Mathematical and Computational Modelling Across the Scales: Lecture Notes of the XX Jacques-louis Lions Spanish-french School, Barcelona, Spain, July 3-7, 2023

Many physical and engineering systems deal with micro-, meso-, macro-, and multi-scale phenomena. The accurate description and the reliable simulation of such phenomena entail major challenges from the point of view of both mathematical modelling and computational engineering. This book covers a selection of challenges related to "Mathematical and Computational Modelling Across the Scales”, stemming from the lecture notes of the XX edition of the Jacques-Louis Lions Spanish-French School in Numerical Simulations in Physics & Engineering. The thematic focus is broad, encompassing mathematical models of complex physical problems, theoretical results on their derivation, and development of numerical methods for their efficient simulation. The contributions of the book include: uncertainty quantification for phenomena at different scales such as epidemic dynamics, medical imaging, and geophysical exploration; structural health monitoring integrating small-scale sensor data in large-scale computational models; frontier numerical methods for the simulation of geophysical and heliophysical dynamics accounting for multi-scale, heterogeneous media; multi-physics, multi-scale models for the mechanobiology of atheroma plaques formation; locomotion models for swimming at the micro-scale; mathematical foundations of quantum mechanics phenomena at the micro-scale. The book is addressed to scientists and engineers, from both academia and industry, interested in the mathematical modelling and numerical simulation of a variety of complex systems in physics and engineering characterised by multiple scales.

Matteo Giacomini is Associate Professor of computational engineering at Universitat Politècnica de Catalunya and Affiliated Researcher at CIMNE, the International Centre for Numerical Methods in Engineering. He is recipient of the 2022 Juan Carlos Simó award by the Spanish Society for Computational Mechanics and Computational Engineering (SEMNI), and he is Fellow of the Serra Hunter Programme of the Government of Catalonia (2021). His research interests span from advanced discretisation methods for partial differential equations to reduced order models and scientific machine learning for computationally demanding problems, such as shape and topology optimisation.

Pedro Diez is Full Professor of applied mathematics at Universitat Politècnica de Catalunya, Affiliated Researcher and Scientific Director of CIMNE, the International Centre for Numerical Methods in Engineering. He received the IACM Fellow Award in 2010, and he is vice-president of ECCOMAS and president of SEMNI. His research focuses on the broad field of mathematical and computational modeling, including credibility (error assessment, adaptivity, and uncertainty quantification) in computational engineering, and reduced order models for real-time computing, optimization and data assimilation.