Overview

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.

Methodology

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.

Credit Recognition

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.

Travel and Accommodation

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.

Erasmus Funding

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.

To registration and deadlines section

Lectures

Lecture 1. Hermenegildo Borges de Oliveira (University of Algarve, Portugal)

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.

Lecture 2. Karolina Kropielnicka (Institute of Mathematics of Polish Academy of Sciences, Poland)

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.

Lecture 3. M. Concepión Muriel Patino (University of Cádiz, Spain)

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.

Lecture 4. Andrijana Curkovic (University of Split, Croatia)

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.

Lecture 5. Saša Krešić-Jurić (University of Split, Croatia)

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.

Contact

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!