Inverse eigenvalue problems for inhomogeneous media: theoretical and numerical advances

25 ene 2023
11:00
Salón de grados, Facultad de Informática UDC
CITMAga

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

Fecha: miércoles 25 de enero

Hora: 11:00 h.

Duración: 40 min

Lugar: Salón de grados, Facultad de Informática UDC y en línea (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.