GUEST LECTURE Adrien Merlini received the M.Sc. Eng. degree from the École Nationale Supérieure des Télécommunications de Bretagne (Télécom Bretagne), France, in 2015 and received the Ph.D. degree from the École Nationale Supérieure Mines-Télécom Atlantique (IMT Atlantique), France, in 2019. Since 2019, he has been an Associate Professor with the Microwave Department, IMT Atlantique. His research interests include preconditioning and acceleration of integral equation solvers for electromagnetic simulations and their application to brain imaging. Dr. Merlini received Young Scientist Awards at the URSI GASS 2020 and the EMTS 2023 meetings. He authored and co-authored award-winning papers, including the third place at the EMTS 2023 best paper competition, the paper that received the 2022 ICEAA-IEEE APWC Best Paper Award, five papers that received honorable mentions (URSI/IEEE–APS 2021, 2022, and 2023), and 3 best paper finalists (URSI GASS 2020, URSI/IEEE–APS 2021 and 2022). He is a member of IEEE-HKN, the IEEE Antennas and Propagation Society, URSI France, and of the Lab-STICC laboratory. He is currently serving as Associate Editor for the Antenna and Propagation Magazine. This course covers the theory and computational methods for electromagnetic modeling of the human head, with applications to electroencephalography (EEG) source imaging. Starting from the physical foundations of bioelectric field propagation, the course introduces boundary and volume integral equation formulations tailored to realistic head geometries, including techniques for handling tissue anisotropy (skull inhomogeneity, white matter fiber structure) and the associated numerical challenges of ill-conditioning at low frequencies and high dielectric contrasts. Topics include the symmetric BEM formulation and its Calderón preconditioning, hybrid surface-volume-wire integral equations, regularized volume integral equations (D-VIE), and quasi-Helmholtz decomposition strategies. The course also addresses the inverse source imaging problem and recent data-driven approaches. The objective is to equip students with the mathematical and computational tools needed to design fast, accurate electromagnetic solvers for brain imaging applications, bridging computational electromagnetics, applied mathematics, and biomedical engineering.

· Fundamentals of electromagnetic theory · Fundamentals of linear algebra

· Part I : Foundations (5h) Neuronal sources and the quasi-static approximation of Maxwell's equations Electrical properties of head tissues: isotropy, anisotropy, conductivity profiles The EEG forward problem: formulation and role in source imaging Green's identities, representation theorems, and boundary integral equations The Boundary Element Method: discretization, basis functions, and comparison with FEM · Part II : Preconditioning and Fast Solvers for Surface Integral Equations (6h) Spectral analysis of BEM operators: dense-discretization and high-contrast breakdowns Calderón preconditioning strategies for the symmetric formulation Refinement-free approaches and fast iterative solvers Quasi-Helmholtz decompositions and low-frequency stabilization Numerical experiments on canonical and realistic head models · Part III : Modeling Tissue Anisotropy: Volume and Hybrid Integral Equations (5h) Wire integral equations for white matter fiber modeling Volume integral equations for inhomogeneous tissues (D-VIE) Regularization of volume formulations: quasi-Helmholtz projectors and permittivity rescaling Hybrid volume-surface-wire formulations for fully anisotropic head models Applications to EEG, TMS, and deep brain stimulation · Part IV : Inverse Problem and Frontiers (3h)

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P.D.2-2 - Maggio