Linear regression-based multifidelity surrogate for disturbance amplification in multiphase explosion

Abstract

When simulations are very expensive and many are required, as for optimization or uncertainty quantification, a way to reduce cost is using surrogates. With multiple simulations to predict the quantity of interest, some being very expensive and accurate (high-fidelity simulations) and others cheaper but less accurate (low-fidelity simulations), it may be worthwhile to use multifidelity surrogates (MFSs). Moreover, if we can afford just a few high-fidelity simulations or experiments, MFS becomes necessary. Co-Kriging, which is probably the most popular MFS, replaces both low-fidelity and high-fidelity simulations by a single MFS. A recently proposed linear regression–based MFS (LR-MFS) offers the option to correct the LF simulations instead of correcting the LF surrogate in the MFS. When the low-fidelity simulation is cheap enough for use in an application, such as optimization, this may be an attractive option. In this paper, we explore the performance of LR-MFS using exact and surrogate-replaced low-fidelity simulations. The problem studied is a cylindrical dispersal of 100-μ m-diameter solid particles after detonation and the quantity of interest is a measure of the amplification of the departure from axisymmetry. We find very substantial accuracy improvements for this problem using the LR-MFS with exact low-fidelity simulations. Inspired by these results, we also compare the performance of co-Kriging to the use of Kriging to correct exact low-fidelity simulations and find a similar accuracy improvement when simulations are directly used. For this problem, further improvements in accuracy are achievable by taking advantage of inherent parametric symmetries. These results may alert users of MFSs to the possible advantages of using exact low-fidelity simulations when this is affordable.

Appendices

Appendix A: Identification of the significant variables

1.1 1.1 Methodology

The objective of this appendix is to present the sensitivity analysis of the quantity of interest studied in the paper (the normalized effective Fourier effective perturbation described in Section 3) with respect to the variables used to parametrized the PVF perturbation and described in Section 2 by (1). We recall that this perturbation is modeled by:

$$ \begin{array}{@{}rcl@{}} \phi^{p} (\theta)&=& {\phi_{0}^{p}} [1 + A_{1} \cos (k_{1} \theta + {\varPhi}_{1} ) + A_{2} \cos (k_{2} \theta + {\varPhi}_{2})\\ &&+ A_{3} \cos (k_{3} \theta + {\varPhi}_{3})], \end{array} $$

with the constraint:

$$ \sqrt{{A_{1}^{2}}+{A_{2}^{2}}+{A_{3}^{2}}}= 0.02. $$

Moreover, as the variables Φ₁, Φ₂, and Φ₃ are used to model the phase shift, one can arbitrary set Φ₁ = 0 and consider the phase shift with respect to the first mode. As a consequence, the problem counts seven independent variables concatenated into the vector $X = \left \lbrace A_{1},\! A_{2},\!k_{1},k_{2},\!k_{3},{\varPhi }_{2},{\varPhi }_{3}\right \rbrace$. The quantity of interest can thus be expressed as:

$$ F = \mathscr{M}(X) $$

in which the mapping ${\mathscr{M}}$ involves the numerical resolution of the multiphase explosion and the post-processing of the solution as explained in Sections 2 and 3.

The methodology used to study the sensitivity of the quantity of interest F to the input parameter X is to perform a global sensitivity analysis by computing variance-based sensitivity indices. Consequently, it is assumed that X is a vector of seven independent random variables (the probability distribution will be discussed in the following) and F is a random variable of unknown probability distribution. Then, the approach proposed in Sudret (2008) and improved in Blatman and Sudret (2010) for the computation of sensitivity indices (Sobol’ indices 2001) by sparse polynomial chaos expansion (PCE) (Blatman and Sudret 2011) is applied. Assuming that F is a second-order random variable, it can be shown (Cameron and Martin 1947) that:

$$ F = \sum\limits_{i=0}^{\infty} C_{i} \phi_{i}(X) $$

where $\left \lbrace \phi _{i} \right \rbrace _{i\in \mathbb {N}}$ is a polynomial basis orthogonal with respect to the probability distribution of X and C_i are unknown coefficients.

Sparse PCE consists in the construction of a sparse polynomial basis $\left \lbrace \phi _{i} \right \rbrace _{\alpha \in {\mathscr{A}}}$, where α = (α₁,⋯ , α_n) is a multi-index used to identify the polynomial acting with the power α_i on the variable X_i and ${\mathscr{A}}$ is a set of indices α. In practice, ${\mathscr{A}}$ is a subset of the set ${\mathscr{B}}$ which contains all the indices α up to a dimension d, i.e., $card({\mathscr{B}})=\frac {(d+n)!}{d!n!}$. The objective of sparse approach is to find an accurate polynomial basis $\left \lbrace \phi _{i} \right \rbrace _{\alpha \in {\mathscr{A}}}$ such that $card({\mathscr{A}})<<card({\mathscr{B}})$. In the present case, this is achieved by Least Angle Regression, i.e., unknown coefficients C_i are computed by iteratively solving a mean square problem and selecting, at each iteration, the polynomial which is the most correlated with the residual (see Blatman and Sudret (2011) for details).

Finally, one gets the following approximation:

$$ F\approx\hat{F} = \sum\limits_{\alpha \in \mathscr{A}} C_{\alpha} \phi_{\alpha}(X) $$

from which the sensitivity index can be derived. Indeed, the orthogonality of the polynomial basis $\left \lbrace \phi _{i} \right \rbrace _{\alpha \in {\mathscr{A}}}$ allows to write the expectation and the variance in the following form:

$$ \begin{array}{@{}rcl@{}} E[\hat{F}] &=& C_{0} \\ Var[\hat{F}] &=& \sum\nolimits_{\alpha \in \mathscr{A}} C^{2}_{\alpha} E[\phi_{\alpha}^{2}(X)] \end{array} $$

In addition, the idea pointed out in Sudret (2008) is to identify the PCE with the ANOVA decomposition, from which one can show that, the first-order sensitivity index of the variable X_i reads:

$$ \hat{S}_{i} = \frac{{\sum}_{\alpha \in L_{i}}C^{2}_{\alpha}E[\phi_{\alpha}^{2}(X)]}{Var[\hat{Y}]} $$

where $L_{i}=\left \lbrace \alpha \in {\mathscr{A}},~ \forall ~j\neq i~\alpha _{j}=0 \right \rbrace$, i.e., only the polynomials acting exclusively on the variable X_i are considered.

The total sensitivity index is also available by:

$$ \hat{S}_{T_{i}} = \frac{{\sum}_{\alpha \in L^{+}_{i}}C^{2}_{\alpha}E[\phi_{\alpha}^{2}(X)]}{Var[\hat{Y}]} $$

where $L^{+}_{i}=\left \lbrace \alpha \in {\mathscr{A}},~ \alpha _{i}\neq 0 \right \rbrace$, i.e., all the polynomials acting on the variable X_i are considered (allows considering interactions between X_i and the other variables).

One can note that the approximation of the sensitivity index obtained by sparse PCE relies on an accurate approximation of the surrogate response by the sparse PCE; however, the link between the accuracy of the PCE approximation and the accuracy of the approximated sensitivity index is not straightforward. In order to access the quality of the sensitivity index computed by PCE, a bootstrap approach proposed in Dubreuil et al. (2014) is set up and detailed in the next section.

1.1.1 1.2 Application to the high-fidelity computation of normalized Fourier effective perturbation

The probabilistic surrogate of the seven independent input parameters is detailed in Table 4. Uniform distributions are assumed for each component of the random vector X and variability ranges are defined, which basically define the domain of interest for the sensitivity analysis.

Table 4 Probabilistic surrogate of seven independent input parameters

Then, a design of experiments of 711 points is drawn by LHS in order to estimate the sensitivity index by sparse PCE. The maximum order of the polynomials is set to d = 4. In order to assess the accuracy of the obtained sensitivity index, the following bootstrap procedure is proposed. Among the 711 points, 611 are used as a training set to compute the PCE approximation (least angle regression approach) and 100 are used as a validation set. The training and validation sets are randomly chosen among the 711 points. The bootstrap approach consists in repeating B times this procedure changing each time the training and the validation set. This leads to B different PCE approximations and thus to a sample of B sensitivity indices. This sample is further used to estimate the coefficient of variation of the sensitivity index estimators obtained by sparse PCE. Moreover, for each bootstrap sample, the relative L₂ norm of the relative error (𝜖²) is computed on the validation set as well as the coefficient of determination (R²) computed on the training set.

$$ \epsilon^{2} = \frac{||\hat{F}_{\text{validation}}-F_{\text{validation}} ||^{2}}{||F_{\text{validation}}||^{2}} $$

$$ R^{2} = \frac{||\hat{F}_{\text{train}}-E[F_{\text{train}}]||^{2}}{||F_{\text{train}}-E[F_{\text{train}}]||^{2}} $$

Over the B bootstrap repetitions, the estimated mean values are E[𝜖²] ≈ 1.54 × 10^− 3 and E[R²] ≈ 9.74 × 10^− 1 and the coefficients of variation are $cv_{\epsilon ^{2}}\approx 1.57\times 10^{-1}$ and $cv_{R^{2}}\approx 1.86\times 10^{-3}$. These first results allowed confidence in the accuracy of the PCE approximation. Note that the relatively large coefficient of variation $cv_{\epsilon ^{2}}$ should be considered with respect to its very low mean value. Concerning the sensitivity index, Table 5 presents the first order and total index with the mean values and coefficient of variation estimated by bootstrap.

Table 5 Estimation of the first-order sensitivity index and total sensitivity index estimated by sparse PCE, mean values, and coefficient of variation estimated by bootstrap with B = 500 repetitions

First of all, the results presented in Table 5 show that when a sensitivity index has a significant value (values highlighted in bold) its estimation is quite accurate as the coefficient of variation is relatively low (less than 3%). One can also note that for low values of sensitivity indices the coefficients of variation are quite large; however, as these sensitivity indices are, at least, two orders of magnitude lower than the significant ones, their poor estimation is not detrimental for the purpose of sensitivity analysis. With respect to the results of the sensitivity analysis, one can conclude that the variance of the quantity of interest is mainly driven by the first five variables namely A₁, A₂, k₁, k₂, and k₃. It is also interesting to note the strong interaction between the amplitude variables (that have almost no first-order effect) with the wavenumbers which is consistent with the shape of the perturbation (in (1) the interactions between A₁, k₁ and A₂, k₂ clearly appear) and the constraint on the amplitude (2), which explains the interaction between A₁, A₂, and k₃, and justifies that $\hat {S}_{T_{k_{3}}}$ has the highest value.

Based on this result, it has been decided to consider only the five variables A₁, A₂, k₁, k₂, and k₃ for the surrogate construction of the quantity of interest F.

Appendix B: Percentage contribution of low-fidelity and high-fidelity data in LR-MFS using up to second-order polynomial basis functions

The contribution of the LF data points to the MFSs is studied in order to understand why the performance of the surrogates that use y_LF instead of the LF surrogate, $\hat {y}_{LF}$, for prediction, worked overwhelmingly better for the metric F. First, notice that in our case n = 21 in (8), which is the number of coefficients of a quadratic polynomial in five variables. Also, note that for prediction, we can choose between using y_LF or $\hat {y}_{LF}$; however, for training purposes, y_LF is used. Therefore, ρ and $\hat {\delta }(x)$ coefficients in (7) and in (14) are identical. Equation (8) shows explicitly the contribution of each surrogate, HF and LF, to the comprehensive MF approximation using the $\hat {y}_{LF}$; however, we can choose to use y_LF, i.e., the LF simulations directly, instead. The first term represents the contribution of the LF simulation, $\rho \hat {y}_{LF}(x)$ (using LF surrogate) or ρy_LF(x) (using LF simulations), to the MFS. The second term, ${\sum }_{1=1}^{p} X_{i}(x) b_{i}$, represents the contribution of the HF simulation to the MFS.

Figure 8 shows the mean contribution in percentage of the LF and HF information to the comprehensive MFS predicted using y_LF, i.e., $\hat {y}_{comp}$. Using y_LF or $\hat {y}_{LF}$ for predicting the MF comprehensive correction gives the same LF and HF mean contributions; therefore, only one plot was included. The mean contribution was calculated averaging the contribution of each of the 92 validation points. The contribution of the LF model is defined as $\rho \hat {y}_{LF}$, while the contribution of the HF model is defined as ${\sum }_{1=1}^{p} X_{i}(x) b_{i}$. A negative percentage indicates, in this case, that the contribution has a negative sign. Values higher than 100% indicate that the mean of the contribution is higher than the metric value on average, i.e., calculating the term $\rho \hat {y}_{LF}$ for each of the validation points and then taking the average. Therefore, ρ does not need to be higher than 1; instead, the term $\rho \hat {y}_{LF}$ needs to be higher than 1 on average. Naturally, the sum of the LF and HF contributions adds to 100%. For more than 50 HF data points, the contribution is dominated by the LF information (≈ 85%). This helps explain why when the LF simulation output is used directly (without constructing a surrogate), the improvement is substantial in Fig. 4. That is, for metric F, the results of the MF comprehensive prediction using y_LF, $\hat {y}_{comp}$, perform substantially better than the ones that use the LF surrogate, $\hat {y}_{\hat {comp}}$. When the source of LF is changed for prediction instead of a surrogate, the results change drastically, which is expected due to the high correlation between HF and LF simulations (0.92).

Fig. 8

Mean contribution of the LF ($\rho \hat {y}_{LF}(x)$) and of the HF (${\sum }_{1=1}^{p} X_{i}(x) b_{i}$ surrogates from (8)) to the comprehensive MFS ($\hat {y}_{comp}$) prediction (using LF simulations) in percentage as a function of the HF data points used. The plots are presented in a linear-log scale. A negative percentage indicates that the contribution has a negative sign. Values higher than 100% indicate that the mean of the contribution is higher than the metric value in average