Systematic methodology for the reduction of uncertainties in transient thermal-hydraulics by using in-bundle measurement data Page: 4 of 13
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The covariance matrices C 'VI describe correlations between the uncertain-
ties in the boundary parameters at distinct time steps, while the Cra's
[cf. Eq. (6)] account for correlations between experimental responses gnd
boundary parameters. Additional features (e.g., methods uncertainties )
can readily be incorporated into the optimization process by replacing the
symbols [... in Eq. (6) with appropriate vectors and corresponding
covariance matrices.
The system of quations (6-7) can be treated as a constrained minimi-
zation problem and solved by the method of Lagrange mutipliers. For
example, if all 6ff-diagonal covariance submatrices are zero, the solution
is
V= -CV, Su v v = CV av (10)
where
v = rCv + SVV C . - h" (11)
rr a1)
with
h = dV - Sv- C0 u . ""X* (12)
u (v
Note the fundamental role12,13 of the Lagrange multipliers av. It appears
not only in the expressions for the adjustments of the boundary parameters
and responses [cf. Eq. (10)], but it can also be shown that the covariance
matrices associated with xv and yv are iven by expressions involving the
covariance matrix associated with aV (CX) in particular:
Cxx = Caa 0 s" Ca- S" Caa (13)
CV CV -CV CV (14)
The covariance matrix CV _ <Aa VdV> is obtained from
Cv = SCv v C uV C VC + sC CV + S~v (15)
C rr as hh [Crr as (1C)
In the above expression Chh = <AhV hV> denotes the covariance matrix asso-
ciated with the inhomogeneous source term of the Lagrange equations, i.e.
Chh = <Addv> - SVu Caa Suu <A!Adv
u (v
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Barhen, J.; Cacuci, D. G.; Wagschal, J. J. & Mullins, C. B. Systematic methodology for the reduction of uncertainties in transient thermal-hydraulics by using in-bundle measurement data, article, January 1, 1980; Tennessee. (https://digital.library.unt.edu/ark:/67531/metadc1060234/m1/4/?rotate=90: accessed July 16, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.