Polynomial approximation by means of the random discrete L2 projection and application to inverse problems for PDEs with stochastic data

The main topic of this thesis concerns the polynomial approximation of aleatory functions by means of the random discrete L2 projection, and its application to inverse problems for Partial Differential Equations (PDEs) with stochastic data. The motivations come from the parametric approximation of the solution to partial differential models. The thesis is arranged in two parts, with an introductory chapter which contains an overview of modern techniques for polynomial approximation of functions depending on random variables. In the former part, from Chapter 1 to Chapter 4, the focus is on the theoretical analysis of the random discrete L2 projection applied to solve the so-called forward problem, e.g. to approximate the moments of an aleatory function given its observations, or to compute the solution to a computational model with stochastic coefficients given initial and boundary data. The stability and optimality of the approximation error evaluated in the L2 weighted norm are addressed. In the latter part of the thesis, composed of Chapter 5 and Chapter 6, the methodology previously developed for the forward problem is applied to inverse problems for PDEs with stochastic coefficients. The factorization method is applied in the framework of Electrical Impedance Tomography, first in the case of inhomogeneous background, and then in the case of piecewise constant background, with values in each region affected by uncertainty. Finally, in Chapter 6 the variants of the Factorization Method proposed in the previous chapter are accelerated exploiting the techniques that have been presented in the first part of the thesis.