# Low-Rank Approximation of Frequency Response Analysis of Perforated Cylinders under Uncertainty
## 2 Preliminaries
### Frequency Response Analysis under Uncertainty
#### 2.2.3 Stochastic Collocation  
There are many options for the selection of the collocation operator. Although a simpler formulation would also suffice for the numerical experiments below, it is advantageous to aim for efficient computation even in high-dimensional parameter spaces. With this in mind, an anisotropic Smolyak-type collocation operator defined with respect to a finite multi-index set (see, e.g., [29; 30; 31; 32]) is taken as the model.  
For the sake of accuracy, it is standard practice to choose the collocation points to be the abscissae of orthogonal polynomials; see [33]. The collocation method is formulated using Legendre polynomials, which are the optimal choice when the input random variables are uniform. In the case of, for example, Gaussian random variables, one should use Hermite polynomials instead, but otherwise the collocation method remains the same.  
Let \(L_{p}\) be the univariate Legendre polynomial of degree \(p\). Denote by \(\{\chi_{k}^{(p)}\}_{k=0}^{p}\) the zeros of \(L_{p+1}\) and by \(\{w_{k}^{(p)}\}_{k=0}^{p}\) the associated Gauss-Legendre quadrature weights. The one-dimensional Lagrange interpolation operators \(\mathcal{I}_{p}^{(m)}\) are defined via  
\[\Big{(}\mathcal{I}_{p}^{(m)}v\Big{)}(\xi_{m})=\sum_{k=0}^{p}v\Big{(}\chi_{k}^{ (p)}\Big{)}\ell_{k}^{(p)}(\xi_{m}), \tag{12}\]  
where \(\{\ell_{k}^{(p)}\}_{k=0}^{p}\) are the related Lagrange basis polynomials of degree \(p\).  
Now, let \(\mathcal{A}\subset\mathbb{N}_{0}^{M}\) be a finite set of multi-indices. For \(\alpha\), \(\beta\in\mathcal{A}\) write \(\alpha\leq\beta\) if \(\alpha_{m}\leq\beta_{m}\) for all \(m=1,\ldots,M\). Further, let us assume that \(\mathcal{A}\) is monotone in the following sense: \(\exists\alpha\in\mathcal{A}\) such that \(\beta\leq\alpha\Rightarrow\beta\in\mathcal{A}\). The sparse collocation operator is defined as  
\[\mathcal{I}_{\mathcal{A}}:=\sum_{\alpha\in\mathcal{A}}\bigotimes_{m=1}^{M} \left(\mathcal{I}_{\alpha_{m}}^{(m)}-\mathcal{I}_{\alpha_{m}-1}^{(m)}\right), \tag{13}\]  
where \(\mathcal{I}_{-1}^{(m)}:=0\). The collocation points are of the form  
\[\chi_{\gamma}^{(\alpha)}=\Big{(}\chi_{\gamma_{1}}^{(\alpha_{1})},\ldots,\chi_ {\gamma_{M}}^{(\alpha_{M})}\Big{)}\in\Gamma \tag{14}\]  
for some \(\gamma\in\mathbb{N}_{0}^{M}\) such that \(\gamma\leq\alpha\in\mathcal{A}\). Similarly, the tensorised quadrature weights \(w_{\gamma}^{(\alpha)}=w_{\gamma_{1}}^{(\alpha_{1})}\cdots w_{\gamma_{M}}^{( \alpha_{M})}\) are defined. Statistics, such as the expected value and variance, for the collocated solution may now be computed by applying the quadrature rule  
\[\mathbb{E}_{\Gamma}\Bigg{[}\bigotimes_{m=1}^{M}\mathcal{I}_{\alpha_{m}}^{(m) }v\Bigg{]}=\sum_{\gamma\leq\alpha}v\Big{(}\chi_{\gamma}^{(\alpha)}\Big{)}w_{ \gamma}^{(\alpha)} \tag{15}\]  
on the terms in (13).  
The accuracy of the collocated approximation is ultimately determined by the smoothness of the solution as well as the choice of the multi-index set \(\mathcal{A}\subset\mathbb{N}_{0}^{M}\). For a detailed analysis, refer to [30; 32]. In this paper, tensor product grids defined by  
\[\mathcal{A}=\{\alpha\in\mathbb{N}_{0}^{M}\mid\alpha_{m}\leq p,\ m=1,\ldots,M \},\quad p\in\mathbb{N}_{0}, \tag{16}\]  
are used.