## 4 Stochastic Finite Element Methods
### Stochastic Collocation  
For the uncertain profile function \(f^{\omega}\), a representation of the form  
\[f^{\omega}(\mathbf{x})=f_{0}(\mathbf{x})-\left(\sum_{m=1}^{\infty}f_{m}( \mathbf{x})Y_{m}(\omega)\right)^{2}, \tag{11}\]  
is assumed. Here, \(\{f_{m}\}_{m\geq 0}\) are spatial stochastic fluctuations decaying in \(L^{\infty}\)-norm as \(m\to\infty\), and \(\{Y_{m}\}_{m\geq 1}\) is a family of random variables assumed to be i.i.d. The series (11) can be viewed as a special form of a Karhunen-Loeve expansion of the random underlying random field, in which case the functions \(f_{m}(\mathbf{x})\) are the eigenfunctions of some covariance function \(C(x_{1},x_{2})\). For a Galerkin SFEM perspective and further references, see [21]. Squaring of the sum of the random input ensures that the perturbation does not change its sign. In some simple geometric settings, these eigenfunctions are known explicitly, but they can also be resolved numerically as eigenvectors of the sampled covariance matrix; see, e.g., ([22], Chapter 8), [23; 24]. For numerical computations, the series is truncated after \(M\) terms. For the model to be applicable as \(m\to\infty\), the decay of of the stochastic fluctuations must be greater than quadratic, which is the critical decay rate (see [14]). Notice that in all truncated formulations, the problem is well posed, even with decay rate \(=2\). In the following, an anisotropic Smolyak-type collocation operator defined with respect to a finite multi-index set (see, e.g., [25; 26; 27; 28]) is considered. In principle, a simpler formulation would also be sufficient for the purpose of our numerical experiments. However, the general form of the operator is kept here, since it allows for efficient computation even in high-dimensional parameter spaces. For the sake of accuracy, it is standard practice to choose the collocation points to be the abscissae of orthogonal polynomials; see [29]. Here, the collocation method is formulated using Legendre polynomials, which are the optimal choice when the input random variables are uniform. If a lognormal random field model had been adopted instead of the squared one, 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 \(\big{(}\mathcal{I}_{p}^{(m)}v\big{)}(Y_{m})=\sum_{k=0}^{p}v\big{(}\chi_{k}^{(p )}\big{)}\ell_{k}^{(p)}(Y_{m})\), where \(\{\ell_{k}^{(p)}\}_{k=0}^{p}\) are the related Lagrange basis polynomials of degree \(p\). Observe that \(\int_{-1}^{1}\big{(}\mathcal{I}_{p}^{(m)}v\big{)}(Y_{m})\ \frac{dY_{m}}{2}=\sum_{k=0}^{p}v\Big{(}\chi_{k}^{(p)} \Big{)}w_{k}^{(p)}\). Now, let \(\mathcal{A}\subset\mathbb{N}_{0}^{M}\) be a finite set of multi-indices. For \(\alpha,\beta\in\mathcal{A}\), one can write \(\alpha\leq\beta\) if \(\alpha_{m}\leq\beta_{m}\) for all \(m=1,\ldots,M\). It is implicitly assumed that \(\mathcal{A}\) is monotone in the following sense: \(\exists\alpha\in\mathcal{A}\) such that \(\beta\leq\alpha\quad\Rightarrow\quad\beta\in\mathcal{A}\). The sparse collocation operator is defined as \(\mathcal{I}_{\mathcal{A}}:=\sum_{\alpha\in\mathcal{A}}\bigotimes_{m=1}^{M} \big{(}\mathcal{I}_{\alpha_{m}}^{(m)}-\mathcal{I}_{\alpha_{m}-1}^{(m)}\big{)}\), where \(\mathcal{I}_{-1}^{(m)}:=0\). This may be rewritten in a computationally more convenient form:  
\[\mathcal{I}_{\mathcal{A}}=\sum_{\alpha\in\mathcal{A}}\sum_{\beta\in\mathcal{G }_{\alpha}}(-1)^{\|\alpha-\beta\|_{1}}\bigotimes_{m=1}^{M}\mathcal{I}_{\beta _{m}}^{(m)}, \tag{12}\]  
where \(\mathcal{G}_{\alpha}:=\{\beta\in\mathbb{N}_{0}^{M}\mid\alpha-1\leq\beta\leq\alpha\}\).