Resúmenes:

Victorita Dolean, TU Eindhoven

Title: Parallel methods to solve PDEs: an introduction to Domain Decomposition Methods and their implementation 

Abstract:  The course serves as an introduction to domain decomposition (DD) solvers and preconditioners. The following outline describes the content of the lectures and the tutorial. The first two parts will follow a few chapters of the book ”An introduction to domain decomposition methods: algorithms, theory and parallel implementation”, SIAM 2015. This book is freely downloadable together with the Freefem codes used to illustrate the methods. Installation of the open source software Freefem (https://freefem.org/) is recommended. (available for all platforms both in binary and compiled version) 
Part 1: A basic introduction to DD methods 
Objectives: This presentation will be kept at a basic level, both continuous and algebraic versions of the methods will be given in their most common variants and the main ingredients of domain decomposition methods will be presented. The content will follow the lines of the chapters 1 and 3 from the domain decomposition book. 
A short introduction to Freefem software will be given which will allow the students to use quickly the codes illustrating the methods.
Outcomes: At the end of this first part, students will have a basic understanding of the methods but also of their implementation. 
Part 2: Two-level methods and preconditioners: how to bring things to scale 
Objectives: Domain decomposition methods are meant to be used as parallel solvers and scal- ability (behaviour independent of the number of subdomains/processors) and robustness with respect to the physical parameters are very important issues. An introduction to coarse spaces and two-level methods for symmetric positive definite (SPD) problems will be given together with the presentation of a few variants of domain decomposition preconditioners (AS, RAS, ORAS, SORAS). The content will follow chapters 4 and 5 from the book, although more recent research results will also be included. 
Outcomes: Students will be able to understand the use and the impact of the two-level methods both for scalability and robustness (even if at this stage the codes are sequential). 
Part 3: ffddm tutorial 
The last part of this mini-course will be dedicated to a short tutorial on a new domain decomposition library ffddm1. The exercise session will be dedicated to the important key topics in the theory and implementation of the methods. All the methods and algorithms will be illustrated by numerical examples and simulations. References to past and current work will be indicated for those who want to deepen the theoretical part. 
Outcomes: At the end of this part, students will be able to run their first parallel code to illustrate the methods introduced in the first two parts. 

Virginie Ehrlacher, ParisTech

Title: Reduced-order models linear and nonlinear approaches

Abstract: Reduced Order Modeling (ROM) methods aim to simplify complex high-dimensional systems while preserving their essential characteristics. The main goals of ROM methods is to reduce the computational cost of simulations, making them feasible for real-time applications, optimization, and control while preserving accuracy and predictive capability of the original high-fidelity model while using fewer degrees of freedom.
The aim of this course is to present an introduction to linear and nonlinear approaches used to build reduced order models, as well as some of the main theoretical results known in this field. In particular, the aim of the course is to introduce the notion of Kolmogorov width of the set of solutions to some parametric mathematical model and illustrate the key role of this concept in the design of new efficient reduced-order modeling approaches.

Francisco Periago, U. Cartagena 

Title: An introduction to deep-learning-based methods for optimization and control of PDEs

Abstract: This mini-course is designed to provide graduate students and researchers with an introduction to state-of-the-art, data-driven algorithms for the numerical approximation of a wide array of control and optimization problems governed by both linear and nonlinear partial differential equations (PDEs). Coverage includes a comprehensive exploration of the theory and convergence analysis of Physics-Informed Neural Networks (PINNs) and Deep Operator Networks (DeepONet). PINNs are applied to problems where initial and boundary conditions remain fixed, while DeepONet is used to approximate operators that map these conditions to optimal controls or designs. A significant portion of the course will focus on the numerical implementation of these algorithms using Python, with two hours of hands-on, computer-based lessons. While prior experience with Python is recommended, it is not a prerequisite for participation. 

Jesús María Sanz-Serna, U. Carlos III de Madrid

Title: Sampling from probability distributions and numerical differential equations

Abstract: Many applications of mathematics, including artificial intelligence, Bayesian statistics, statistical physics and chemistry, require the generation of samples from a given probability distribution, often in spaces of extremely high dimensionality. Markov chain Monte Carlo algorithms are the method of choice for such a task; thousands of them have been suggested in the literature and, frequently, are based on the numerical integration of a suitable deterministic or stochastic differential equation. In the lectures I will describe relations between sampling algorithms and numerical differential equations. Even though many scientific disciplines will feature in the course, I will endevour to make the presentation understandable to everybody.
 

Charles-Edouard Bréhier, U. Pau et des Pays de l'Adour

Title: Some structure preserving schemes for stochastic PDEs

Abstract: I will present several situations where standard numerical schemes applied stochastic ordinary and partial differential equations fail to preserve some important qualitative properties of the exact solutions: for instance regularity of trajectories, positivity, trace formulas, invariant distributions, averaging or diffusion approximation principles, etc... I will then describe a series of techniques that can be used to design some structure preserving schemes in these situations, in particular using splitting methods. Convergence results will be given for those methods and numerical experiments will illustrate the behavior of standard and proposed numerical schemes.

Jesus Cortés, U. Castilla-La Mancha

Title: A certified POD-greedy method based on Legendre collocation for a Rayleigh-Bénard problem 

Abstract: This work presents a certified reduced basis method for studying bifurcations in a Rayleigh-Bénard problem using a Legendre collocation spectral method as high-fidelity discretization. The method is designed to handle independently each branch of the bifurcation diagram, producing for each one a certified reduced-order model that is formulated as a least-squares problem minimizing the restriction of the high-fidelity residual to a reduced basis. To certify the method, the different reduced bases are constructed iteratively through a POD-greedy algorithm driven by rigorous a posteriori error estimates, which are defined  as the quotient of a high-fidelity residual and a stability factor. A notable contribution of this work is the use of another reduced basis approach to approximate the stability factor. Furthermore, the proposed methodology does not require prior knowledge of the location of bifurcation points, accurately locating
them in the online stage using reduced-order indicators. Overall, this method allows for accurate, fast, and reliable computation of the bifurcation diagram of the Rayleigh-Bénard problem.

Soledad Le Clainche, U. Politécnica de Madrid

Title: Bridging Physics and AI: Hybrid Reduced-Order Models for Sustainable Innovation

Abstract: Addressing climate change requires innovative strategies and technological advancements to reduce atmospheric pollution. Fluid mechanics plays a crucial role in this challenge, with applications spanning from optimizing combustion efficiency to controlling urban air pollution and improving aircraft aerodynamics for lower fuel consumption.
This study focuses on developing hybrid reduced-order models (ROMs) that integrate fundamental physical principles with modern computational techniques. By employing modal decomposition methods such as singular value decomposition (SVD) and higher-order dynamic mode decomposition (HODMD), along with machine learning approaches like neural networks, this research aims to create predictive models that balance accuracy and efficiency in fluid dynamics applications.
The proposed methodologies have diverse applications. In the energy sector, they enhance combustion processes, reducing emissions. In urban settings, they support the development of more effective air quality control strategies. In aerodynamics, they contribute to the design of more fuel-efficient aircraft. Furthermore, in industrial flow control, these techniques enable advanced solutions for optimizing fluid behaviour, reinforcing global efforts toward sustainability and environmental responsibility.
 

Frederic Magoules,  U. Paris Saclay

Title: t.b.a.

Abstract: t.b.a.