Chapter 1
Introduction to Spectral Methods
for Uncertainty Quantification
João F. Reis, Giulio Gori, Pietro M. Congedo, and Olivier Le Maître
Abstract 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–Loéve 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 onedimensional heat diffusion problem.
Keywords Uncertainty quantification · Monte Carlo methods · KL expansion ·
Non-intrusive spectral method · Galerkin method
1.1 Motivation
One of the most challenging questions in science is to make predictions about a
physical phenomenon of interest. We face this challenge in everyday life, since
predictions are useful to plan our actions in advance and to optimise our choices
to simplify our lives. Indeed, meteorological predictions help us deciding whether
to organise a holiday trip in the countryside or not. Predictions regarding traffic at
peak hours let us decide for different routes to avoid a long wait in a queue. In
Wall Street, predictions drive brokers in choosing the most profitable investment.
J. F. Reis () · G. Gori · P. M. Congedo
DeFI Team, CMAP Lab (École Polytechnique, Inria Saclay Île-de-France), Palaiseau, France
e-mail: joao.reis@inria.fr; giulio.gori@inria.fr; pietro.congedo@inria.fr
O. Le Maître
CMAP, CNRS, Inria, École Polytechnique, Palaiseau, France
e-mail: olm@limsi.fr
© Springer Nature Switzerland AG 2021
M. Vasile (ed.), Optimization Under Uncertainty with Applications to Aerospace
Engineering, https://doi.org/10.1007/978-3-030-60166-9_1
1
Introduction to Spectral Methods
for Uncertainty Quantification
João F. Reis, Giulio Gori, Pietro M. Congedo, and Olivier Le Maître
Abstract 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–Loéve 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 onedimensional heat diffusion problem.
Keywords Uncertainty quantification · Monte Carlo methods · KL expansion ·
Non-intrusive spectral method · Galerkin method
1.1 Motivation
One of the most challenging questions in science is to make predictions about a
physical phenomenon of interest. We face this challenge in everyday life, since
predictions are useful to plan our actions in advance and to optimise our choices
to simplify our lives. Indeed, meteorological predictions help us deciding whether
to organise a holiday trip in the countryside or not. Predictions regarding traffic at
peak hours let us decide for different routes to avoid a long wait in a queue. In
Wall Street, predictions drive brokers in choosing the most profitable investment.
J. F. Reis () · G. Gori · P. M. Congedo
DeFI Team, CMAP Lab (École Polytechnique, Inria Saclay Île-de-France), Palaiseau, France
e-mail: joao.reis@inria.fr; giulio.gori@inria.fr; pietro.congedo@inria.fr
O. Le Maître
CMAP, CNRS, Inria, École Polytechnique, Palaiseau, France
e-mail: olm@limsi.fr
© Springer Nature Switzerland AG 2021
M. Vasile (ed.), Optimization Under Uncertainty with Applications to Aerospace
Engineering, https://doi.org/10.1007/978-3-030-60166-9_1
1