Mathematics

Incorporating random terms in mathematical models is crucial for capturing and quantifying uncertainties in real-world systems, as the presence of randomness can have a significant impact on the solution behavior, and more realistic and informative results can thus be provided. On the other hand, the simulation of stochastic models can require significant computational resources due to the need of generating numerous realizations of the system. This makes the development of reduction tools, such as surrogate models, essential for enabling efficient and scalable simulations.
In this talk, polynomial chaos expansion (PCE) is firstly presented, being a powerful technique that provides accurate surrogate representations to given stochastic processes. Then, its inclusion in the finite element (FE) setting is described, arriving to the formulation of the enhanced stochastic FE method. Therefore, a way of introducing Bayesian inference in the framework is presented, aimed to provide a posterior knowledge of the probabilistic solution, as well as capable of producing a powerful setting for hyper-parameters learning and models fidelity comparison. Possible relevant applications are finally explained. The talk is based on a joint work with Federico Pichi and Moaad Khamlich.