The objective of this module of the course is to illustrate the basic aspects of numerical approximation and efficient solution of parametrized PDEs for computational mechanics problems (heat and mass transfer, linear elasticity, viscous and potential flows) using reduced order methods and deep learning techniques. 

0. Introduction to numerical methods for the solution of parametric partial differential equations
1. Linear reduced order modeling techniques
1.1 Introduction to the reduced basis method, offline-online computing
1.2 Sampling, greedy algorithm, Proper Orthogonal Decomposition (POD)
1.3 Time-dependent problems: POD-greedy sampling
1.4 Geometrical parametrization
1.5 Reference worked problems
1.6 Examples of Applications in computational fluid dynamics (CFD)
2. Deep Learning Strategies
2.1 Introduction to data-driven methods for reduced order modeling
2.2 Basics of physics informed neural networks

Knowledge and understanding.  Learn the techniques for the numerical programming of numerical methods for complexity reduction and reduced order models for parametric PDEs. At the end of the course, the student will have acquired a good knowledge of the mathematical topics covered in the classes. 

Applying knowledge and understanding. Acquiring the ability to implement numerical methods for reduced order modeling techniques. Developing the ability to program, testing interpreting the results correctly. Acquiring the ability to solve mathematical problems using problem-solving environment.  

Making judgments. acquiring the ability to find the most suitable numerical method for the complexity reduction of differential problems.

Communication skills. acquiring the ability to rigorously define the mathematical problem studied in the course and to expose its numerical methods, outlining its fundamental properties

Learning skills. ability to study and solve problems similar but not necessarily the same as those dealt with during lessons.

The teaching material prepared by the lecturer in addition to recommended textbooks (such as for instance slides, lecture notes, exercises, bibliography) and communications from the lecturer specific to the course can be found inside the Moodle platform › blended.uniurb.it

Not available

Teaching

- Frontal lessons

- Examples and exercises using Python in the computer laboratory

- The whole material uploaded in the Moodle platform http://blended.uniurb.it

Course books

J. Hesthaven, G. Rozza, B. Stamm 'Certified reduced basis methods and a posteriori error bounds for parametrized PDEs', Springer 2015.