Research Interests

Polytopal Methods for PDEs

The potential utility of the mathematical models in neurogeneration is strictly connected to the design of numerical methods to simulate PDEs in realistic geometries.  In many applications, the geometrical complexity complicates the application of classical FEM strategies, because extremely computationally expensive. In this context, the possibility of using polygonal or polyhedral meshes (polytopal, for short) to describe the computational geometry is beneficial. Indeed, a polyhedral grid allows a reduction of the degrees of freedom (DOFs) of the simulation without a loss of details in the representation of boundaries and interfaces. 

My contributions

My contributions to the topic include the construction and theoretical analysis of polytopal discontinuous Galerkin methods for different types of PDEs, such as the multiple-network poroelastic model [Cor+23a], the Fisher-Kolmogorov model [Cor+23b], the heterodimer system [Ant+24], and thermo/poro-viscoelastic models [BC25]. Moreover, I worked on the agglomeration strategies of three-dimensional grids in heterogeneous domains by means of geometric machine-learning [ACM24].

Structure-Preserving Methods for PDEs

In many applications, the obtained numerical solution should respect the same physical and mathematical properties of the continuous solution. For this reason, the construction of structure-preserving methods is a fundamental topic. For example, in the protein propagation models, the importance of being positive is also connected with the physical meaning of the solution.

My contributions

Concerning this topic, I worked on the construction of a structure-preserving interior-penalty discontinuous Galerkin method for the Fisher-Kolmogorov model on polytopal grids [CBA24]. Recently, I worked on another structure-preserving method for the same equation, based on a local discontinuous Galerkin approach, based on a change of variable, guided by the entropy variable of the system [Ant+26].

Neurodegenerative diseases represent a significant societal challenge. Indeed, due to the aging of the global population, the number of people affected by these pathologies is constantly increasing. Some neurodegenerative diseases, known as proteinopathies, are characterized by the misfolding of proteins. This misfolding leads to the generation of toxic proteins that spread and aggregate inside the Central Nervous System (CNS). Diseases such as Alzheimer’s, Parkinson’s, Amyotrophic Lateral Sclerosis (ALS), and Lewy body dementia follow this pattern, with specific proteins showing stereotypical progression. To better highlight the differences between all these pathologies, which often coexist, several mathematical models for the dynamics of prion-like proteins have been proposed. 

Mathematical modeling of proteinopathies

I am interested in the construction of novel mathematical models to describe the physical processes appearing in brain pathology. Moreover, the goal of my work is the development of patient-specific numerical simulations of prion-like spreading associated with different pathologies. This requires designing some calibration techniques for the physical parameters of the mathematical models. 

My contributions

Among my contributions in the field, I mainly worked on the numerical simulations of the spreading of misfolded proteins inside the brain by means of polytopal discontinuous Galerkin (dG) methods. In particular, I focused on α-synuclein [Cor+23b,Ant+24], amyloid-β [CBA24], and tau protein [AC25]. Moreover, I worked on the parameters calibration starting from biological post-mortem measurements in the brain cortex [Cor24]. Moreover, I worked on the construction of patient-specific simulations starting from PET measurement based on inverse uncertainty quantification methods coupled with reduced-order methods [Cor+24]. Finally, concerning the mathematical models, I want to mention the construction of a coupled model of the spreading of a tau protein in Alzheimer’s disease and brain tissue atrophy [Ped+25].