# Active Learning Strategy for Surrogate-Based Quantile Estimation of Field Function
## 2 Surrogate Model for Function with Mono-Dimensional Domain Random Field Output
### Model Order Reduction  
The surrogate model \(\hat{\mathcal{Q}}_{\mathbf{X}}(\cdot)\) is defined based on a limited size set of black-box model evaluations \(\mathcal{Q}_{\mathbf{X}}(\cdot)\) constituting a training set. For that matter, \(M\) samples are obtained by a Monte-Carlo Simulation (MCS). The realizations of the input vector \(\mathcal{U}_{M}=[\mathbf{u}_{1},\ldots,\mathbf{u}_{M}]\) are generated by sampling according to the joint PDF \(\phi_{\mathbf{U}}(\cdot)\). For each sample \(\mathbf{u}_{k\in[1,\ldots,M]}\) of \(\mathcal{U}_{M}\), the exact function \(\mathcal{Q}_{\mathbf{X}}(\cdot)\) is evaluated giving a realization of the output stochastic process \(\mathbf{X}(t,\mathbf{u}_{k})\). These \(M\) evaluations correspond to different realizations of the stochastic process defining the set \(\mathcal{X}_{M}=\{\mathbf{X}(t,\mathbf{u}_{1}),\ldots,\mathbf{X}(t,\mathbf{u}_ {M})\}\). From a numerical point of view, the stochastic process realizations are discretized over a mesh in time \(\mathcal{T}_{n_{v}}\) composed of \(n_{v}\) vertices \([\mathbf{X}(t_{1},\mathbf{u}_{k}),\ldots,\mathbf{X}(t_{n_{v}},\mathbf{u}_{k})]\). In general, the number of vertices \(n_{v}\) is large to get precision in the behavior of the field output. The resulting discretized stochastic process is composed of \(n_{v}\) time-correlated random variables. Because of the high dimension \(n_{v}\) of the stochastic process discretization, the surrogate model of the function \(\mathcal{Q}_{X}(\cdot)\) is based on a model order reduction technique. The Karhunen-Loeve (KL) decomposition [19; 20; 21; 23] allows to represent the stochastic process by a linear combination of orthogonal functions. The orthogonal functions are the eigenfunctions of the autocovariance function of the stochastic process. In order to determine the set of deterministic orthogonal functions an eigenvalue problem is solved. In that purpose, the second kind Fredholm equation associated to the autocovariance function \(\mathcal{C}_{\mathbf{X}}(\cdot,\cdot)\) (noted \(C(\cdot,\cdot)\) for simplicity) is solved to estimate the eigenvalues and eigenfunctions:  
\[\int_{\mathcal{T}}C(s,t)\mathcal{L}_{k}(t)\mathrm{d}t=\lambda_{k}\mathcal{L}_ {k}(s)\ \ \ \forall s\in\mathcal{T} \tag{3}\]  
where \((\mathcal{L}_{k})_{k\geq 1}\) is the sequence of eigenfunctions and \((\lambda_{k})_{k\geq 1}\) are the associated eigenvalues. A complete orthogonal basis of \(L^{2}(\mathcal{T})\) is defined by \((\mathcal{L}_{k})_{k\geq 1}\). Consequently, any realization of the stochastic process may be expanded over the estimated basis:  
\[\mathbf{X}(t,\mathbf{u})=\mu(t)+\sum_{k=1}^{\infty}\sqrt{\lambda_{k}}\xi_{k}( \mathbf{u})\mathcal{L}_{k}(t) \tag{4}\]  
with \((\xi_{k})(\mathbf{u})_{k\geq 1}\) the coordinates of the stochastic process sample with respect to the deterministic function \(\mathcal{L}_{k}(\cdot)\). A set of orthonormal random variables \((\xi_{k})_{k\geq 1}\) is defined by all the possible realization of the stochastic process. Each random variable \(\xi_{k}(\mathbf{u})\) involved in the KL decomposition is defined by a linear transform (using the orthonormality of the eigenfunctions):  
\[\xi_{k}(\mathbf{u})=\frac{1}{\sqrt{\lambda_{k}}}\int_{\mathcal{T}}(\mathbf{X} ^{*}(t,\mathbf{u})-\mu(t))\mathcal{L}_{k}(t)\mathrm{d}t \tag{5}\]  
For numerical purposes, the KL expansion is truncated to the first \(N_{KL}\) most significant modes. These latter correspond to the most significant eigenvalues: \(\mathbf{X}(t,\mathbf{u})\simeq\mu(t)+\sum_{k=1}^{N_{KL}}\sqrt{\lambda_{k}}\xi_ {k}(\mathbf{u})\mathcal{L}_{k}(t)\). The optimality of the eigenfunction basis \(\mathcal{L}_{k}(\cdot)\) is obtained in the sense that, compared to any alternative \(L^{2}(\mathcal{T})\) basis, the mean square error integrated over \(\Omega\) given the limitation of the KL expansion to the first \(N_{KL}\) terms is minimal.