Oriented to Master, PhD students and young researchers in Mathematics, Science or Engineering.

Erasmus Blended Intensive Program:

Sept. 2024:

2-6 Sept (online),

9-13 (Split, Croatia).

9-13 (Split, Croatia).

Prerequisites: Mathematics and computer programming skills.

3 courses & 5 lectures in a friendly environment for enjoying computational mathematics and applications.

This school is designed for young students, up to master and doctorate level, seeking to acquire skills in the **latest trends** in **mathematical** and **computational modeling**. We place a strong emphasis on practical applications in science and engineering, with a focus on modeling spatiotemporal variations of physical quantities using partial differential equations (PDEs) and related topics such as **finite element methods**, **neural networks** and **numerical optimization**, playing a vital role today in fields like **deep learning**.

The school is organized into **three courses**. Two courses will cover the widely used finite element method (FEM) and other emerging techniques such as physically informed neural networks (PINN) for computing numerical solutions of PDEs. The third course will focus on numerical methods for constrained and unconstrained optimization. In these courses, we introduce fundamental mathematical structures, provide illustrative examples, review algorithms, program them using high-level open-source tools, and post-process the solutions for high-quality plotting.

The essential mathematical and computer skills will be standardized in **introductory online sessions**, with expert proffesors in the field overseeing the entire learning process. The courses will be complemented by **five lectures** aimed at introducing recent research directions relevant to the field.

The school begins with a phase dedicated to homogenizing the students’ knowledge to help them reach a basic level before the classroom courses begin to develop in Split. It will be designed as **5 online days** where students will be able to access material specifically prepared for them to acquire basic math and computer skills. During these days they will have the support and tutoring of the curses instructors, through video meetings and forums in a virtual campus.

In the **second week**, the School will be held physically on the premises of the **University of Split**. It will consist on three courses where theoretical lectures will be given and numerous examples, algorithms and exercises will be reviewed. In computer classes, students will be provided with the necessary software to implement the algorithms and analyze the results. The courses will be complemented with lectures where invited professors will stimulate the curiosity of the students, stowing recent research topics related to computational mathematics.

After the end of the classes in Split, the students interested in receiving ECTS credits will submit a final work in small groups related to the school topics. For this, they will be assisted in a final tutoring session.

Participants who complete the course will receive **3 ECTS credits**. Evaluations for students interested in credit recognition will be based on daily attendance to the school and a on final offline group work.

The city of Split, Croatia, is served by an international airport which is located approximately 20 km from the city center of Split, on the west side of Kaštela Bay. Regular taxis at Split Airport are available during the airport’s operating hours, with the average cab fare to the center of Split being about 30 euros. Other options for travelling include shuttle bus (Pleso Prijevoz) and local bus, with respective costs of 6€ and 2.5€.

The School will provide students with up to 20 dormitory rooms in the university residence located on the university campus, a 5-minute walk from the Faculty of Science, where the activities will take place. Breakfast and lunch are included.

Up to 20 Students coming either from a SEA-EU University or from any EU Member State or third country associated to the Erasmus+ program can apply for funding through Blended Intensive Program (BIP). This will allow them to get support to attend the classroom week in Split.

*Organizer & Speaker:* **Francisco Ortegón Gallego** (Universidad de Cádiz)

*Classroom time:* 10 hours

*Topics:*

- Basic notions: differential operators and integral identities.
- Some PDEs arising in physics and engineering.
- Variational formulation. Functional spaces. The finite element method.
- Introduction fo
`Freefem++`

: numerical resolution of PDEs. - Working on a 3D problem: 3D tetrahedralization, resolution and post-processing.

*Organizers & Speakers:* **J. Rafael Rodríguez Galván, M. Victoria Redondo Neble** (Universidad de Cádiz)

*Classroom time:* 10 hours

*Topics:*

- Introduction and mathematical foundations of neural networks (NN). Fundamentals, architectures, activation and loss functions, differentiation and chain rule for functions of several variables
- Training and optimization in NN. Backpropagation, optimization algorithms
- Computational aspects and software libraries. Significance of hardware acceleration and parallelization for efficient training
- A perspective on neural networks for PDE models: physics informed neural networks (PINN). PINN software libraries. Modeling diffusion and convection for linear and non-linear processes
- Governing equations in fluid dynamics. The finite element method (FEM) for approximating non-turbulent flows
- PINN in fluid dynamics: comparing to FEM, exploring the pros and cons in a challenging case with applications to real-world scenarios

*Organizer & Speaker:* **Malte Braack** (University of Kiel)

*Classroom time:* 6 hours lectures, 4 hours exercises

*Topics:*

- Numerical methods for unrestricted optimization: Newton, steepest decent, Armijo step length control
- Restricted Optimization Problems: equality and inequality constraints, types of restricted optimization problems
- Sequential Unrestricted Minimization Technique (SUMT): penalty method, SUMT for equality constraints, SUMT for inequality constraints
- Stationary points for Restricted Optimization Problems: first-order necessary condition, active sets, linearized tangential cones, Abadie constraint qualification, Lagrange function, Karush-Kuhn-Tucker (KKT) system, Farkas lemma
- Convex optimization and Slater condition: convex constraints, relation between Slater condition and KKT
- Numerical methods based on Karush-Kuhn-Tucker system: Lagrange-Newton for restricted optimization, Sequential Quadratic Programing (SQP)

*Theoretical and numerical analysis of Navier-Stokes equations arising in fluid confinement*

In this lecture, we consider the incompressible Navier-Stokes equations with the forcing term assumed to be (non-linearly) dependent on the velocity field. Some applications of this problem will be shown. For the considered problem, we characterize feedback forces fields that are able of confining the fluid flow. We use the Continuous/Discontinuous Finite Element Method with interior penalty terms to solve the resulting nonlinear fourth-order problem. For the associated continuous and discrete problems, we prove the existence of weak solutions and establish the conditions for their uniqueness. Consistency, stability, and convergence of the numerical method are also shown analytically. To validate the numerical model regarding its applicability and robustness, several test cases are carried out.

*Split in Split*

Splitting methods are well established and widely used techniques for
finding approximate solutions of linear DEs of the form
*u’=(A+B)u*. They can be also used for the case of time dependent
component *B(t)*.

In this lecture I will give a short introduction to this subject on the example of celebrated Strang splitting for possibly time dependant component. I will show, that surpassingly it can be derived from Duhamel (Variation-of-Constant) formula. Based on this approach I will present a new proof of convergence of this scheme and elaborate on the possibilities brought by this approach.

*Solvable Structures and C-infinity Structures for Differential Equations*

We present some new integration methods for ordinary differential equations, based on the existence of -structures,
a recent generalization of the concept of solvable structure.
Both notions are established in the more general context of the
integrability of involutive distributions *Z* of vector fields.

Several illustrative examples show how both objects (solvable structures and -structures) can be found and used in practice to obtain exact solutions for problems modelled by ordinary differential equations.

*Introduction to Asymptotic Methods in Fluid Mechanics*

The application of asymptotic analysis to partial differential equations is presented using the example of a fluid flow through a thin pipe that is heated. The expansion of the pipe is considered and the small system parameters are the expansion coefficient of the pipe and the radius of the circular pipe.

*Introduction to Symplectic Methods for Hamiltonian Systems*

Hamilton’s equations describe the time evolution of a mechanical system defined on a symplectic manifold. Such systems have a wide range of applications from celestial mechanics, rigid body motion to molecular dynamics.

When discretizing Hamilton’s equations, one wishes to preserve the geometric properties important for qualitative behavior of the system. In this talk we give a short introduction to symplectic integrators for Hamiltonian systems based on preservation of the symplectic form under the flow of a Hamiltonian vector field. We discuss the symplectic Euler and Stromer-Verlet algorithms as well as composition methods for symplectic maps. We show that these algorithms outperform standard methods in terms of accuracy and stability for long time integrations. These methods are illustrated with several interesting examples from physics.

Universidad de Cádiz, Spain

Universidad de Cádiz, Spain

Universidad de Cádiz, Spain

Kiel University, Germany

University of Split, Croatia

Universidade do Algarve and CMAFcIO, Portugal

Institute of Mathematics of the Polish Academy of Sciences

Universidad de Cádiz, Spain

Uniwersytet Gdańsk, Poland

University of Split, Croatia

Universidad de Cádiz, Spain

= 5

Universities from the SEA-EU alliance

Universities from the SEA-EU alliance

= 4 + 5

Speakers in courses + lectures

Speakers in courses + lectures

= 5 + 5

Online + face-to-face days

Online + face-to-face days

≥ 20

Tutored hours of computer practice exercises

Tutored hours of computer practice exercises

**Monday 2**, 9:00h. Opening (video conference). Introduction to Course 1 (video conference).

10:00h*Francisco Ortegón Gallego*. Individual work, tutored in forums on the virtual campus**Tuesday 3**, 9:00h. Introduction to Course 2 (video conference).*J. Rafael Rodríguez Galván & M. Victoria Redondo Neble*. Individual work, tutored in forums on the virtual campus**Wednesday 4**, 9:00h. Introduction to Course 3 (video conference).*Malte Braack*. Individual work, tutored in forums on the virtual campus**Thursday 5**and**Wednesday 6**. Guided exercises.

An insight of mathematical and computational tools for simulation of models from Science and Engineering, specifically those which are written as PDE.

With a first course on the finite element method (FEM) for numerical approximation of PDE, a second one on Artificial Intelligence for learning solutions to PDE, in particular by delving into physically informed neural networks (PINN), and a third course on numerical methods for constrained and unconstrained optimization.

Computer Simulation of PDE Models

40%

Artifical Intelligence

27%

Numerical optimization

33%

**Application deadline**:**June 10**, 2024.**Confirmation of acceptance**:**June 11**, 2024.

Please, complete the following form *before the application deadline* (in case you are unable to visualize it correctly, you can complete it through this link):

Prof. Sasa Kresic-Juric, `sea-eu.comp.math@pmfst.hr`.

Department of Mathematics,
Faculty of Science,
University of Split.

Rudjera Boskovica 33.
21000 Split, Croatia.

Phone: +385 21 619 228

* * Email: sea-eu.comp.math@pmfst.hr

*See you in Split!*