Introduction to spectral methods for uncertainty quantification

Spectral methods (SM) for uncertainty quantification are introduced. We start by introducing the transition between the deterministic and the stochastic frameworks, using the one-dimensional heat equation as an example. A simple Monte Carlo (MC) technique to solve the stochastic equation is introduced, together with its main advantages and drawbacks. The Karhunen-Loeve expansion, a crucial tool to construct other (SM), is presented. Non-intrusive spectral projection (NISP) and Galerkin methods are introduced, and comparisons against the MC approach are discussed. The main differences between NISP and Galerkin methods are also highlighted. All the sections in the chapter are consistently illustrated with the one-dimensional heat diffusion problem.