Disputation: Oussama Amine
Doctoral candidate Oussama Amine at the Department of Mathematics, Faculty of Mathematics and Natural Sciences, is defending the thesis "Regularization Effects for Certain Dynamical Systems through Gaussian Noises" for the degree of Philosophiae Doctor.
The University of Oslo is closed. The PhD defence and trial lecture will therefore be fully digital and streamed directly using Zoom. The host of the session will moderate the technicalities while the chair of the defence will moderate the disputation.
Ex auditorio questions: the chair of the defence will invite the audience to ask ex auditorio questions either written or oral. This can be requested by clicking 'Participants -> Raise hand'.
The meeting opens for participation just before the disputation starts, and closes for new participants approximately 15 minutes after the defence has begun.
Submit the request to get access to the thesis.
"The Wasserstein distance and some applications."
- Prerecorded trial lecture
Main research findings
It is well known that randomness can be used as an effective tool to turn a priori ill-posed problems into well-posed ones. This is useful both for answering questions at the theoretical as well as the practical levels. Examples of the effectiveness of such an approach are abundant in the fields of optimization, numerical analysis, inverse problems, AI and machine learning, to name a few.
On the other hand, continuous-time dynamical systems in the form of Ordinary Differential Equations and the related transport and continuity Partial Differential Equations, the main object of study in this thesis, appear in the modelling of several natural phenomena. A common characteristic of many such models is the lack of regularity of their input-data (i.e. vector fields) which makes the use of classical results, relying for the most part on smoothness assumptions, ineffective.
In this work, we study the effect of applying novel regularization maps to certain classical continuous-time dynamical systems with discontinuous vector fields and hence ill-posed a priori. We prove, among other results, that these regularization maps induce a modified dynamics enjoying well-posedness as well as increased stability, with respect to initial conditions. A key aspect of our main result is the preservation of the notion of solution when passing from the original problem to the regularized one.