This PhD thesis is concerned with inverse problems arising in 1d scattering theory. An inverse problem can be thought of as a mathematical framework that describes the process of estimating an unknown quantity from some data related to this unknown and some a-priori assumptions on the governing physics (for example a differential equation). The governing physics in our case is an equation describing scattering of waves in the frequency domain. In particular, in this thesis we study inverse problems of estimating coefficients of differential operators appearing in 1d quantum or acoustic scattering theory.
The main objective of this dissertation is to propose direct alternatives to highly nonlinear methods commonly used to solve such inverse scattering problems. To do that, we start by exploring the possibilities that the classical inverse Schrödinger scattering theory offers. In particular, we study the Gelfand-Levitan-Marchenko (GLM) point of view for solving the inverse Schrödinger scattering problem and we attempt to extend that approach to the Helmholtz case.
We continue by following a GLM inspired approach, the so-called reduced order model (ROM) approach. The ROM method serves roughly as a finite dimensional counterpart of the GLM type inversion methods. In all proposed methods (pure GLM and ROM) we investigate the issues of the well-posedness and we also perform several computational examples.
- Start date and time
- End date and time
- Academiegebouw, Domplein 29 & online (livestream link)
- PhD candidate
- Andreas Tataris
- Gelfand-Levitan-Marchenko and model order reduction methods in inverse scattering
- PhD supervisor(s)
- prof. dr. Tristan van Leeuwen
- prof. dr. Sjoerd Verduyn Lunel
- dr. Ivan Pires de Vasconcelos