PERIOD: JANUARY - FEBRUARY Prof. Andrea Manzoni - Politecnico di Milano The goal of the course is to introduce and discuss the most recent developments related to reduced order models for parametrized linear and nonlinear (PDE) systems, with special emphasis on interpolation and projection‐based techniques. The increasing computer power is nowadays essential to deal with the numerical solution of large dimension problems, however a computational reduction is still determinant whenever interested in real‐time simulations and/or repeated output evaluations for different values of some inputs of interest. In order to reduce this computational burden, we may rely on suitable reduced order models (ROMs), which are faster and cheaper to simulate, yet accurately represent the original high‐fidelity (or full‐order) system behavior. Within this class of techniques, parametric model reduction has emerged only in the last decade as a very active research area. Parametric ROMs target the broad class of problems for which the equations governing the system behavior depend on a set of parameters, such as in the case of parameterized PDEs. These parameters may represent, for instance, material properties, system geometry, boundary or initial conditions. The goal of parametric ROMs is thus to generate rapid but reliable models that characterize the system’s response over the whole parametric range. Such a strategy proves to be necessary in order to speedup design, control, optimization, and uncertainty quantification settings, which require several

This course surveys the state‐of‐the‐art of parametric reduced order models, describing the most popular approaches related to projection‐based techniques (such as reduced basis methods and proper orthogonal decomposition) and interpolation‐based techniques (such as empirical interpolation and discrete interpolation techniques), providing also a comparative discussion that lend insights to potential advantages/disadvantages entailed by their application. Implementation of basic techniques, as well as their application to several problems of interest, will be possible thanks to a Matlab library co‐developed by the lecturer providing a user‐friendly framework for their implementation.

The lectures will take place on the following dates: Jan 25, Feb 1, Feb 8 Every lecture day will be structured in two parts. First part: 10-12:30 Second part: 13-15:30 All lectures will be delivered in Room Buzano (DISMA - Politecnico di Torino), except the lecture in the afternoon of Jan 25, that will tke place in Room 1D.

TEACHING ORGANIZATION The course will focus on both the numerical methodology and applications to problems of real‐life interest (e.g., fluid dynamics, uncertainty quantification problems). Regular lectures and Matlab Sessions can be scheduled to present the most relevant techniques and to show their implementation. TEACHING MATERIALS Write max 5 lines A. Quarteroni, A. Manzoni and F. Negri, Reduced Basis Methods for Partial Differential Equations. An Introduction. Springer, 2016. Lecture notes and further recent research papers will be provided for the classes; additional material including Matlab codes will be offered for the practical work and laboratory activities. LEARNING EVALUATION Write max 3 lines The evaluation will be based on a project related with the lectures’ topics, focusing on either applicative or methodological aspects. The project will include numerical simulations to be developed within the Matlab library provided.

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Gli studenti e le studentesse con disabilità o con Disturbi Specifici di Apprendimento (DSA), oltre alla segnalazione tramite procedura informatizzata, sono invitati a comunicare anche direttamente al/la docente titolare dell'insegnamento, con un preavviso non inferiore ad una settimana dall'avvio della sessione d'esame, gli strumenti compensativi concordati con l'Unità Special Needs, al fine di permettere al/la docente la declinazione più idonea in riferimento alla specifica tipologia di esame.