] [ ]22

, ,Var ( ) Var ( )

i itoti iS S

F F

ββ∈Ω ∈Ω= = ∑∑ kk k k

u u (20)

where , : 0 and 0, 1,..., i i j ik k j d≠′Ω = ∈ Ω ≠ = =k

is the set of terms having dependencies on variable

iu only; : 0i ik′Ω = ∈ Ω ≠k ( 1,...,i d= ) is the set

of terms containing dependence on parameter iu

and, possibly, on other parameters (i.e. one takes cross-terms into account).

The knowledge of iS and totiS provides

sufficient information to determine the sensitivity of the analyzed function to the individual input variables. For example, in the following extreme situations:

• 0toti iS S= = means that ( )F u does not

depend on iu ;

• toti iS S= corresponds to the absence of

interactions between variable iu and other

variables (namely, 1toti iS S= = means that

( )F u depends only on iu ).

In practice, the dependence on a particular state parameter iu can be neglected if the corresponding

sensitivity index totiS is smaller than some

predefined value model

LOF ,i∆ which describes the

approximation error we allow for the sake of model simplification.

As in the case of shrinkage of the approximation function, the reduction of the model parameter set can be done in a fast and controlled way, requiring no extra function evaluations or regression iterations.

Note that the automatic identification of the most important dependencies, with the associated advantages of library shrinkage, should not be seen as an automated state-parameter generator. The user must define the state parameters expected to play a role, which will then be eliminated by shrinkage if not needed. Missing state-parameters can not be identified.

3. Problem specification and approach

The cross-section dependencies required for the parameterization of water moderated reactor systems are generally well known. This statement is however not necessarily true for the more complex modern fuel designs and new generation reactor concepts under development today. It is therefore interesting to apply the Randomized Quasi-Regression (RQR) method to the Pebble Bed Modular Reactor (PBMR) design where the effect of the environment plays an important part in the few-group cross sections. The OECD PBMR 400MW benchmark (Reitsma et al., 2006) problem was selected to perform an initial evaluation of the RQR method.

In the benchmark specification, neutronic cross-sections for steady-state and transient analysis are parameterized against five prescribed state parameters, namely:

1. Fuel Temperature (300 to 2400 [K]), 2. Moderator Temperature (300 to 2200 [K]) , 3. Fast Buckling, which describes the fast

spectrum environmental effect or leakage (from 41 10−− × to 34 10−× [cm-2]),

4. Thermal Buckling, which describes the thermal spectrum environmental effect (from 32.5 10−− × to 35 10−× [cm-2]),

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5. Xenon Number Density, which explicitly captures the Xe feedback effect for startup, operational and transient conditions ( 151 10−× to 108 10−× [#/barn·cm]).

The benchmark specification utilizes a multi-dimensional cross-section table (4 × 7 × 3 × 3 × 3) dependent upon the five described state parameters (which includes cross-term dependencies) for 115 different materials, with linear interpolation laws applied between points. The prescribed sets of table points were generated via the Tispec code.

In order to evaluate the RQR method for this problem, the QUARX test code (QUAsi-Regression Cross-Section parameterization) was developed and coupled to the Tispec code (to supply cross-section values at the required LDS points).

A QUARX library was produced for all materials specified in the OECD PBMR-400 bench-mark. An independent control set of random points was used to compare the reconstructed cross-sections from the QUARX library and the linearly interpolated OECD PBMR-400 benchmark library to the reference solution from the Tispec code. The aim of this comparison is four-fold:

1. determine whether the developed methodo-logy succeeds in describing, automatically and accurately, OECD PBMR-400 cross-section dependencies,

2. determine the validity of the built-in error estimation within the randomized quasi-regression method,

3. report the error associated with the OECD PBMR-400 linear interpolation library and comment on the practicality of using the QUARX library for such an application, and

4. utilize the ANOVA technique to prioritize state parameters for the potential optimiza-tion of the model.

It should be kept in mind that the cross section dependencies studied, are determined by the selection of state parameters made in the OECD PBMR-400 benchmark specification and also on the Tispec code limitations. The study and results reported can therefore not be seen as a definitive study of the PBMR cross section dependencies. However, since the OECD/PBMR-400 library does capture the primary cross section dependencies, the abilities and potential of the automated QUARX library generation and quasi-regression methodo-logy are illustrated adequately.

4. Results

Prior to comparing the accuracy and performance of the QUARX library, it is insightful to investigate the structure of such a library. Table 1 illustrates an extract of the library content for the thermal ν-fission cross-section for fuel material 101 (as numbered in benchmark specification). The library was generated with the input requirement that all cross-sections have estimated average relative errors (ARE) of less than 0.5% and maximum relative error (MaxRE) of less than 4%. These accuracy requirements were chosen in order to produce a sufficiently accurate library to analyze the dependencies, and not necessarily to use as an official library for core calculations. In order to achieve this 39159 × 25 (LDS points × replications) sample evaluations (Tispec runs) had to be performed

Table 1 Structure of QUARX library: example based on the thermal ν-fission cross-section. Ten of the most important (from 37 retained after shrinkage) terms are listed

Thermal ν-fission cross-section ARE: 0.20% MaxRE: 2.55% Terms

Coefficient

Relative Error (%)

Importance (%)

0 0 0 0 0 3.62 × 10-3 1.39 × 10-6 0.00

0 0 0 1 0 -5.01 × 10-4 6.57 × 10-6 50.88

0 1 0 0 0 -4.76 × 10-4 6.38 × 10-6 45.75

0 1 0 1 0 9.49 × 10-5 3.61 × 10-5 1.82

0 0 0 0 1 -4.95 × 10-5 1.01 × 10-5 0.50

0 2 0 0 0 3.91 × 10-5 3.41 × 10-6 0.31

0 0 0 2 0 3.18 × 10-5 4.77 × 10-6 0.20

0 1 0 0 1 2.88 × 10-5 4.27 × 10-5 0.17

0 3 0 0 0 -1.92 × 10-5 9.86 × 10-6 0.07

0 0 0 1 1 1.58 × 10-5 4.41 × 10-5 0.05

The data in Table 1 may be used to reconstruct the thermal ν-fission cross-section, which was approximated with an estimated ARE of 0.2% and MaxRE of 2.54%. These values differ from the input criteria since the thermal ν-fission cross-section converged faster than the Xe absorption cross-section (see Table 2), which was the slowest among the 15 two-group cross-sections to converge. The reconstruction is done by interpreting each term in the first column as index (order of term) ik of the

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univariate basis function (see Section 2.2) and the second column as the corresponding regression coefficient .βk Therefore the term (0 1 0 1 0)

indicates a bi-linear cross-term between moderator temperature and thermal buckling (the order of state-parameters as described in the previous section). Furthermore, columns three and four

provide an error measure 1/2ˆ ˆ(Var[ ]) /β βk k and the

relative importance (contribution) 2 / Var[ )]Fβk (u

of each term, respectively. From the results summarized in Table 1 we may

conclude that the linear dependence on thermal buckling is most important (50.88%), with a significant linear contribution from moderator temperature (45.75%). The bi-linear cross-term between moderator temperature and thermal buckling contributes as much as 1.82%. Moreover, it follows that the thermal ν-fission cross-section dependence on both fuel temperature and fast buckling is negligible. This information is useful in constructing e.g. reduced look-up tables, or simply for understanding the nature of the model.

Note that the importance (or contribution) of terms in an approximation (and later on of the state parameters in the model) are global and they are introduced in the least-square sense for the whole range of all the state parameters, used in the model. This means that they take into account not only cross-section dependences on state parameters, but also the limits in which the analysis has been done. Therefore a local (e.g. at the nominal condition) importance of a given state parameter can differ

from the indicators discussed here. In order to characterize the accuracy of the

approximation presented in Table 1, we summarize the average and the maximum relative errors, as well as the LOF for the QUARX library and the linear interpolation of the OECD PBMR-400 benchmark library in Table 2. Table 2 includes results for all the cross-sections reconstructed for benchmark fuel material 101. Columns two, three and four depict the ARE, MaxRE and LOF predicted during the QUARX library creation process. Columns four to seven, and eight to ten show the same error estimates for QUARX and OECD/PBMR-400 libraries respectively, after an independent error assessment. These errors are calculated as a result of a comparison of cross-sections reconstructed from the previously generated libraries with the reference solution from the Tispec code for the total of 20 000 control points. Maxima in all the columns are marked.

Results from Table 2 illustrate that the QUARX methodology succeeds to capture relevant depen-dencies to within the required accuracy limits over all evaluated cross-sections. The predicted error estimates during the library creation agree well with the error calculated from the control set for all three error measures. The observed discrepancy between the on-line and the independent error estimation based on a random set of points is expected and has a statistical nature (i.e. they converge as a number of samples tends to infinity).

The library based error and importance estima-tes are developed to provide diagnostic information,

Table 2 Summary of error predictions for fuel material 101

QUARX Library Prediction Control Set: QUARX Control Set: OECD

Cross Section ARE MaxRE LOF ARE MaxRE LOF ARE MaxRE LOF

Fast D 0.01 0.06 8.5 × 10-4 0.01 0.05 1.6 × 10-3 0.14 0.23 1.7 × 10-1 Fast absorption 0.08 0.60 2.7 × 10-3 0.08 0.55 2.7 × 10-3 3.75 6.56 3.7 Fast ν-fission 0.02 0.17 4.4 × 10-4 0.02 0.11 3.0 × 10-4 2.42 3.88 3.8 Fast fission 0.02 0.17 4.3 × 10-4 0.02 0.11 2.8 × 10-4 2.43 3.90 3.6 Fast scattering 0.05 0.44 4.8 × 10-4 0.05 0.44 3.7 × 10-4 7.50 12.27 6.6 Thermal D 0.05 0.93 3.2 × 10-1 0.05 0.87 3.2 × 10-1 0.06 0.27 3.8 × 10-1 Thermal absorption 0.18 2.84 6.6 × 10-2 0.18 2.84 6.5 × 10-2 0.56 2.26 4.5 × 10-1 Thermal ν-fission 0.20 2.55 2.1 × 10-2 0.19 2.30 1.9 × 10-2 1.17 3.04 4.6 × 10-1 Thermal fission 0.20 2.53 2.1 × 10-2 0.20 2.30 1.9 × 10-2 1.18 3.15 4.5 × 10-1 Thermal Xe absorption 0.37 3.75 8.6 × 10-3 0.37 4.11 9.0 × 10-3 2.34 6.25 3.0 × 10-1

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which can be utilized to make decisions regarding the applicability of the chosen set of state parameters. To illustrate this application of the method, Table 3 gives the calculated importance (i.e. sensitivity indices iS and tot

iS ) of each state

parameter for the thermal ν-fission cross-section for material 101. The importance of each state parameter is given in two forms: (1) total contribution to the function variance (tot

iS ) and (2)

contribution excluding cross-term interactions (iS ).

Table 3 allows valuable conclusions to be drawn concerning potential optimization of the cross-section model, and indicates whether any given state parameter should be removed from the model. As follows from Table 3 “Fuel Temperature” and “Fast Buckling” could be removed from the cross section model for thermal ν-fission with no loss of accuracy while also removing “Xe Number Density” will add only a loss of 0.77% in terms of the LOF estimator. In this way a 5-dimensional problem may be reduced to a 2-dimensional one which would dramatically improve cross-section reconstruction time and library storage space.

Table 3 Individual state parameter importances for ν-fission cross-section of material 101

State Parameter Importances (% of Var)

State Parameters Total No interactions

Fuel Temperature 0.00 0.00

Moderator Temperature 48.34 46.20

Fast Buckling 0.00 0.00

Thermal Buckling 53.08 51.09

Xe Number Density 0.77 0.50

This improvement becomes even more dramatic if one can demonstrate that a particular state parameter can be discarded not only for a specific cross-section in some energy group but for all cross-sections in a given group or for all groups and cross sections of a given material. In the case of OCDE PBMR-400 our analysis showed that 5-dimensional dependencies can be reduced to 2-dimensional for the fast spectrum (“Moderator Temperature”, “Thermal Buckling” and “Xe Number Density” may be discarded) and 3-dimensional ones for the thermal spectrum for all fuel materials (“Fuel Temperature” and “Fast Buckling” may be rejected). For the reflector material one of three parameters

(“Fast Buckling”) may be neglected in both fast and thermal spectrum.

As a final illustration of the potential application of the methodology, Figure 1 shows the importance of the moderator temperature to the thermal ν-fission cross-section for all the fuel material in the OECD PBMR-400 core model. Note that for illustration purposes only the moderator temperature importance for the materials is presented as a map reflecting the material positions in the reactor model. Information as presented in Figure 1 may potentially be used to further optimize the cross-section model based on material compositions. The importance were evaluated for the total state-parameter space and thus not dependent on the specific state parameter conditions in the location of the material in the reactor. The variation seen is thus exclusively due to the different material compositions (variations in burnup and isotopic concentrations).

Fig. 1. Moderator temperature importance to ν-fission cross-section across the OECD

benchmark PBMR core

5. Discussion

The major advantage of the automated parameterization process as implemented in QUARX is a black-box solution to provide a library representation with predefined accuracy over the state-parameter domain. Moreover, the methodology is not restricted to an approximation technique only, but includes tools for the analysis and optimization

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of the approximation function, and subsequently for model reduction. This property offers an opportunity to utilize it in conjunction with other parameteriza-tion approaches. For example, the QUARX evaluation of importance already provides some clues on the state parameter dependence that could introduce the large errors in the OECD library. Thus, for the thermal ν-fission the moderator tempe-rature shows higher order (more than linear) depe-ndencies that can only be captured in the OECD library by selecting additional state parameter values (increasing the seven already selected).

The QUARX library also shows a significant improvement in accuracy over the control set OECD PBMR-400 linear interpolation library since the latter was not necessarily optimized and the specific state parameter values or number of values could be adjusted to obtain similar accuracy. This can be done by inspection and/or data evaluations and would increase the storage needs whereas the QUARX library did not require any additional hu-man effort (but only additional sample evaluations).

The storage size of the QUARX library (and hence the memory usage) is substantially (few orders of magnitude) less than for an accurate linear table-lookup library; nevertheless the reconstruction time of the QUARX library may be somewhat greater depending on the problem considered (at this stage we do not have a reliable quantitative result for a comparison in similar conditions).

The major disadvantage in the method is the number of sample evaluations required to obtain realistic errors and accurate coefficients, which was close to a million in this example for material 101. Such a large number of assembly calculations will cause a limitation in practice, but it reflects the complexity of dependencies presented in the benchmark cross-section library (which has to be faced and dealt with by any competitive methodo-logy): a high number of input parameters leading to the well known curse of dimensionality; non-smooth dependence of some cross-sections (as our analysis has demonstrated). These complexities require high order terms in the approximation in order to achieve good accuracy. Moreover, we observe that such a large number of samples is necessary to assess accurately high-order regression coefficients, whereas the appropriate terms in the approximation are identified much earlier in the iteration process.

Therefore, despite its limitations, the proposed methodology constitutes a powerful diagnostic tool for automated parameterization of unknown multidimensional dependencies.

6. Conclusions

The quasi-regression based cross-section para-meterization technique described and applied in this work allows the building and quantifying of un-known model dependencies for a given set of state-parameters. The methodology includes a procedure for model simplification via the identification and rejection of the irrelevant state parameters. The technique further has the ability to generate different error measures associated with the library creation process. In this paper, the method is applied to the OECD PBMR-400MW benchmark problem and compared to the table-based cross-section library supplied with the benchmark. Results show that the method successfully describes the relevant model dependencies and accurately predicts the error associated with the generated quasi-regression library. The methodology can be considered in general as a powerful analysis and diagnosis tool for multidimensional cross-section parameterization.

References

An, J., Owen A.B., 2001. “Quasi-Regression”. Journal of complexity 17(4), 588-607.

Bokov P.M., Prinsloo R., 2007. “Cross-Section Parameterization Using Quasi-Regression Approach”. Joint International Topical Meeting on Mathematics & Computation and Supercomputing in Nuclear Applications (M&C + SNA 2007) Monterey, California, April 15-19, 2007.

Lemieux, Ch., Owen, A.B., 2002. “Quasi-Regression and the Relative Importance of the ANOVA Components of a Function” in: K.T. Fang, F.J. Hickernell, H. Niederreiter (Eds.), Monte Carlo and Quasi-Monte Carlo Methods, Springer-Verlag, Berlin, 331-344.

Kucherenko, S., Shah N., 2007. “The Importance of being Global. Application of Global Sensitivity Analysis in Monte Carlo Option Pricing“. Wilmott Magazine, 4, 2-10.

Reitsma, F. et al., 2006. The OECD/NEA/NSC PBMR Coupled Neutronics/Thermal Hydrau-lics Transient Benchmark: The PBMR-400 Core Design, PHYSOR2006.

Sobol’, I.M., 1993, “Sensitivity Estimates for Non-Linear Mathematical Models”. Mathematical Modelling and Computational Experiment. 1, 407-414.