Inverse eigenvalue problems for inhomogeneous media: theoretical and numerical advances

"Inverse eigenvalue problems for inhomogeneous media: theoretical and numerical advances", organizado polo CITMAga. Será impartido por Nikolaos Pallikarakis,  Departamento de Matemáticas (National Technical University of Athens)

Data: mércores 25 de xaneiro

Hora: 11:00 h.

Duración: 40 min

Lugar: Salón de graos, Facultade de Informática e en liña (MS Teams)

Abstract:

Research on inverse eigenvalue problems has been a very active subject both from a theoretical and a computational point of view. In this talk, after a brief introduction on the inverse Sturm-Liouville eigenvalue problem, we discuss the transmission eigenvalue problem. This problem arises in the inverse scattering of inhomogeneous media and is a non-standard and non-self-adjoint eigenvalue problem. We present some older and recent uniqueness results for the inverse spectral problem in the case when the refractive index is spherically symmetric, as well as when the refractive index is discontinuous. Next, we present a computational spectral-Galerkin method to solve the discrete direct transmission eigenvalue problem and define the corresponding inverse problem. Some examples of reconstructing constant and piecewise constant refractive indices of discs, using the lowest transmission eigenvalues are given. Finally, we propose a new idea of solving the above inverse eigenvalue problems using supervised Machine Learning regression methods. The models we use are k-Nearest Neighbours, Random Forest and Multi-Layer Perceptron (Artificial Neural Network). We verify the effectiveness of these models with some numerical examples.