Lecturer 1 | Lecturer 2 |
---|---|

Scuola Normale Superiore, Pisa |
Dipartimento di Matematica, Università di Pisa |

Large-scale systems of equations in saddle-point form arise in many different areas of science and engineering, from the numerical solution of important PDEs to problems in constrained optimization and data science. Many techniques have been developed to solve such systems but a number of challenges remain, and research in this field continues to be very active. The course will describe some of the applications where these systems arise and offer a panoramic view of the best solution methods available, up to recent advances. Theoretical foundations will also be addressed.

Topics will include:

- A brief introduction to Krylov subspace methods and preconditioning;
- Spectral properties of saddle-point matrices;
- Saddle-point problems arising from finite element discretizations of incompressible flow problems and elliptic PDEs;
- PDE-constrained optimization problems;
- Solution of saddle-point problems arising in constrained optimization;
- Approximate Schur complement and augmented Lagrangian techniques;
- Constraint preconditioning;
- Spectral analysis of preconditioned matrices.

**Organization:**

The course will be divided in two parts of 20 hrs each. In the morning lectures, Prof. Benzi will discuss fundamental principles, theoretical aspects, and the general algorithmic approaches available for the solution of saddle-point systems. In the afternoon lectures, Dr. Durastante will cover in detail some of the main applications, solution algorithms, and implementation and software aspects. Whenever possible, concrete examples will be discussed and hands-on exercises will be assigned to the participants.

Reading material will be distributed to the participants, who are expected to take an active role in the learning process.

**Obligations of partecipants**

Each participant is required to attend both modules and the afternoon seminars.

**Application**

Applications can be submitted exclusively online, on the SMI website.

Applications start: 1 March, 2022;**Deadline for applications is APRIL 15, 2022.**

A selection committee will meet shortly afterwards and all accepted foreign applicants will be notified of the result in due time.

Applications should contain a **BRIEF CURRICULUM VITAE** and a **DETAILED CURRICULUM STUDIORUM**. The exact e-mail address to which all correspondence concerning the Summer courses has to be mailed should be clearly stated.

Selected people will be informed by e-mail.

To run the *examples* discussed during my lectures the code is available
on the GitHub repository blazingsaddles.
The instructions for cloning the repository can be read on GitHub. If you are
not working on a Unix system, there are several interfaces available for working
with GitHub, read the documentation.

- Material on Finite Elements - Slides
- Material on Optimal Control Problems - Slides
- Material on Algebraic Multigrid Methods - Slides

Prof. Benzi material on *special topics*:

- Fast Iterative Solution Methods for the Incompressible Navier–Stokes Equations
- Solving Linear Systems of the Form $(A + \gamma U U^T)\mathbf{x} = \mathbf{b}$
- Iterative Solution of Double Saddle-Point Problems
- Block Preconditioners for the Coupled Stokes-Darcy Problem