This thesis focuses on PDEs in which some of the parameters are not known exactly but affected by a certain amount of uncertainty, and hence described in terms of random variables/random fields. This situation is quite common in engineering applications.

A common goal in this framework is to compute statistical indices, like mean or variance, for some quantities of interest related to the solution of the equation at hand (“uncertainty quantification"). The main challenge in this task is represented by the fact that in many appli- cations tens/hundreds of random variables may be necessary to obtain an accurate represen- tation of the solution variability. The numerical schemes adopted to perform the uncertainty quantification should then be designed to reduce the degradation of their performance when- ever the number of parameters increases, a phenomenon known as “curse of dimensionality".

A method that acts in this direction is Monte Carlo sampling. Such method is known to be dimension independent and very robust but it is also known for is very slow convergence rate. In this work we describe Monte Carlo sampling together with a solid error analysis and we provide a test for its robustness by integrating different functions with different regularitues and by solving different PDE problems with random coefficients.

Later on we introduce the technique of variance reduction and a further application of this idea that goes with the name of Multilevel Monte Carlo (MLMC) method. The asymptotic cost of solving the stochastic problem with the multilevel method is proved to be significantly lower than that of the standard method and, in certain circumstances, grows only proportionally with respect to the cost of solving the deterministic problem. Numerical calculations demonstrat- ing its effectiveness are presented and more complex problems such as elliptic PDEs in a do- main with random geometry are also presented, this last task has been performed to test a code designed for the simulation of flow through porous media at the pore scale. The results are promising considering the very complex geometries that require extremely expensive dis- cretizations.

This work is the final outcome of the participation to the Visiting Student Research Pro- gram at the King Abdullah University of Science and Technology, a project that allows students to conduct research with faculty mentors in selected areas of pure and applied sciences, and a Visiting Research Fellowship at the University of Texas in Austin.