Mathematics

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).

 

TOPICS:

• 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).

SOFTWARES:

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. We will provide 5 worked problems.