Stochastic-Collocation#
Description#
Stochastic-Collocation is a non-intrusive method to propagate uncertainties through a given code. Collocation treats the given code as a black box which is evaluated at a fixed set of realizations. Outputs at these realizations are then used to approximate quantities such as expectation or variance.
Examples#
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Example illustration of stochastic-Collocation
Let \(\xi\) be a random variable with probability density \(f_{\Xi}:\Theta\rightarrow\mathbb{R}_+\) and \(f\) be a function \(f:\Theta\rightarrow\mathbb{R}\). We wish to determine the expectation of \(f\), which reads \(\mathbb{E}[f]=\int_{\Theta}f(\xi)f_{\Xi}\,d\xi\). The expectation is approximated with quadrature points \(\xi_k\) where \(k=1,\cdots,N_q\) and corresponding quadrature weights \(w_k\) via \(\mathbb{E}[f]\approx \sum_{k=1}^{N_q}w_k f(\xi_k)f_{\Xi}(\xi_k)\). The values of \(f(\xi_k)\) are determined by treating the implementation of \(f\) as a black box, which is evaluated at all \(N_q\) collocation points.
Applications#
Ambiguities and field-specific meaning#
There are different interpretations of what a stochastic-Collocation method is. Several authors call any method which evaluates a given code at certain points Stochastic-Collocation. In that sense, Monte-Carlo is also a stochastic-Collocation method. Others only refer to a method as stochastic-Collocation if it evaluates the code at Gauss quadrature points.
Contributors#
Maqsood Rajput