Carlos Jerez-Hanckes: From Calderón Calculus to Random Domains: Boundary Integral Methods for Maxwell Scattering

Abstract:

Modern technologies increasingly rely on electromagnetic fields interacting with complex environments, with applications ranging from radar and space systems to telecommunications, imaging, and medicine. Their simulation leads to challenging Maxwell scattering and transmission problems, from homogeneous scatterers to composite and uncertain geometries.

In this talk, I will follow this progression in geometric complexity. Starting with homogeneous scatterers, I will introduce Maxwell boundary integral operators and the associated Calderón calculus. For composite scatterers with multiple subdomains and interfaces, I will discuss multiple trace formulations and how their operator structure can be exploited for preconditioning and efficient iterative solution.

Finally, I will consider parametric and random geometries, where the interfaces themselves vary. I will present recent results on the shape holomorphy of electromagnetic boundary integral operators and solution maps, and discuss how this regularity enables efficient high-dimensional approximation and many-query strategies for uncertainty quantification, design optimization, and inverse problems.

The common theme is how analytical structure can be exploited to construct efficient numerical methods as electromagnetic scattering problems become increasingly complex.