ABSTRACT
As has been discussed in Chapters IV and V, the scope of local sensitivity analysis is to calculate exactly and efficiently the sensitivities of the system's response to variations in the system's parameters, around their nominal values. As has also been shown in those chapters, the sensitivities are given by the first Gâteaux-differential of the system's response, calculated at the nominal value of the system's dependent variables (i.e., state functions) and parameters. Two procedures were developed for calculating the sensitivities, namely the Forward Sensitivity Analysis Procedure (FSAP) and the Adjoint Sensitivity Analysis Procedure (ASAP). Once they became available, the sensitivities could be used for various purposes, such as for ranking the respective parameters in order of their relative importance to the response, for assessing changes in the response due to parameter variations, or for performing uncertainty analysis by using the propagation of errors (moments) procedure presented in Section III.F. In particular, the changes in the response due to parameter variations can be calculated by using the multivariate Taylor series expansion given in Eq. (III.F.3), which is reproduced, for convenience, below: https://www.w3.org/1998/Math/MathML"> R ( α 1 , ..., α k ) ≡ R ( α 1 0 + δ α 1 , ..., α k 0 + δ α k ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429210099/fe595b98-8ca7-4733-9fbf-816042942552/content/math721.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> https://www.w3.org/1998/Math/MathML"> = R ( α 0 ) + Σ i 1 = 1 k ( ∂ R ∂ α i 1 ) α 0 δ α i 1 + 1 2 Σ i 1 , i 2 = 1 k ( ∂ 2 R ∂ α i 1 ∂ α i 2 ) α 0 δ α i 1 δ α i 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429210099/fe595b98-8ca7-4733-9fbf-816042942552/content/math722.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> https://www.w3.org/1998/Math/MathML"> + 1 3 ! Σ i 1 , i 2 , i 3 = 1 k ( ∂ 3 R ∂ α i 1 ∂ α i 2 ∂ α i 3 ) α 0 δ α i 1 δ α i 2 δ α i 3 + ⋯ ( I I I . F .3 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429210099/fe595b98-8ca7-4733-9fbf-816042942552/content/math723.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> https://www.w3.org/1998/Math/MathML"> + 1 n ! Σ i 1 , i 2 , ... , i n = 1 k ( ∂ n R ∂ α i 1 ∂ α i 2 ... ∂ α i n ) α 0 δ α i 1 ... δ α i n + ⋯ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429210099/fe595b98-8ca7-4733-9fbf-816042942552/content/math724.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
