Mathematical Analysis, Modelling, and Applications

The activity in mathematical analysis is mainly focussed on ordinary and partial differential equations, on dynamical systems, on the calculus of variations, and on control theory. Connections of these topics with differential geometry are also developed.The activity in mathematical modelling is oriented to subjects for which the main technical tools come from mathematical analysis. The present themes are multiscale analysis, mechanics of materials, micromagnetics, modelling of biological systems, and problems related to control theory.The applications of mathematics developed in this course are related to the numerical analysis of partial differential equations and of control problems. This activity is organized in collaboration with MathLab for the study of problems coming from the real world, from industrial applications, and from complex systems.

A reduced-order model for segregated fluid-structure interaction solvers based on an ALE approach

Ngan VEric Brice, Stabile G, Mola A, Rozza G. A reduced-order model for segregated fluid-structure interaction solvers based on an ALE approach.; 2022.

POD–Galerkin Model Order Reduction for Parametrized Time Dependent Linear Quadratic Optimal Control Problems in Saddle Point Formulation

Strazzullo M, Ballarin F, Rozza G. POD–Galerkin Model Order Reduction for Parametrized Time Dependent Linear Quadratic Optimal Control Problems in Saddle Point Formulation. Journal of Scientific Computing. 2020 ;83.

Space-time POD-Galerkin approach for parametric flow control

Ballarin F, Rozza G, Strazzullo M. Space-time POD-Galerkin approach for parametric flow control. Handbook of Numerical Analysis. 2022 ;23.

A Data-Driven Partitioned Approach for the Resolution of Time-Dependent Optimal Control Problems with Dynamic Mode Decomposition

Donadini E, Strazzullo M, Tezzele M, Rozza G. A Data-Driven Partitioned Approach for the Resolution of Time-Dependent Optimal Control Problems with Dynamic Mode Decomposition. In: 13th International Conference on Spectral and High Order Methods, ICOSAHOM 2021. 13th International Conference on Spectral and High Order Methods, ICOSAHOM 2021. ; 2023.

Non-linear manifold reduced-order models with convolutional autoencoders and reduced over-collocation method

Romor F, Stabile G, Rozza G. Non-linear manifold reduced-order models with convolutional autoencoders and reduced over-collocation method. Journal of Scientific Computing [Internet]. 2023 ;94(3). Available from: https://link.springer.com/article/10.1007/s10915-023-02128-2

Isoperimetric problems in curved spaces and applications

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A time-adaptive algorithm for pressure dominated flows: a heuristic estimator

Prusak I, Torlo D, Nonino M, Rozza G. A time-adaptive algorithm for pressure dominated flows: a heuristic estimator. [Internet]. 2024 . Available from: https://arxiv.org/abs/2407.00428

Exact pattern formation and symmetry breaking for a family of isotropic local/nonlocal interaction system

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Numerical Solution of Partial Differential Equations with deal.II

The course "Numerical Solution of PDEs with deal.II" offers a focused exploration of solving Partial Differential Equations (PDEs) using the Finite Element Method (FEM), employing the deal.II software library. Key components of the course include an introduction to PDEs, basics of numerical methods and FEM analysis, practical training using deal.II, and hands-on projects. The course will also cover High-Performance Computing (HPC) techniques for parallelizing, optimizing, and load balancing FEM simulations for real-world applications.

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Variational methods in symplectic dynamics

Program:

- Introduction to critical point theory (Morse theory, Lusternik-Schnirelmann theory, equivariant methods and Fadell-Rabinowitz index)

- Introduction to symplectic dynamics: Hamiltonian flows, including geodesic flows and, more generally, Reeb flows

- The periodic orbits problems: introduction, and proof of a selection of results (multiplicity of periodic orbits, spectral characterization of Besse and Zoll Reeb flows, etc.).

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Mathematics