# Function-Valued Uncertainties

The geometrical uncertainties also depend on the geometry itself, so they are modeled as a Gaussian random field

у: Г x Q ^ R, (3)

defined on a probability space (Q, Y,P) and on the shape of the airfoil Г. In each point x of the shape Г, the uncertainty is described by a normally distributed random variable y(x, ■) : Q ^ R. Additionally, the second order statistics, the mean value and the covariance function, are given to fully describe the random field. According to the scalar-valued uncertainties, the mean value of the random field у is equal to 0, since we expect no perturbations and the squared exponential covariance function describes the interaction between the random variables on the shape:

E(y(x, Z)) = У0 (x) = 0 Ух є Г (4)

Cov(x, y) = b2 • exp Ух, у Є Г (5)

The parameter l determines how quickly the covariance falls off and b controls the magnitude of the bumps. A squared exponential covariance function is chosen, since the resulting perturbed geometry is smooth due to the smoothness of the random field.

Then, a perturbed geometry is given as

v (x, Z) = x + y(x, Z) ■ n (x) Ух є Г, Z є Q (6)

where n is the unit vector in x normal to the profile Г. As we need to compute statistics of the flow depending on the uncertainty in our optimization algorithm, we have to approximate and discretize the probability spaces. In the next chapter, we will introduce the Karhunen-Loeve-Expansion which provides an approximation of the random field у for the numerical evaluation of such statistics and efficient discretization techniques of the probability space.