## 2 Theory and Methodology  
### Stochastic Parametric Models  
Without loss of generality, let us assume that the behavior of the physical system under consideration can be modeled by a set of parametric partial differential equations (PDEs). The general form of the corresponding mathematical model is given by:  
\[\mathcal{D}_{\mathbf{y}}(u)=0, \tag{1}\]  
where \(\mathcal{D}=\mathcal{D}_{\mathbf{y}}(u)\) is a differential operator, \(\mathbf{y}\in\Xi\subset\mathbb{R}^{N}\) a \(N\)-dimensional parameter vector, \(u=u_{\mathbf{y}}\) the parameter-dependent solution of Equation (1), and \(\Xi\) a measurable set called the image space. Typically, we are interested in a certain quantity, that is the QoI, which is computed by post-processing the solution \(u=u_{\mathbf{y}}\). For simplicity, we assume that the QoI is a real scalar, however, the methodology discussed in Section 2.4 can be applied to complex or vector-valued QoIs with only minor modifications. We further assume that the functional dependence between the parameter vector and the QoI is given by the map \(g:\mathbf{y}\mapsto g\left(\mathbf{y}\right)\), which is assumed to be deterministic, i.e., the exact same output \(g\left(\mathbf{y}\right)\) is observed each and every time the QoI is evaluated for the same parameter vector \(\mathbf{y}\). Parametric problems in the form of Equation (1) arise in both deterministic and stochastic settings. An example of the former case is, e.g., an optimization study, while an example of the latter is, e.g., a UQ study as considered in this work. In the stochastic setting, the parameter vector \(\mathbf{y}\) corresponds to a realization of a \(N\)-dimensional RV \(\mathbf{Y}=\left(Y_{1},Y_{2},\ldots,Y_{N}\right)\), alternatively called a random vector. Assuming that the RVs \(Y_{n}\), \(n=1,\ldots,N\), are mutually independent, the joint PDF is given as a product of univariate PDFs, such that \(\varrho(\mathbf{y})=\prod_{n=1}^{N}\varrho_{n}(y_{n})\). While the case of dependent RVs is not rigorously addressed here, we mention the option of using generalized Nataf or Rosenblatt transformations to transform the dependent RVs into independent RVs [39; 40]. The sparse interpolation method presented in Section 2.4 can then be applied to the transformed RVs. Due to its dependence on stochastic input parameters, the deterministic map \(g\) results now in a random output. In other words, the input uncertainty propagates through the deterministic model and renders the QoI uncertain. The main goal of UQ studies is to quantify this output uncertainty by computing statistics, such as moments, event probabilities, or sensitivity metrics. Denoting the expectation operator by \(\mathbb{E}\left[\cdot\right]\), any statistic can be written in the general form:  
\[\mathbb{E}\left[\phi\left(g\right)\right]=\int_{\Xi}\phi\left(g\left(\mathbf{ y}\right)\right)\varrho\left(\mathbf{y}\right)\mathrm{d}\mathbf{y}, \tag{2}\]  
where the functional \(\phi\) corresponds to the statistic of interest, e.g. \(\phi\left(g\right)=g\) for the expected value of the QoI and \(\phi\left(g\right)=\left(g-\mathbb{E}\left[g\right]\right)^{2}\) for its variance. The integral of Equation (2) cannot in general be computed analytically. Therefore, one relies on numerical integration methods, such as MC integration or quadrature. Considering the case of MC integration, if the original input-output map \(g\) corresponds to an expensive numerical model, then it is desirable to replace it by an inexpensive albeit accurate surrogate model \(\widetilde{g}\approx g\). In this work, this surrogate is obtained by interpolating \(g\). To capture accurately the behavior of \(g(\mathbf{y})\), given that \(\mathbf{y}\) are realizations of a random vector, the interpolation must be based on nodes tailored to the PDF \(\varrho(\mathbf{y})\). The surrogate model can also be used for tasks other than computing statistics, e.g., for optimization. Moreover, by integrating the interpolant, one can derive a quadrature scheme for the computation of Equation (2).