PCE Surrogate Model Formulation and Global Sensitivity Analysis

III. Forward Uncertainty Quantification of Spent Fuel

3.3. PCE Surrogate Model Formulation and Global Sensitivity Analysis

PCE expands a model response 𝑌(𝝃) as a function of orthogonal basis of uncertain input parameter vector 𝝃 = (𝜉₁, 𝜉₂, … , 𝜉_𝑛𝑢):

𝑌(𝝃) = ∑^∞_𝒏=0𝑎_𝒏𝜓_𝒏(𝝃) (3.17)

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where 𝝃 are the uncertain input parameters, 𝑛_𝑢 is the number of uncertain input parameters, n is the expansion order, 𝑎_𝒏 are the coefficients of expansion, and {𝜓_𝒏(𝝃) }₀^∞ is a complete set of multivariate orthogonal basis in the 𝐿²(𝝃) space, being 𝝃 ∈ ℝ^𝑛𝑢, with inner product:

⟨𝜓_𝒏, 𝜓_𝒎⟩ = ∫ 𝜓_{𝝃 𝒏}(𝝃)𝜓_𝒎(𝝃)𝑝(𝝃)𝑑𝝃 (3.18) where 𝑝(𝝃) is the joint probability density function (PDF) of all uncertain input parameters. To be shown later, the index of the multivariate basis function 𝜓_𝒎 , has multiple indices 𝒎 = (𝑚₁, 𝑚₂, … , 𝑚_𝑛𝑢) if expressed in the univariate basis 𝜙_𝑚𝑛. This captures the polynomial order and the random variable 𝜉_𝑛 considered in the univariate basis. The variables of the uncertain input parameter vector, the coefficients of expansion, and order of the multivariate basis are in bold italicized terms. Non-bold terms represent the uncertain input parameters and the univariate basis expansion order. In Eq. (3.17), the model response is projected into a polynomial basis that is orthogonal to a weighting function which is the PDF of the random independent variable. The uncertain input parameters in Eq. (3.17) have individual PDF 𝑝_𝑛(𝜉_𝑛) and joint PDF shown in Eq.

(3.19), and no correlation is considered to exist between the input parameters.

𝑝(𝝃) = ∏^𝑛_𝑛=1^𝑢 𝑝_𝑛(𝜉_𝑛) (3.19) The multivariate basis 𝜓_𝒎(𝝃) is built as the tensor product of orthogonal univariate functions 𝜙_𝑚𝑛(𝜉_𝑛):

𝜓_𝒎(𝝃) ≝ ∏^𝑛_𝑛=1^𝑢 𝜙_𝑚𝑛(𝜉_𝑛) (3.20) (𝜓_𝒎(𝝃))

𝒎=0

∞ ≅ (𝜙_𝑚1(𝜉₁)𝜙_𝑚2(𝜉₂) … 𝜙_𝑚𝑛𝑢(𝜉_𝑛𝑢))

𝑚=0 𝑞

(3.21)

The orthogonality condition for the multivariate basis leads to Eq. (3.22).

⟨𝜓_𝒏, 𝜓_𝒎⟩ = 𝛿_𝒏𝒎 (3.22) where 𝛿_𝒏𝒎 is the Kronecker delta symbol. The basis function selected for the uncertain parameters depends on the probability distribution. Random variables which are assumed to follow uniform

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and normal distribution commonly employ Legendre and Hermite polynomials as basis functions, respectively. Eq. (3.17) is an infinite expansion which is truncated at a given expansion order q, leading to the model response approximation as in Eq. (3.23):

𝑌(𝝃) ≅ ∑^𝑞_𝒏=0𝑎_𝒏𝜓_𝒏(𝝃) (3.23) The model response mean, and variance are determined from the coefficients of expansion 𝑎_𝒏 (when known). The coefficients of expansion are evaluated by linear regression. Eq. (3.17) can be cast as the sum of Eq. (3.23) and a truncation error 𝜀_𝑞+1, and then written as a linear system:

𝑌(𝝃) = ∑^∞_𝒏=0𝑎_𝒏𝜓_𝒏(𝝃)= ∑^𝑞_𝒏=0𝑎_𝒏𝜓_𝒏(𝝃)+ 𝜀_𝑞+1 (3.24)

𝑌(𝝃) ≡ 𝒂^𝑇𝜓(𝝃) + 𝜀_𝑞+1 (3.25) The least square minimization of the mean residual error is set up as:

𝒂

̂ = 𝑎𝑟𝑔 𝑚𝑖𝑛 𝔼[(𝒂^𝑇𝜓(𝝃) − 𝑌(𝝃))²] (3.26) where 𝜓(𝝃) = (𝜓₀(𝝃), … , 𝜓_𝑞(𝝃))^𝑇 is the matrix of Legendre polynomials of the uniform random variables and 𝒂 = (𝑎₀, … , 𝑎_𝑞)^𝑇 is the coefficient vector. From the uncertain parameters in Table 3.1 as input random vector 𝑬 = (𝝃⁽¹⁾, … , 𝝃^(𝑁))^𝑇 from sample of size N and 𝑹 = (𝑦⁽¹⁾, … , 𝑦^(𝑁))^𝑇 representing the model responses (i.e., STREAM simulation results) associated with the inputs, the least square solution of Eq. (3.26) is given as:

𝒂

̂ = (Ψ^𝑇∙ Ψ)⁻¹(Ψ^𝑇∙ 𝑹) (3.27) where:

Ψ_𝑖𝑗 = 𝜓_𝑗(𝝃^(𝑖)) 𝑖 = 1, … , 𝑁; 𝑗 = 0, … , 𝑞 (3.28) And Ψ is an N x (q+1) matrix of Legendre polynomial basis constructed from Eq. (3.20). The number of unknowns in the linear system is given by (𝑛_𝑢+ 𝑞)!/(𝑛_𝑢! 𝑞!). This is the minimum number of model evaluations needed to calculate the coefficients from the solution of the linear

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system. For cases in which the number of available model evaluations exceed the number of unknowns, then it yields an overdetermined system which can be solved by least square minimization. The implementation details of the PCE method and can be obtained from [116].

Employing Eq. (3.22) and basis functions with 𝜓₀ = 1, the model response mean, and variance can be obtained from the coefficients of the PCE as:

𝜇 = ⟨𝑌(𝝃)⟩ = 𝑎₀ , 𝜎² = ⟨(𝑌(𝝃) − 𝜇)²⟩ = ∑^𝑞_𝒏=1𝑎_𝒏² (3.29) SA apportions the response uncertainty of a calculation code among the input parameters.

Local SA determines the response uncertainty due to minor input variation around specific input parameters based on the partial derivative of the model. GSA considers the complete range of variation and statistical distribution of input parameters and evaluate the response uncertainty in a statistical framework. GSA in this thesis decomposes the model and its associated according to the Sobol method [117]. The contribution of each input parameter to the response variance is referred to as first order Sobol’ indices or the main effect. The total Sobol’ indices or total effect accounts for each input parameter contribution plus the high order effects i.e., the interactions between the other input parameters. Sobol’ indices can split the effect of individual input on the output response.

Therefore, we can account for the case in which the perturbed modeling parameters considered in the uncertainty analysis are independent. The first order and total Sobol’ indices are sufficient to identify the important input parameters. The Sobol’ indices can be obtained by post-processing the coefficients of the PCE as outlined in [118]:

If we define 𝐼_{𝑖,…,𝑠} as indices in {𝒏 ∈ ℕ^𝑛𝑢: 0 ≤ |𝒏| ≤ 𝑞} such that only the indices {𝑖, … , 𝑠}

are nonzero:

𝐼_{𝑖,…,𝑠} = {𝒏 ∈ ℕ^𝑛𝑢 ∶ 0 ≤ |𝒏| ≤ 𝑞 , ∀𝑘 ∈ {1, … , 𝑛_𝑢}\{𝑖, … , 𝑠} , 𝑛_𝑘 = 0} (3.30) 𝐼_{𝑖,…,𝑠} represents the basis functions 𝜓_𝒏 depending only on the input parameters (𝜉_𝑖, … , 𝜉_𝑠).

In Eq. (3.30), the terms in basis functions depending on input parameters other than (𝜉_𝑖, … , 𝜉_𝑠) have been excluded. With this notation we can rearrange the PCE terms in Eq. (3.23) according to their dependent variables:

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𝑌(𝝃) = 𝑎₀+ ∑^𝑛_𝑖=1^𝑢 ∑_{𝒏∈𝐼𝑖}𝑎_𝒏𝜓_𝒏(𝜉_𝑖)+ ∑^𝑛_{1≤𝑖<𝑗≤𝑛}^𝑢 ∑_{𝒏∈𝐼𝑖,𝑗}𝑎_𝒏𝜓_𝒏(𝜉_𝑖, 𝜉_𝑗)

𝑢 + ⋯ +

∑^𝑛1≤𝑖<⋯<𝑠≤𝑛^𝑢 _𝑢∑_{𝒏∈𝐼𝑖,…,𝑠}𝑎_𝒏𝜓_𝒏(𝜉_𝑖, … , 𝜉_𝑠)+ ⋯ + ∑_{𝒏∈𝐼1,…,𝑛𝑢}𝑎_𝒏𝜓_𝒏(𝝃) (3.31) In Eq. (3.31), each multivariate polynomial is shown to depend on a subset of the input parameters. Eq. (3.31) is the Sobol’ decomposition of the PCE. The orthogonality of the basis functions makes the constant term 𝑎₀ the mean of the model response and the terms in the summation satisfy the condition:

∫_𝐷 𝑌_{𝑖,…,𝑠}(𝜉_𝑖, … , 𝜉_𝑠)𝑝_𝑘(𝑥_𝑘)𝑑𝜉_𝑘

𝜉𝑘 = 0 1 ≤ 𝑖 < ⋯ < 𝑠 ≤ 𝑛_𝑢 𝑘 ∈ {𝑖, … , 𝑠} (3.32) Thus, the term ∑_{𝒏∈𝐼𝑖,…,𝑠}𝑎_𝒏𝜓_𝒏(𝜉_𝑖, … , 𝜉_𝑠) in the summation of Eq. (3.31) can be denoted as:

𝑌_{𝑖,…,𝑠}(𝜉_𝑖, … , 𝜉_𝑠) = ∑_{𝒏∈𝐼𝑖,…,𝑠}𝑎_𝒏𝜓_𝒏(𝜉_𝑖, … , 𝜉_𝑠) (3.33) Sobol decomposition breaks the output variance down into constituents which are supplied by individual inputs or a group of inputs. For example, the contribution of each input parameter to the response variance, i.e., first order Sobol’ indices or the main effect, can be written as:

𝑆_𝑖 =^{∑ 𝑎𝒏}

𝒏∈𝐼𝑖 2

∑^𝑞_𝒏=1𝑎_𝒏² (3.34) where the index of the summation refers to basis functions depending only on one input parameter i. The high order and total Sobol’ indices can be obtained in a similar fashion. Eq. (3.34) is the variance of the second term in Eq. (3.31) normalized to the total variance in Eq. (3.29). When the model response is decomposed into summands as in Eq. (3.31), each term of the summation can be viewed as a truncated PCE depending on different subsets of the input parameters [118]. Then the coefficients can be grouped by the input parameters that each polynomial basis depends on [119].

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