Mathematical Analysis, Modelling, and Applications | Mathematics Area - SISSA

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.

Topics in Continuum Mechanics

Reminders on linear algebra and tensor calculus
Kinematics of deformable bodies
Eulerian and Lagrangian descriptions of motion
Balance laws of continuum mechanics: conservation of mass, balance of linear and angular momentum, energy balance and dissipation inequality
Constitutive equations
Fluid dynamics: the Navier Stokes equations
Solid mechanics: nonlinear and linearized elasticity
Selected topics from the mechanics of biological systems

Books:

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Driving bifurcating parametrized nonlinear PDEs by optimal control strategies: application to Navier–Stokes equations with model order reduction

Pichi F, Strazzullo M, Ballarin F, Rozza G. Driving bifurcating parametrized nonlinear PDEs by optimal control strategies: application to Navier–Stokes equations with model order reduction. ESAIM: M2AN [Internet]. 2022 ;56(4):1361 - 1400. Available from: https://doi.org/10.1051/m2an/2022044

MicroROM: An Efficient and Accurate Reduced Order Method to Solve Many-Query Problems in Micro-Motility

Giuliani N, Hess MW, DeSimone A, Rozza G. MicroROM: An Efficient and Accurate Reduced Order Method to Solve Many-Query Problems in Micro-Motility. [Internet]. 2020 . Available from: https://arxiv.org/abs/2006.13836

The Neural Network shifted-proper orthogonal decomposition: A machine learning approach for non-linear reduction of hyperbolic equations

Papapicco D, Demo N, Girfoglio M, Stabile G, Rozza G. The Neural Network shifted-proper orthogonal decomposition: A machine learning approach for non-linear reduction of hyperbolic equations. Computer Methods in Applied Mechanics and Engineering [Internet]. 2022 ;392. Available from: https://www.scopus.com/inward/record.uri?eid=2-s2.0-85124488633&doi=10.1016%2fj.cma.2022.114687&partnerID=40&md5=12f82dcaba04c4a7c44f8e5b20101997

Summer school on Reduced Order Methods in Computational Fluid Dynamics

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Non-linear MOR: non-linear manifold method

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Data Enhanced Reduced Order Models for Turbulent Flows

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Deep-learning based ROMs for fast transient dynamics

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Application of Neural Networks in the optimisation of a hybrid energy system

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Digital Twins: state of the art and future challenges

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