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Reduced Order Methods for Computational Mechanics

Course Type: 
PhD Course
Academic Year: 
20 h

In this course we present reduced basis (RB) approximation and associated a posteriori error estimation for rapid and reliable solution of parametrized partial differential equations (PDEs). The focus is on rapidly convergent Galerkin approximations on a subspace spanned by "snapshots'"; rigorous and sharp a posteriori error estimators for the outputs/quantities of interest; efficient selection of quasi-optimal samples in general parameter domains; and Offline-Online computational procedures for rapid calculation in the many-query and real-time contexts. We develop the RB methodology for a wide range of (coercive and non-coercive) elliptic and parabolic PDEs with several examples drawn from heat transfer, elasticity and fracture, acoustics, and fluid dynamics. We introduce the concept of affine and non-affine parametric dependence, some elements of approximation and algebraic stability. Finally, we consider application of RB techniques to parameter estimation, optimization, optimal control, and a comparison with other reduced order techniques, like Proper Orthogonal Decomposition. Some tutorials are prepared for the course based on FEniCS and Python within the training/educational library RBniCS (open-source based on python and FEniCS) and featured on Colab with self installation. 


  • Introduction to RB methods, offline-online computing, elliptic coercive affine problems
  • Parameters space exploration, sampling, Greedy algorithm, POD
  • Residual based a posteriori error bounds and stability factors
  • Primal-Dual Approximation
  • Time dependent problems: POD-greedy sampling
  • Non-coercive problems
  • Approximation of coercivity and inf-sup parametrized constants
  • Geometrical parametrization
  • Reference worked problems
  • Examples of Applications in CFD and flow control
  • Tutorials (5 worked problems)

Learning materials for lectures

Slides and other learning materials are available at this link.

Software for the exercise sessions

We will employ the RBniCS package, which is an open-source code developed within SISSA mathLab in the framework of the AROMA-CFD ERC CoG project.
Each exercise session will be divided in two parts. During the first part, we will present and discuss an RBniCS tutorial, and run it on Google Colab. You will need a Google account (either associated with your institutional email address, or even a personal Gmail account) to be able to run notebooks. This part will take place in the Lecture room on Microsoft Teams.
During the second part, students will be asked to carry out some assignments on Google Colab. A one-on-one helpdesk will delivered through the Helpdesk channel #1 and Helpdesk channel #2 on Microsoft Teams, held by Dr Francesco Ballarin and Dr Giovanni Stabile.
A short video on how to run RBniCS on Google Colab is available. Students are asked to view the video before the first exercise session.


Lectures and exercise sessions will be delivered in a Lecture room on Microsoft Teams. You can join the lecture room either from any browser or by installing the Microsoft Teams app.
Lectures and Exercise sessions will be offered online by MS Teams.

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