Recent years have seen an explosive growth in the recording of increasingly
complex and high-dimensional data. Classical statistical methods are often unfit
to handle such data, whose analysis calls for the definition of new methods
merging ideas and approaches from statistics and applied mathematics. My talk
will in particular focus on spatial and functional data defined over non-Euclidean
domains, such as linear networks, two-dimensional manifolds and non-convex
volumes. I will present an innovative class of methods, based on regularizing
terms involving Partial Differential Equations (PDEs), defined over the complex
domains being considered. These physics-informed regression methods enable
the inclusion in the statistical model of the available problem specific information,
suitably encoded in the regularizing PDE. The proposed methods make use of
advanced numerical techniques, such as finite element analysis and isogeometric
analysis. A challenging application to neuroimaging data will be illustrated.
29 Aprile 2022
Laura M. Sangalli
MOX - Dipartimento di Matematica, Politecnico di Milano