OutlineUNCERTAINTY QUANTIFICATION IN MODELING
Short course, Mauritius, January 2019
Laurent
Dumas (Versailles University, France)
The
systematic quantification of the uncertainties affecting the dynamics
of a system and the characterization of the uncertainty of its
outcomes is critical for engineering design and analysis, where risks
must be reduced as much as possible. Uncertainties stem naturally
from our limitations in measurements, predictions and manufacturing,
and we can say that any dynamical system used in engineering is
subject to some of these uncertainties.
The first part of the lectures presents an overview of the mathematical framework used in Uncertainty Quantification (UQ) analysis and introduces the use of Polynomial Chaos approximation in UQ. Some applications of UQ in medicine will then be presented.
Data Assimilation (DA) basically combines different sources of information in order to produce the best possible estimate of the state of a system. These sources generally consist of observations of the system and of physical laws describing its behaviour, often represented in the form of a numerical model. In a broad sense, how to predict the evolution of a system, knowing its past when the model and the measurements are known with a finite accuracy.
In the second part of the series of lectures, the use of DA using the mathematical approach of Extended Kalman Filter will be discussed. An application to drone tracking will also be presented.
The short course will be addressed to Research Students involved in Optimization, dynamical systems and Computational Fluid dynamics. It is also suitable for undergraduate students having interest in Numerical Analysis (Year 2 and Year 3), Mechanical Engineering (Year 3 and Year 4 students).
Two pictures of all students attending the course are available here and also here.
Day 1 (3h30)
UNCERTAINTY QUANTIFICATION (slides.pdf)
Day
2 (3h30) DATA ASSIMILATION (slides.pdf)
Some exercices (in french) around Legendre polynomials : exercices.pdf
UQ with Monte Carlo method on the toy problem (uniform uncertainty) : script 1
Best polynomial approximation for the function x-> e^(-x) : script 2
UQ with Legendre polynomial chaos on the toy problem (uniform uncertainty) : script 3
UQ with Monte Carlo method on the toy problem (gaussian uncertainty) : script 4
UQ with herlite polynomial chaos on the toy problem (gaussian uncertainty): script 5