A simple method for obtaining an upper bound to the uncertainty of any linear computed response

Ann. nucl. Energy, Vol. 12, No. 9, pp. 509-510, 1985 Printed in Great Britain. All rights reserved

0306-4549/85 $3.00+0.00 Copyright © 1985 Pergamon Press Ltd

TECHNICAL

NOTE

A SIMPLE METHOD FOR OBTAINING AN UPPER BOUND TO T H E UNCERTAINTY OF ANY LINEAR COMPUTED RESPONSE Y. RONEN Department of Nuclear Engineering, Ben-Gurion University of the Negev, Beer Sheva 84120, Israel

(Received 12 April 1985) Obtaining the uncertainty of a calculating response by a large computer code is a very complicated procedure. This problem is particularly difficult when there are many input parameters whose uncertainties are the source of the response's uncertainty. Probably the most effective way to calculate the uncertainty of a response due to the uncertainties of the input parameters is the uncertainty analysis based on sensitivity methodology. Consider that we have a response R(~I . . . . . aN) which is a function of the input parameters col,..., ccN.The variance of R is given by

~R/R ?,R/R

= ~

SiS~puaiaj,

(1)

i,j= 1

where a~ is the relative standard deviation of the input parameter ~ ; the relative standard deviation of the response is a Rand the relative sensitivity coefficient of the input parameter at is S~, the correlation matrix elements are P0, and - - 1 ~< PO ~< 1.

(2)

There are several difficulties in applying equation (1) in uncertainty analysis. First, equation (1) is limited to linear problems or to cases where first-order derivatives are sufficient, namely, linear approximations. Second, it is a very difficult task to obtain the sensitivities St (i = 1,..., N). A third problem is that the correlation matrix elements Pu, in many cases, are unknown. Although a priori the sensitivity coefficients are unknown, we usually know their sign. Namely, by knowledge of the physics behind the code we usually know a priori whether a change in an input parameter will increase or decrease the response. Knowing the signs of the sensitivity coefficients, we can apply a very simple method for obtaining an upper bound to the response. Let us change each of the input parameters by its standard deviation : whenever S~ is positive, we change the input parameter by + a~, if St is negative we change the input parameter by - - a i. Thus the relative change in the response AR/R is given by AR

- ~ Siai;

R

Due to our choice of the sign of a~ we have

SiS~poaia~ ~ SiSFia j

a~ <~ - -

.

(6)

Table 1. Standard deviations of the fission cross-sections of 2 3 5 U

(3)

the square of AR/R is given by ~ . S i S j 6 r i a j. i,j= 1

(5)

So, by a simple procedure in which we change each of the input parameters, either by the positive or negative value of its standard deviation, we obtain a change in the response. The absolute value of this change is an upper bound to the standard deviation of the response. The advantage of this method lies in the fact that only two computer runs are needed in order to obtain the upper bound. The first run is with the input parameters at their nominal value, the second run is with the changed input parameters. There is no need to calculate sensitivities and there is no need to know the values of the correlation matrix elements. The main disadvantage of this method lies in the fact that only an upper bound to the uncertainty of the response is obtained. Since it is an upper bound, we do not know how close it is to the real value. However, if the bound is small enough to be considered as the uncertainty, we have achieved our aim of obtaining an uncertainty to the response in a very simple way. In order to demonstrate this method, we have considered the effect of the uncertainty in fission cross-sections on the effective multiplication factor kef f. We have calculated the k~ff of a 235U sphere. The radius of the sphere was 10.0 cm. The calculations were done in 12 energy groups with the transport code ANISN.The energy-group structure is given in Table 1.

i=1

=

i,j = 1. . . . . N,

thus

(4) 509

Group

Energy range (eV)

Relative standard deviation (%)

1 2 3 4 5 6 7 8 9 10 11 12

1.00E + 07-3.68 E + 06 3.68E + 0~1.35 E + 06 1.35E +06-4.98E + 05 4.98E + 05-1.83E + 05 1.83E +05 1.lIE+05 1.11E +05-6.74E+04 6.74E + 04-2.48E + 04 2.48E +04-9.12E +03 9.12E + 03-3.36E + 03 3.36E + 03-1.23E + 03 1.23E + 034.54E + 02 4.54E + 02-1.00E - 05

3.2 2.4 2.7 2.8 3.3 2.7 2.9 3.8 5.0 5.7 3.2 0.3

510

Technical N o t e Table 2. Elements of the correlation matrix of the fission cross-sections of z35U Energy group :

1

2

3

4

5

6

7

8

9

10

l1

12

1 2 3 4 5 6 7 8 9 10 11 12

1.000 0.522 0.165 0.037 0.001 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.522 1.000 0.485 0.228 0.092 0.021 0.000 0.000 0.000 0.000 0.000 0.000

0.165 0.485 1.000 0.725 0.409 0.010 0.000 0.000 0.000 0.000 0.000 0.000

0.037 0.228 0.725 1.000 0.497 0.435 0.389 0.273 0.000 0.000 0.000 0.000

0.001 0.092 0.409 0.497 1.000 0.653 0.415 0.291 0.000 0.000 0.000 0.000

0.000 0.021 0.010 0.435 0.653 1.000 0.941 0.689 0.686 0.598 0.507 0.119

0.000 0.000 0.000 0.389 0.415 0.941 1.000 0.838 0.887 0.784 0.707 0.159

0.000 0.000 0.000 0.273 0.291 0.689 0.838 1.000 0.940 0.868 0.871 0.204

0.000 0.000 0.000 0.000 0.000 0.686 0.887 0.940 1.000 0.955 0.746 0.155

0.000 0.000 0.000 0.000 0.000 0.598 0.784 0.868 0.955 1.000 0.701 0.135

0.000 0.000 0.000 0.000 0.000 0.507 0.707 0.871 0.746 0.701 1.000 0.250

0.000 0.000 0.000 0.000 0.000 0.110 0.159 0.204 0.155 0.135 0.240 1.000

Table 3. The relative sensitivities of kcff with respect to the fission cross-section of 235U Group

Sensitivity

1 2 3 4 5 6 7 8 9 10

0.05997 0.17641 0.17858 0.10764 0.02450 0.01229 0.00872 0.00158 0.00025 0.00008 0.00000

11 12

0.00000

A priori, we k n o w t h a t all the sensitivities of the groups' fission cross-sections are positive. Any increase in the fission cross-section increases the keff. I n c r e a s i n g all the fission crosssections by the s t a n d a r d d e v i a t i o n s in T a b l e 1, we o b t a i n e d the value of 0.0158 for the relative c h a n g e of keff. W e also c a l c u l a t e d the sensitivities. W e m a d e 13 c o m p u t e r runs to o b t a i n these sensitivities. O n e r u n with the n o m i n a l values of the cross-sections and 12 runs where, in each run, we c h a n g e d the fission cross-section in one e n e r g y group. The sensitivities o b t a i n e d are given in T a b l e 3. U s i n g e q u a t i o n (1) with the sensitivities of T a b l e 3 a n d the s t a n d a r d d e v i a t i o n s a n d the c o r r e l a t i o n elements of T a b l e s 1 a n d 2, we c a l c u l a t e d tr R which is the s t a n d a r d d e v i a t i o n of keff. The value o b t a i n e d was 0.0111, which is quite close to the u p p e r b o u n d of 0.0158 o b t a i n e d by our method. Acknowledgement--Thanks are due to Y. F a h i m a for his help.

The 235U cross-sections for AN1SN were o b t a i n e d from the WIMS-D code a n d library. The s t a n d a r d d e v i a t i o n s of the fission cross-sections a n d the c o r r e l a t i o n m a t r i x were e v a l u a t e d by S m i t h a n d B r o a d h e a d (1981) a n d are given in T a b l e s 1 a n d 2.

REFERENCES Smith J. D. I I I and ORNL/TM-7389.

Broadhead

B. L. (1981) R e p o r t