Polynomial Chaos Expansions¶
Introduction¶
In their most basic use, polynomial chaos expansions (PCE) create an emulator for a quantity that depends on a random variable. For simplicity, we will assume that this random variable is finite-dimensional. Let \(\xi\) denote a \(d\)-dimensional random variable. For some function \(f: \mathbb{R}^d \rightarrow \mathbb{R}\), the PCE approach performs the approximation,
where \(\{\phi_n\}_{n=1}^\infty\) are polynomial functions of the random variable \(\xi\), and \(\{hat{f}_n\}_{n=1}^\infty\) are coefficients. If such an emulator can be constructed, then statistics are evaluated as statistics of the emulator.
For example, the (approximation to the) mean is
which can be efficiently evaluated by manipulation of the coefficients \(\hat{f}_n\). The terms \(\mathbb{E}[\phi_n(\xi)]\) can be evaluated exactly using properties of the \(\phi_n\) polynomials.
All of the above extends to the case when the function \(f\) depends on other variables, such as space \(x\) or time \(t\). For example, if \(f = f(x,t,\xi)\), then the PCE approach becomes
so that the coefficients depend on \((x,t)\). Then the space- and time-varying expectation can be evaluated in a manner similar to (2).
For PCE approaches, most of the computation involves computing the coefficients \(\{\hat{f}_n\}_{n=1}^\infty\). Non-intrusive PCE strategies accomplish this by computing values of \(f\) on a specific sampling grid or experimental design in stochastic space: \(\{\xi_m\}_{m=1}^M\). The procedures used in UncertainSCI typically require the number of samples \(M\) to scale with the degrees of freedom in the emulator \(N\).
In order to utilize the PCE approaches in UncertainSCI, two items must be provided:
The distribution of the random variable \(\xi\). See Distributions for how to generate this distribution.
The type of polynomial functions in (1). This amounts to defining a particular polynomial subspace. See Polynomial Spaces for how to generate this subspace.