# Multidimensional scattering for biharmonic operator with quasi-linear perturbations

## Thesis event information

### Topic of the dissertation

Multidimensional scattering for biharmonic operator with quasi-linear perturbations

### Doctoral candidate

Master of Science Jaakko Kultima

### Faculty and unit

University of Oulu Graduate School, Faculty of Science, Research Unit of Mathematical Sciences

### Subject of study

applied mathematics

### Opponent

professor Nuutti Hyvönen, Aalto University

### Custos

Professor Emeritus Valery Serov, University of Oulu

## Multidimensional scattering for biharmonic operator with quasi-linear perturbations

Scattering theory concerns with perturbations to wave motion. A plane wave is sent toward the object of interest and the interaction between the object and this incident wave causes the wave to form ripples, i.e., a new wave called scattered wave is formed. Investigating the pattern of this scattered wave allows us to gather some crucial information about the object of interest. Scattering theory has a wide range of practical applications such as medical imaging, radar systems and quality control in industry.

The study begins by considering the mathematical model that governs the behaviour of the wave motion and analysing the direct scattering problem. We prove that certain class of objects will always produce a unique scattering pattern when a plane wave is sent toward it. This uniqueness plays a crucial role as we strive to establish a connection between the scattering pattern and the object.

Once we have established what kind of objects will form a unique scattering pattern, we turn our attention to the inverse problem. In the inverse scattering problem, we switch the roles of the initial conditions (object of interest) and the outcome (scattering pattern) of the scattering process, and we attempt to show that the knowledge about the scattering pattern will allow us to gather some information about the object. To this end, we study three separate inverse scattering problems each corresponding to a different set of scattering data. In the case of full scattering data, meaning that the scattering pattern is measured for all possible directions of the incident wave and for all measurement angles with arbitrarily large wave numbers. Additionally, we study the backscattering and fixed-angle scattering problems, and we show that the main singularities of the object may be recovered with these limited datasets. To support the theoretical results we have provided some numerical examples that show these methods in action.

Last updated: 22.5.2023