# 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
As the stochastic process is known according to \(M\) realizations \(\mathcal{X}_{M}=(\mathbf{X}_{1}=\mathbf{X}(t,\mathbf{u}_{1}),\ldots,\)\(\mathbf{X}_{M}=\mathbf{X}(t,\mathbf{u}_{M}))\), the covariance function associated to the optimal process is not explicitly known. Based on classical statistics, for a centered stochastic process, the covariance is estimated by \(C=\frac{1}{M}\hat{X}\hat{X}^{T}\) with \(\hat{X}=(\mathbf{X}_{1}|\ldots|\mathbf{X}_{M})\) the snapshot matrix constituted of the \(M\) realizations of the stochastic process. The eigenvalues and eigenfunctions involved in KL expansion may be determined by solving the second kind Fredholm equation. A Galerkin-type technique [19; 23; 24] may be used for such purpose. Considering a basis \(h_{i}(\cdot)\) of the Hilbert space \(L^{2}(\mathcal{T})\), each eigenfunction \(\mathcal{L}_{k}(\cdot)\) may be expressed in this basis:  
\[\mathcal{L}_{k}(t)=\sum_{i=1}^{N_{KL}}d_{i}^{k}h_{i}(t) \tag{6}\]  
with some \(d_{i}^{k}\) the unknown coefficients. According to the truncation level, the aim of the [PERSON] method is to estimate the suited approximation of \(\mathcal{L}_{k}(t)\). The residual resulting from the truncation is determined using an error function \(\epsilon_{N_{KL}}(\cdot)\) along with Equation (3):  
\[\epsilon_{N_{KL}}(t)=\sum_{i=1}^{N_{KL}}d_{i}^{k}\left[\int_{\mathcal{T}}C(t, t^{\prime})h_{i}(t^{\prime})\mathrm{d}t^{\prime}-\lambda_{k}h_{i}(t)\right] \tag{7}\]  
To define the approximating basis, it is imposed that the residual is orthogonal to that approximating basis, leading to:  
\[\int_{\mathcal{T}}\epsilon_{N_{KL}}(t)h_{j}(t)\mathrm{d}t=0\quad\forall j=1, \ldots,N_{KL} \tag{8}\]  
This residual expression may be combined with Equation (8) to give:  
\[\sum_{i=1}^{N_{KL}}d_{i}^{k}\left[\int_{\mathcal{T}}\left[\int_{\mathcal{T}}C (t,t^{\prime})h_{i}(t^{\prime})\mathrm{d}t^{\prime}\right]h_{j}(t)\mathrm{d} t-\lambda_{k}\int_{\mathcal{T}}h_{i}(t)h_{j}(t)\mathrm{d}t\right]=0 \tag{9}\]  
Such an equation has a matrix form corresponding to a generalized eigenvalue problem [19; 23]:  
\[\mathbf{C}\mathbf{D}=\mathbf{\Lambda}\mathbf{E}\mathbf{D} \tag{10}\]  
with:  
\[\mathbf{C}_{ij} = \int_{\mathcal{T}}\int_{\mathcal{T}}C(t,t^{\prime})h_{i}(t)h_{j}( t^{\prime})\mathrm{d}t\mathrm{d}t^{\prime} \tag{11}\] \[\mathbf{D}_{ij} = d_{i}^{k}\] (12) \[\mathbf{X}(t,\mathbf{u})=\mu(t)+\sum_{k=1}^{\infty}\sqrt{\lambda_{k}}\xi_{k}( \mathbf{u})\mathcal{L}_{k}(t) \tag{4}\]0 (13) \[\mathbf{X}(t,\mathbf{u})=\mu(t)+\sum_{k=1}^{\infty}\sqrt{\lambda_{k}}\xi_{k}( \mathbf{u})\mathcal{L}_{k}(t) \tag{4}\]1  
with \(\delta_{ij}\) the Kronecker symbol. This matrix form equation is solved to get the eigenvectors \(\mathbf{D}\) and the eigenvalues \(\mathbf{\Lambda}\). A quadrature technique is employed to estimate the integrals that define the terms \(\mathbf{C}\) and \(\mathbf{D}\). The vertices of the mesh \(\mathcal{T}_{n_{v}}\) are used as the quadrature nodes. In addition, to ease the solving of Equation (10) a Singular Value Decomposition approach [24] is implemented. The resulting KL decomposition of the output stochastic process allows to separate the random dependency expressed through \(\xi_{k}(\cdot)\) and the time dependency expressed through \(\mathcal{L}_{k}(\cdot)\). This decomposition ease the construction of a surrogate model of \(\mathcal{Q}_{X}(\cdot)\), by replacing \(\xi_{k}(\cdot)\) by Gaussian processes which are particularly interesting for uncertainty quantification analyses.