it is typical in gas turbine engine testing to replicate measurements at these locations in the tangential direction (into the page in Figure 4.2), as a safeguard against thermocouple failure. Often, only two replicatesNd = 2are available at a given axial-radial position due to the cost of instrumentation and measurement channel limitations.
Since synthetic data is used in this analysis,Nd= 2will be compared withNd= 10.
U1∈Rn×npc form a new orthonormal basis for the reducednpc-dimension subspace over theRnoutput space, which become the PCs,
Truncation error is the difference between the original dataset and the truncated dataset, which is reconstructed to recover the approximate original outputs. Reconstruction is achieved by back-transforming the truncated set ˆ
ypcto the original space,yˆ0 = ˆypcU1T, whereU1 ∈ RN×npc is the reduced set of eigenvectors. Then, the error is computed as(y0−yˆ0)⊙Sy, whereSy = [σy1′ ,σy2′ , . . . ,σ′yn]∈RN×nis a matrix withNrows, each containing a copy of the standard deviations of the original outputs, and⊙is the Hadamard product (element-wise multiplication).
Finally, since calibration will be performed in the PC-space, the test data is also transformed to the PC-space of the model. The dataydare first standardized with the model output column means and standard deviations to obtain standardizedyd0, then transformed using the eigenvectors from the model transformation similar to Eq.
4.4)
ydpc=yd0U (4.5)
which may again be truncated by usingU1in place ofU.
Figure 4.4: Percentage contribution of PCs to the total variance (red), and maximum output reconstruction error of temperatures (T, K) and time constants (τ, s).
4.3.2 Input transformation with active subspace (AS)
The transformed model outputs result in a mappingF : Rp → Rof the parameter setθ ∈ Rp to a given PC (ykpc), to which the AS method [65] is applied (F is used here for a generic function to simplify notation in this section). If this mappingFis differentiable and square integrable, a symmetric positive semi-definite matrix may
be defined with the eigenvalue decomposition,
C= Z
θ
∇F(θ)∇F(θ)Tρ(θ)dθ=WΞWT (4.6)
whereW = [w1, . . . , wp]∈Rp×pis an orthogonal matrix of the eigenvectors,Ξ =diag[ξ1, . . . , ξp]∈Rp×pare magnitude-ordered eigenvalues,ρ(θ)is the sampling density. The parametersθare scaled to a[−1,1]phypercube.
Since LHS has been used,ρ(θ)is interpreted as a uniform distribution of the random variablesθ, as in prior work [76–79]. Another interpretation of Eq. 4.6, based on the definition of expectation E[x] = R
xρ(x)dx[80], is that,
C=E
∇F ∇FT
(4.7) which is to say,Cis the expected value (average) of the gradient outer product. The eigenspace ofC defines important directions in the domain ofF. Ordering the eigenpairs[ξi,wi]of this result in decreasing magnitude of ξiindicates thatw1is the most important direction, followed byw2, etc. Identifying these important directions has the potential for dramatic computational implications when considering quadrature rules for integration [81], optimization to minimize or maximizeF [76–79, 82], or approximation ofF [76, 78, 79, 82, 83]. By means analogous to PCA above, a heuristic based on eigenvalue magnitudes is used to truncate topa < psuch that a pa-dimensional partition ofW, the active subspaceWa = [w1, . . . ,wa], captures the majority of the change in the function. The remaining columns ofW are the inactive subspace. The function approximationFa:Rpa →R is called a ridge approximation over the active subspace [84],
F(θ)≈ Fa(WaTθ) (4.8)
Plots of the model parameters and PC outputs based on Eq. 4.8 are known asshadow plots[85]. In the case that strong trends over the first one or two important directions are observed [75–78, 82, 83], the ability to visualize how the function changes in these important directions allows better selection of the most appropriate type of approximation, i.e., providing strong empirical evidence that a function is predominately linear, quadratic, or more complicated. Moreover, this change of variables is simplypa linear combinations of theporiginal parameters.
Therefore, the entries of the eigenvectors may be considered as weights indicating the importance of a particular parameter ordered by the corresponding eigenvalue. Thus, the magnitude of the entries of the first eigenvector offers a sensitivity analysis. More information on the sensitivity analysis interpretations can be found in [86].
The definition of Eq. 4.6 depends on gradients, which are often not available in standard FE tools. Therefore, gradient approximations will be based on the space-filling model samples. In this problem, a global linear gradient approximation is assumed for a single eigenvector (see Algorithm 1.3 in [65]). This eigenvectorw1is calculated
from the linear gradientsbasw1=b/||b||, resulting in active parameterswTkθ, where the subscript of this single eigenvector is repurposed from here on to represent PCk.
4.3.3 The resulting PC-AS surrogate model
By this combination of PCA and AS, the surrogate modeling problem has been significantly simplified: the 5-input and 48-output problem is compressed into five 1-input and 1-output quadratic surrogate models. Further- more, AS dimension reduction makes visualization of the surrogate models feasible as shown by the five shadow plots in Figure 4.5. The shadow plots are the PC-space outputsykpcplotted as a function of the active subspace parameterswkTθ(blue dots), which are fit with quadratic polynomial regression surrogate models (red curves).
The measurements are also transformed into the model’s PC-space using Eq. 4.5 (green horizontal lines). The goal of calibration is to find the best values ofwkTθ which result in a surrogate model output equivalent to the data, i.e. the intersection of the green and red lines for allkPC-AS shown in Figure 4.5. Re-fitting the PC-AS surrogate model for different numbers ofN (retaining 50 test points) andnpcresulted inR2>0.95forN >20 andnpc≥5.
Figure 4.5:Active subspace shadow plots for the first 5 PCs (blue dots), surrogate model fits (red line), and measure- ments in PC-space (green lines).
4.3.4 Sensitivity analysis of PCs vs eigenvector
Another benefit discussed in Section 4.3.2 is that the eigenvectors of the active parameters provide first-order parameter sensitivity analysis for each PC. These eigenvectors are plotted in Figure 4.6 and are compared to first order Sobol’ indices. The first order Sobol’ index forθi and PCkwas generated using a modularized sample- based method [87],
Ski =Var
E[ypck |θi]
Var[ykpc] (4.9)
where the expectation is taken over samples within bins of eachθi to determine the variance ofypck due toθi. Both approaches use the available 200 LHS samples from the FE model. The magnitude of these two first order sensitivity measures are in good agreement, but the AS approach requires no additional calculation.
Note that in both methods, the sensitivity results are computed for outputs in PC-space. Alternatively, sensi-
tivity results could be calculated for all physical outputs. Calculating sensitivities for all outputs results in many conflicting relationships when the objective is dimension reduction, i.e., parameters that are important to one out- put may not be important to another output. By computing sensitivity results for the PCs, the affect of a parameter is considered across all outputs. And, in this case there are only 5 PCs versus 48 physical outputs, potentially improving interpretability of the results. How to best use the sensitivity analysis of multivariate outputs merits further investigation (e.g., [88, 89], particularly as the dimensionality of the output space increases (i.e. more thermocouples). An approach toward combining these into a single Pareto chart of importance ranking is shown later in Figure 7.7. Furthermore, interaction effects may be of interest, which requires the calculation of higher order Sobol’ indices.
Figure 4.6:First-order Sobol’ indices (top) compared to active subspace eigenvectors (bottom) for each PC.