Spectral Methods for Uncertainty Quantification

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Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients (2011, background): “Consequently stochastic Galerkin and stochastic collocation approaches, based for example on polynomial chaos expansion [19, 23], are impractical since their cost grows exponentially with the number of stochastic degrees of freedom, and truncating to any feasible number leads to large systematic err”
Wiener's Polynomial Chaos for the Analysis and Control of Nonlinear Dynamical Systems with Probabilistic Uncertainties [Historical Perspectives] (2013, background): “The principal objectives of uncertainty quantification and propagation include [9] » Model checking: In model-based analysis and control, models must be validated or invalidated by assessing their consistency with measurements/observations of the actual system.”
Survey of multifidelity methods in uncertainty propagation, inference, and optimization (2018, background): “We refer to [130, 208, 209, 139, 207] for an introduction to polynomial chaos expansion and further details.”
A multiscale method with patch for the solution of stochastic partial differential equations with localized uncertainties (2013, background): “Numerous developments have been realized for the propagation using functional approaches (see reviews [1, 2, 3, 4]).”