# Low-Rank Approximation of Frequency Response Analysis of Perforated Cylinders under Uncertainty
## 2 Preliminaries
### Frequency Response Analysis under Uncertainty
#### 2.2.1 The Stochastic Eigenproblem  
Let \(D\subset\mathbb{R}^{2}\) denote the computational domain (see Appendix A). Assume further that the Young's modulus is a random field expressed in a series of the form  
\[E(x,y,\xi)=E_{0}(x,y)+\sum_{m=1}^{M}E_{m}(x,y)\xi_{m},\quad(x,y)\in D, \tag{4}\]  
where \(M\in\mathbb{N}\) and \(\xi=(\xi_{1},\xi_{2},\ldots,\xi_{M})\) is a vector of mutually independent random variables. For simplicity, it is assumed here that each \(\xi_{m}\) is uniformly distributed in some interval of \(\mathbb{R}\) and, therefore, after suitable rescaling, the random vector \(\xi\) takes values in the domain \(\Gamma:=[-1,1]^{M}\). A parametrisation of the form (4) can be seen as derived from a Karhunen-Loeve expansion of the underlying random field \(E(x,y,\xi)\); see, e.g., [25], in which case the functions \(E_{m}(x,y)\) are the eigenfunctions of some covariance function \(C(x,y)\). These eigenfunctions are known explicitly only in some simple geometric settings, but can also be resolved numerically as eigenvectors of the sampled covariance matrix; see, e.g., Chapter 8 in Ref. [26] and [25; 27]. The series has to be truncated for numerical computations, of course. In the sequel, \(M\) terms are used in applications.  
The standard assumption is that \(E(x,y,\xi)\) is strictly uniformly positive and uniformly bounded, i.e., there exist \(E_{\min},E_{\max}>0\) such that  
\[E_{\min}\leq\operatorname*{ess\,inf}_{(x,y)\in D}E(x,y,\xi)\leq\operatorname* {ess\,sup}_{(x,y)\in D}E(x,y,\xi)\leq E_{\max}\quad\forall\xi\in\Gamma. \tag{5}\]  
Given an integrable function \(v=v(\xi)\), denote by \(\mathbb{E}_{\Gamma}[v]=2^{-M}\int_{\Gamma}v(\xi)\,d\xi\) its expectation taken over the parameter space \(\Gamma\).  
The uncertainty is introduced to the problem via material properties--more specifically, by representing the Young's modulus using the stochastic model (4) above. As a consequence, the eigenpairs of the free vibration problem depend on \(\xi\in\Gamma\). Obtaining the stochastic version of the eigenvalue problem becomes simple: one only has to replace the differential operators in (2) with their stochastic counterparts and assume that the resulting equation holds for every realisation of \(\xi\in\Gamma\). In variational form, the problem is stated as: Find functions \(\mathbf{u}(t,\cdot)\colon\Gamma\to V\) and \(\omega^{2}(t,\cdot)\colon\Gamma\to\mathbb{R}\) such that for all \(\xi\in\Gamma\)\[ta_{m}(\xi;\mathbf{u}(t,\xi), \mathbf{v})+ta_{s}(\xi;\mathbf{u}(t,\xi),\mathbf{v})+t^{3}a_{b}( \xi;\mathbf{u}(t,\xi),\mathbf{v})\] \[=\omega^{2}(t,\xi)m(t;\mathbf{u}(t,\xi),\mathbf{v})\quad\forall \mathbf{v}\in V. \tag{6}\]  
Here, \(a_{m}(\xi;\cdot,\cdot)\), \(a_{s}(\xi;\cdot,\cdot)\) and \(a_{b}(\xi;\cdot,\cdot)\) are stochastic equivalents of the deterministic bilinear forms in (3).  
The stochastic stiffness matrix \(\mathbf{S}(\xi)\) is obtained after integration and assembly. Using (4), one has  
\[\mathbf{S}(\xi)=\mathbf{S}^{(0)}+\sum_{m=1}^{M}\mathbf{S}^{(m)}\xi_{m},\quad \xi\in\Gamma, \tag{7}\]  
where each matrix \(\mathbf{S}^{(m)}\) corresponds to a term in the series. Notice that there is no dependence on the parameter vector \(\xi\in\Gamma\) in the mass matrix \(\mathbf{M}\).