Mathematics

Lectures Prof Gianluigi Rozza, Tutorials coordinated by Dr Giovanni Stabile,  Dr Francesco Ballarin, Dr Maria Strazzullo and Dr Federico Pichi

Learning outcomes and objectives

The course aims to provide the basic aspects of numerical approximation and efficient solution of parametrized PDEs for computational mechanics problem (heat and mass transfer, linear elasticity, viscous and potential flows) using reduced order methods.

Description

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).

Please contact gianluigi.rozza@sissa.it  if you need more info. If you plan to attend in order to provide you links to lectures, exercise sessions as well as  to material and software instructions, please send an email to confirm your participation  in advance (no later than April 2, 2022).

Topics/Syllabus

• 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)

Schedule

Location:  Lectures  and Exercise sessions will be offered in presence at SISSA (Bonomea campus) and online on Zoom platform

Dates: 

1. Monday, April 4, 2022, 128-129 - 14:15 to 16:00 - lecture

2. Monday, April 4, 2022, 128-129 - 16:30 to 18:15 -  exercise session -intro- 

3. Tuesday, April 5, 2022,133  - 11:30 to 13:00 - lecture

4. Tuesday, April 5, 2022, 134 - 15:15 to 17:00 - exercise session

5. Wednesday, April 6, 2022, 134 - 10:15 to 13:00 - lecture

6. Wednesday, April 6, 2022, 136 - 14:15 to 16:00 - lecture 

7. Wednesday, April 6, 2022, 136 - 16:30 to 18:15 - exercise session

8. Thursday, April 7, 2022, 128-129 - 10:15 to 13:00 - lecture

9. Thursday, April 7, 2022, 133 - 14:15 to 18:00 - exercise sessions

Software for the exercise sessions

We will employ the RBniCS package, which is an open-source code developed at SISSA mathLab (and at the Catholic University of the Sacred Heart) in the framework of the AROMA-CFD ERC CoG project. We will provide 5 worked problems. The tutorials will run on the cloud service provided by Google Colab, so that no local installation is required. See our tutorials page for more information on the RBniCS library and on how to use the Google Colab platform.