## Section:
Scientific Foundations2>
### Uncertainty quantification3>

Participants : RĂ©mi Abgrall, Mario Ricchiuto, Pietro Marco Congedo.

Another topic of interest is the quantification of uncertainties in non linear problems. In many applications, the physical model is not known accurately. The typical example is that of turbulence models in aeronautics. These models all depend on a number of parameters which can radically change the output of the simulation. Being impossible to lump the large number of temporal and spatial scales of a turbulent flow in a few model parameters, these values are often calibrated to quantitatively reproduce a certain range of effects observed experimentally. A similar situation is encountered in many applications such as real gas or multiphase flows, where the equation of state form suffer from uncertainties, and free surface flows with sediment transport, where often both the hydrodynamic model and the sediment transport model depend on several parameters, and my have more than one formal expression.

This type of uncertainty, called *epistemic*, is associated
with a lack of knowledge and could be reduced by further experiments and investigation.
Instead, another type of uncertainty, called *aleatory*, is related to the
intrinsec aleatory quality of a physical measure and can not be reduced.
The dependency of the numerical simulation from these uncertainties can be studied by propagation of chaos
techniques such as those developped during the recent years via
polynomial chaos techniques. Different implementations exists,
depending whether the method is intrusive or not. The accuracy of these
methods is still a matter of research, as well how they can handle an
as large as possible number of uncertainties or their versatility with
respect to the structure of the random variable pdfs.
Our objective is to develop some non-intrusive or semi-intrusive methods, trying to define
an unified framework for obtained a reliable and accurate numerical solution
at a moderate computational cost.
Dealing with high dimensional representation of stochastic inputs in design optimiza-
tion is computationally prohibitive. In fact, for a robust design, statistics of the fitness
functions are also important, then uncertainty quantification (UQ) becomes the predom-
inant issue to handle if a large number of uncertainties is taken into account. Several
methods are proposed in literature to consider high dimension stochastic problem
but their accuracy on realistic problems where highly non-linear effects could exist is not
proven at all.
We developed several efficient global strategies for robust optimization: the first class of method is based on the extension of simplex stochastic collocation to the optimization space, the second one consists in hybrid strategies using ANOVA decomposition.

This part of our activities is supported by the ERC grant
`ADDECCO` , the ANR-MN project `UFO` and the associated
team `AQUARIUS` .