Statistical Finite Element Methods

The finite element method (FEM) is one of the great triumphs of applied mathematics, numerical analysis and software development. Recent developments in sensor and signalling technologies enable the phenomenological study of complex natural and physical systems. The connection between sensor data and FEM has been restricted to solving inverse problems placing unwarranted faith in the fidelity of the mathematical description of the system under study. If one concedes mis-specification between generative reality and the FEM then a framework to systematically characterise this uncertainty is required. This talk will present a statistical construction of the FEM which systematically blends mathematical description with data observations by endowing the Hilbert space of FEM solutions with the additional structure of a Probability Measure. This initiative is part of the “Ph.D. Lectures” activity of the project "Departments of Excellence 2023-2027" of the Department of Mathematics of Politecnico di Milano. This activity consists of seminars open to Ph.D. students, followed by meetings with the speaker to discuss and go into detail on the topics presented during the talk.