Mathematics Area - SISSA - MsC Course

2026 - Spring

Guglielmo Padula — Wed, 03 Jun 2026 08:38:45 +0000

Research Group:

MsC Course

Year:

2026 - Spring

English Text:

Written exam_2026.pdf

Advanced Mathematical Physics Mod. A

Ariel Surya Boiardi — Thu, 11 Sep 2025 09:15:34 +0000

Lecturer:

Pavlo Gavrylenko

Course Type:

Master Course

Academic Year:

2025-2026

Period:

October-December

Duration:

48 h

Research Group:

MsC Course

Next Lectures:

Advanced Geometry II

Ariel Surya Boiardi — Thu, 11 Sep 2025 09:13:50 +0000

Lecturer:

Nicolò Sibilla

Course Type:

Master Course

Academic Year:

2025-2026

Period:

October-December

Duration:

48 h

Research Group:

MsC Course

Next Lectures:

Numerical Solution of Partial Differential Equations with deal.II

Irene Anello — Fri, 13 Sep 2024 08:00:58 +0000

Lecturer:

Pasquale Claudio Africa

Course Type:

PhD Course

Master Course

Academic Year:

2024-2025

Period:

January

Duration:

24 h

Description:

The course "Numerical Solution of PDEs with deal.II" offers a focused exploration of solving Partial Differential Equations (PDEs) using the Finite Element Method (FEM), employing the deal.II software library. Key components of the course include an introduction to PDEs, basics of numerical methods and FEM analysis, practical training using deal.II, and hands-on projects. The course will also cover High-Performance Computing (HPC) techniques for parallelizing, optimizing, and load balancing FEM simulations for real-world applications. Targeted towards engineering, physics, and applied mathematics, it requires a foundational knowledge of differential equations, numerical methods and the C++ programming language.This is a joint course offered by SISSA and the MHPC program.

Research Group:

Mathematical Analysis, Modelling, and Applications

mathLab

MsC Course

Location:

ICTP, Leonardo da Vinci Building, Lecture Room D

Next Lectures:

Topics in advanced analysis II

Guglielmo Padula — Fri, 06 Sep 2024 17:10:34 +0000

Lecturer:

Stefano Bianchini

Course Type:

PhD Course

Master Course

Academic Year:

2024-2025

Period:

October - January

Duration:

50 h

Description:

Topics in advanced analysis II

The course is focused on evolutionary PDEs of first oder: these will comprise

- first order PDEs and the method of characteristics, local existence and uniqueness

- linear transport equation, in particular continuity equation, Lagrangian representation, renormalization and mixing estimates

- hyperbolic systems of conservation laws, well posedness for scalar equation and systems in 1d, non uniqueness for Euler equation in multid

- Hamilton-Jacobi equation, viscosity solutions, Lax formula, regularity, control problems

Research Group:

Mathematical Analysis, Modelling, and Applications

MsC Course

Location:

A-134

Next Lectures:

Functional analysis

Guglielmo Padula — Fri, 06 Sep 2024 16:51:20 +0000

Lecturer:

Alberto Maspero

Course Type:

PhD Course

Master Course

Academic Year:

2024-2025

Period:

March - June

Duration:

48 h

Description:

Aim of the course is to introduce the basic tools of linear and nonlinear functional analysis, and to apply these techniques to problems in PDEs. The course is divided into two parts: the first one concerns spectral theory of linear operators, whose goal is to extend the classical notion of spectrum of a matrix to an infinite dimensional setting. The second part of the course introduces the methods of nonlinear analysis to find the zeros of a nonlinear functional on a Banach space. In particular it gravitates around the implicit function theorem and its variants.

Course contents:

Part 1: Linear analysis

- Spectrum of a bounded linear operator

- Compact operators, their spectrum and Fredholm theory

- Functional calculus and spectral theorem of selfadjoint operators

- Unbounded operators

Part 2: Nonlinear analysis

- Differential Calculus in Banach Spaces

- Implicit Function Theorem and applications

- Lyapunov–Schmidt Reduction and Bifurcation

- Degree theory

References

[1] Ambrosetti, Prodi: A primer of nonlinear analysis. Cambridge Studies in Advanced Mathematics, 34. Cambridge University Press, 1995

[2] Bogachev, Smolyanov: Real and Functional Analysis. Moscow Lectures, Springer 2020

[3] Brezis: Functional analysis, Sobolev spaces and partial differential equations.Universitext. Springer, New York, 2011.)

[4] Eidelman, Milman, Tsolomitis. Functional analysis. Graduate studies in Mathematics, 66. American Mathematical Society, 2004

[5] Reed, Simon: Methods of modern mathematical physics. I. Functional analysis. Academic Press, Inc., New York, 1980

[6] Teschl: Topics in Linear and Nonlinear Functional Analysis, Graduate Studies in Mathematics, Volume XXX, Amer. Math. Soc., Providence, link

Research Group:

Mathematical Analysis, Modelling, and Applications

MsC Course

Next Lectures:

Advanced geometry 2

Guglielmo Padula — Fri, 06 Sep 2024 16:46:31 +0000

Lecturer:

Luca Rizzi

Course Type:

PhD Course

Master Course

Academic Year:

2024-2025

Period:

October - January

Duration:

50 h

Description:

Smooth manifolds and differential topology.

Research Group:

Mathematical Analysis, Modelling, and Applications

MsC Course

Next Lectures:

Topics in computational fluid dynamics

Guglielmo Padula — Fri, 06 Sep 2024 16:42:44 +0000

Lecturer:

Gianluigi Rozza

Course Type:

PhD Course

Master Course

Academic Year:

2024-2025

Period:

May

Duration:

20 h

Description:

Topics/Syllabus

Introduction to CFD, examples.
Constitutive laws
Incompressible flows.
Numerical methods for potential and thermal flows
Boundary layer theory
Thermodynamics effects, energy equation, enthalpy and entropy
Vorticity equations
Introduction to turbulence
Numerical methods for viscous flows: steady Stokes equations
Stabilisation (SUPG) and inf-sup condition
Numerical methods for nonlinear viscous flows: steady Navier-Stokes equations (Newton and Fixed Point vectorial methods)
FEM Discretisation techniques for steady and unsteady Navier-Stokes equations.
Worked problems/CFD exercises session based FEM by FEniCS(-X) by Dr. Davide Oberto

Lecture details:

Tuesday May 6, 2025: 14-16, room 133

Wednesday May 7, 2025: 12-13 and 15-16, room 133

Thursday May 8, 2025: 14-16, room 134

Friday May 9, 2025: 11-13, room 133

Monday May 12, 2025: 15-17, room 133

Wednesday May 14, 2025: 16:30-18:30, room 133

Thursday May 15, 2025: 16-18, room 133

Friday May 16, 2025: 11-13, room 005

Friday May 23, 2025: 10-12, room 005

Tuesday May 27, 2025: 14-16 (Exercise Session), room 133

Thursday May 29, 2025: 11-13 (Exercise Session), room 133

Friday May 30, 2025: 11-13 (Exercise Session), room 134

Some dates/Hours are subject to changes due to institutional duties.

Research Group:

Mathematical Analysis, Modelling, and Applications

MsC Course

Next Lectures:

Computational mechanics by reduced order methdos

Guglielmo Padula — Fri, 06 Sep 2024 16:40:44 +0000

Lecturer:

Gianluigi Rozza

Course Type:

PhD Course

Master Course

Academic Year:

2024-2025

Period:

April

Duration:

20 h

Description:

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 and exercises material, as well as software instructions, please send an email to confirm your participation.

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)

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. The first half of the exercise sessions will be dedicated to proving the necessary basic knowledge and components of FeniCS and RBniCS libraries for simple problems which is necessary for understanding more complex problems treated in the second part. 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.

Info and updates: https://www.math.sissa.it/course/phd-course-master-course/computational-mechanics-reduced-order-methdos

Theoretical lectures:

31/3, 16-18, Room A-133
01/4, 10-12, Room A-129
15/4, 16-18, Room A-133
23/4, 14-16, Room A-134
24/4, 11-13, Room A-133
29/4, 16-18, Room A-133
30/4, 11-13, Room A-134

Practical session:

15/4, 14-16, Room A-134
24/4, 14-16, Room A-133
29/4, 14-16, Room A-134
30/4, 14-16, Room A-134

Research Group:

Mathematical Analysis, Modelling, and Applications

MsC Course

Next Lectures:

Applied mathematics: An introduction to scientific computing by numerical analysis

Guglielmo Padula — Fri, 06 Sep 2024 16:35:46 +0000

Lecturer:

Gianluigi Rozza

Michele Girfoglio

Course Type:

PhD Course

Master Course

Academic Year:

2024-2025

Period:

October - January

Duration:

48 h

Description:

Lectures

Prof. Gianluigi Rozza (gianluigi.rozza@sissa.it)
Dr. Michele Girfoglio (mgirfogl@sissa.it)

Tutorial and exercise session

Dr. Ariel Surya Boiardi (aboiardi@sissa.it)

Practical Information on the course

This is a Joint course, between SISSA PhD in Mathematical Analysis, Modeling, and Applications, Laurea Magistrale in Matematica, and the Laurea Magistrale in Data Science and Scientific Computing

Course materials are available on the repository.

All recordings for lectures of 2021-2022 are available on this YouTube playlist.

Lectures from the academic year 2020-2021 are available at this YouTube playlist.

Syllabus 2023-2024

Four Modules of 12h each (1.5 CFU for each module), for a total of 48h, 6 CFU

Frontal Lectures

Module 1 (Basis of Numerical Analysis - Part I - Dr. Michele Girfoglio)

Well posedness, condition numbers
Polynomial based approximations (Power basis interpolation, Lagrange interpolation, Weierstrass approximation theorem)
Interpolatory Quadrature rules
Orthogonal polynomials and Gauss Quadrature Formulas
L2 projection
Review of elementary PDEs
Introduction to Finite Difference Methods
Introduction to Finite Element Methods

Module 2 (Basis of Numerical Analysis - Part II - Prof. Ganluigi Rozza)

Least square methods
Solution methods for Linear Systems: direct and iterative solvers
Eigenvalues/Eigenvectors
Solution methods for non-Linear systems
Review of ODEs

Module 3 (Basis of Numerical Modeling - Prof. Gianluigi Rozza)

Data assimilation in biomechanics, statistics, medicine, electric signals
Model order reduction of matrices
Linear models for hydraulics, networks, logistics
State equations (real gases), applied mechanics systems, grow population models, financial problems
Applications of ODEs
example in electric phenomena, signals and dynamics of populations (Lotke-Volterra)
Models for prey-predator, population dynamics, automatic controls
Applications of PDEs, the poisson problem

Elastic rope
Bar under traction
Heat conductivity
Maxwell equation

Module 4 (Numerical Analysis with Python - Dr. Michele Girfoglio)

Introduction to Python, Numpy, Scipy
Exercises on Condition numbers, interpolation, quadratures
Using numpy for polynomial approximation
Using numpy for numerical integration
Using numpy/scipy for ODEs
Working with numpy arrays, matrices and nd-arrays
Solving non-linear systems of equations
Using numpy/scipy for simple PDEs

Students projects

Application of the Finite Element Method to the solution of models taken from the course

Further material provided during lectures by Prof. Gianluigi Rozza [https://people.sissa.it/~grozza/amnasc/]

References and Text Books:

A. Quarteroni, R. Sacco, and F. Saleri. Numerical mathematics, volume 37 of Texts in Applied Mathe- matics. Springer-Verlag, New York, 2000.
[E-Book-ITA] [E-Book-ENG]
A. Quarteroni. Modellistica Numerica per problemi differenziali. Springer, 2008.
[E-Book-ITA]
A. Quarteroni. Numerical Models for Differential Problems. Springer, 2009.
[E-Book-ENG]
A. Quarteroni and A. Valli. Numerical approximation of partial differential equations. Springer Verlag, 2008.
[E-Book-ENG]
S. Brenner and L. Scott. The mathematical theory of finite element methods. Springer Verlag, 2008.
[E-Book-ENG]
D. Boffi, F. Brezzi, L. Demkowicz, R. Durán, R. Falk, and M. Fortin. Mixed finite elements, compatibility conditions, and applications. Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy June 26–July 1, 2006. Springer Verlag, 2008.
[E-Book-ENG]
D. Arnold. A concise introduction to numerical analysis. Institute for Mathematics and its Applications, Minneapolis, 2001.
[E-Book-ENG]
A. Quarteroni, F. Saleri, P. Gervasio. Scientific Computing with Matlab and Octave. Springer Verlag, 2006.
[E-Book-ENG]
B. Gustaffson Fundamentals of Scientific Computing, Springer, 2011
[E-Book-ENG]
Tveito, A., Langtangen, H.P., Nielsen, B.F., Cai, X. Elements of Scientific Computing, Springer, 2010
[E-Book-ENG]

Note that, when connecting from SISSA, all of the text books above are available in full text as pdf files.

Research Group:

Mathematical Analysis, Modelling, and Applications

mathLab

MsC Course

Location:

A-005

Location:

A-005 and A-128 (only on 1/10/24 and 9/10/24)

Next Lectures: